Talk:Euclidean distance

7158:, for an article that at least in part needs to be readable by schoolchildren. It removes relevant and sourced information. And things like quadratic forms and Bregman divergence are not even close to being Euclidean distances. We have separate articles already where they belong much better. Hilbert spaces are already present (in a very non-technical way) in the mention of infinite dimensions in the "generalizations" section, with links for those looking for real information. In contrast, the material on squared Euclidean distance that you removed could plausibly be split off into a second article, but I am not convinced there is enough notability for it to stand alone as a separate article (certainly not under the "quadrance" title of the part you removed, as that is mostly just Wildberger reinventing the wheel) and in the meantime this article is the place for it. As for your removal of geodesy: it would be seriously misleading to say in this article that Euclidean distance was the only distance used even by non-mathematicians prior to the 19th century. And finally, reverting a set of bad edits once does not even come close to edit warring. — 5674:

squared not distance, norm not distance, and gradient. Is this level of removal still sufficiently relevant to be included in this article? (2) How does the detailed calculation do anything but intimidate readers? Please remember in particular that students likely learn about Euclidean distance somewhere in middle school or high school, and that therefore to the extent possible we should try to keep material in this article to that level. The squared distance section already does go beyond that but we don't need to make it more so. (3) Where are the published sources for this calculation? We would in any case also need sources in order to say anything about why this is relevant to the topic of this article. —

6687:. The statement "squared distance cannot be a metric" is (mildly) misleading because it does not explicitly say that it has in mind the metric in the mathematical sense. It is certainly not true for the Euclidean space and metric as defined in relativity. The reference after this statement gives as an example the number line on which the Euclidean distance satisfies the triangle inequality while the squared Euclidean distance does not. However, when choosing basis vectors for the tangent space on a Riemannian manifold, it is specified at the start that the basis vectors should be linearly independent, which excludes them being on the same straight line. So these are indeed different concepts. 7455:. I want this article to be readable by people who are not experts in real analysis. I want it to be readable by people who have not yet really learned what a function is. I think it should be possible to understand the distance between points without understanding the nature of functions. I think this possibility is clearly demonstrated by the ancient nature of Euclidean distance and the much less ancient nature of real analysis: the ancient Greek mathematicians do not appear to have had an adequate notion of functions as a general class of mathematical objects, but still were well acquainted with lengths of line segments. — 7250:

Wildberger is comparatively humble in that he is only claiming a new approach to an old subject, and is putting out conventionally written articles on the Arxiv whose only crime is to introduce new terminology ("chromogeometry", anyone?). He does not seem to be developing an institutionalized personality cult either, of the sort that has been a problem in some of those fields. So it's pretty harmless and I don't see it as unreasonable to have an article on R.T as a mathematical subject rather than the book. But the other way around is also acceptable.

228: 218: 197: 21: 166: 7051: 107: 7024: 7016: 6987: 6967: 6956: 6927: 6916: 2516: 7750:: the Pythagorean formula for Euclidean distance is commonly studied by schoolchildren, justifying its prominence and early placement in this article. Other material can be included but should not be placed in a way that would make the more-accessible parts of the article harder to find for the part of the audience that might be looking for them. — 7321: 3601: 2179: 5593: 4308: 5872: 7249:

There are entire fields of mathematics that are in large part hype surrounding a prosaic foundation of ordinary theorems about previously unhyped areas. E.g., chaos theory, catastrophe theory, complex systems, the set theory multiverse... and check out some of the math articles in Quanta these days!

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Aside from which additional here might be appropriate here, I'd like to emphasize there is no general need to turn every short or midsized article into long one. Many smaller articles serves the readers just fine and much of the suggested material above might be better in separate entries and/or the

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We currently discuss generalizations to manhattan distance and chebyshev distance. These are generalizations based on the concept of "Euclidean distance" explicitly thought of as "distance in Cartesian coordinates for a Euclidean space". That is, these are ideas you get by 'keep the idea of distance

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Before your reply, I edited my comment to make clear that I meant the squared norm. You removed that correction from my comment. I assume it was accidental but please do not do that. Anyway, yes, you can get a similar formula for distance, but only in an artificial way by holding one point fixed and

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Where is that section? I think it was accidentally removed with the approximations, put it back. Comparing distances without using the square root is a simple but very important fact in software/hardware development since the square root is a very expensive operation. Raising the awareness of such a

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To be clear, I think the name "Euclidean distance" should clearly be its own article, not combined with others. I just mean that discussing other concepts built on top, alternatives, or generalizations here doesn't seem like it would make the article excessively long, impinge on readers' ability to

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This article is very short, and I feel like it is predominantly focused on lists of formulas instead of explaining how the concept is related to other concepts or how it is used; if I had to give name to the current scope, I'd summarize as something like "formulas for calculating Euclidean distance

7123:

This is wrong as a description of non-Euclidean geometry. It is anachronistic; the notion of "mathematical space" came later, and distance functions on such spaces. It is also misleading: it was a familiar everyday fact that travel distance differs from Euclidean distance (hence the phrase "as the

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I removed the following section as it imho does really fits here for several reasons. First of all the triangle inequality has its own article, so the definition and various derivations should be handled there. Secondly imho scope and style of the section went a bit against the notion "wikipedia is

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What's so unique about the Euclidean metric? What properties does the euclidian metric have that other metric spaces don't have, such as the other Minkowski metrics? What is a rotation in a metric space? Do other metric spaces not have the property that the composition of two rotations is always a

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Why is the approximation section here? Euclidean metrics are very simple concepts, and I don't think you'd find anything about approximation in 99% of the textbooks that cover the topic. It seems like the audience for that material is different from the audience for the rest of the article. At the

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1. Loci of equidistant points. I think it's worth discussing how the locus of points equidistant from a point is a circle, the locus of points equidistant from a circle is a concentric circle, the locus of points equidistant from a straight line is a parallel line, and in general it's possible to

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I figured that if we're going to have an article on rational trigonometry at all, it would be better to mention it briefly in a context that we make very clear that squared Euclidean distance is far from a new idea, rather than avoiding the subject and allowing readers to keep the impression that

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Do you really not understand the significant difference between "distance" (a number, between two points), "distance function" (a thing that takes pairs of points and spits out numbers), and "Euclidean metric" (a space of points together with a distance function on it)? Or the significant gap in

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If you take the metric tensor as the space metric, it is symmetric in all cases. Whether positive (or its determinant is positive) is a matter of convention. In most books the determinant of the metric tensor is negative. About the triangle inequality, I lack imagination how it can be applied to

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and performing an appropriate simplification. (One point is missing here, the criteria of interpolation optimality. Frankly, I have forgotten, what condition I have actually used: whether least area between the curve and line, or maximum distance minimization. I have to find the paper and recall

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is off-topic here. Length and distance are not really the same thing (as evidenced by the millenia-long gap between the study of lengths of non-straight curves and the formulation of non-Euclidean geometry and non-Euclidean distances) and we have separate articles for them. "Various measures of

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Euclidean norm. One can easily transform this equation to include the distance instead of norm. If I search my textbooks maybe I shall find a reference; however, the calculations are obvious for anyone who can differentiate and know what a gradient is. Anyway, I agree to your argument that this

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would have far higher priority but are not even mentioned. Rational trigonometry is a one-man show by its inventor, has not caught on whatsoever (as noted in its Knowledge page), and is already being hyped at a number of other WP locations. No need to promote it where it is at best marginally

5673:

Ok, but (1) this is the gradient of the squared Euclidean norm, not of the squared Euclidean distance. The Euclidean norm is only treated briefly in this article before pointing to a different article on that topic. So it is removed in three ways from the actual topic of this article: distance

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is perfectly enough. Anyone who knows what a "metric" is (a small subset of this article's intended audience!) can reasonably infer what the "Euclidean metric" is. In all likelihood, such a person will have already been introduced to the Euclidean metric on R^n as their first example. WP math

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It is true that people who have some idea what a multivariate function is can reasonably infer that there is a function mapping pairs of points to their distance, and it is also true that that function is sometimes called, for short, "Euclidean distance". But, Fgnievinski, you are still

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If you want to know whether objects A and B are at distance c or less, you can compare ((Ax - Bx)^2 + (Ay - By)^2 + (Az - Bz)^2) with c^2, and similarly for other numbers of dimensions. This avoids the square root entirely. If c is a constant, then c^2 can be precomputed as well. --

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3. Chord length vs. arc length vs. other measurements between points on a curve or surface. In e.g. Greek spherics/astronomy, compasses were used to draw circles on solid spheres and transfer geometric objects between the sphere and an auxiliary plane. Hipparchus' foundation of

7418:"Euclidean distance" is also routinely interpreted as the "Euclidean distance function" between any two arbitrary points. Sometimes mathematicians have a very very narrow interpretation. The term Euclidean metric is used by physicists, programmers, etc. in a less strict manner. 3022: 361:

Where it says "The technique has been rediscovered numerous times throughout history, as it is a logical extension of the Pythagorean theorem.", are they talking about repeated use of the pythagorean theorem to prove the pythagorean theorem? The statement seems disjointed.

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Some years ago I developed a similar distance approximation algorithm using three terms, instead of just 2, which is much more accurate, and because it uses power of 2 denominators for the coefficients can be implemented without using division hardware. The formula is:

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My general reaction is: This is a specific article on Euclidean distance, not a catch-all for any vaguely related notion, and it is important for it to stay focused on its actual topic. None of the things you describe are really "sub-subjects" of Euclidean distance.

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should be pulled into a separate section or subsection. Arc length is geodesic distance along a curve: a direct generalization of Euclidean distance, found by breaking a curve up into infinitesimal parts and then integrating the Euclidean distance over the parts.

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I don't like the categorical statement that "the squared Euclidean distance cannot be a metric because it does not satisfy the triangle ineqality" although this statement is properly sourced. Mathematically, the statement is OK if we understand the metric as

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In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

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I still don't see what your point is. Orthodromic and geodesic distances are still distance functions that define metric spaces. "Metric tensor" and "metric space" use different words so it should not be surprising that they determine different concepts.

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difference between shapes" goes in the "Objects other than points" section, which already briefly mentions two such measures: minimum distance and Hausdorff distance. It might go too far in the way of excessive detail on off-topic subjects, though (

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Yes, you are right about this. Talking about "metric space" and avoiding the ambiguos term "metric" makes it true also for Riemannian space. The shortest path between two points there is a geodesic which is not a straight line but is well defined.

5731: 7592:. Geodesic distance is not the same as Euclidean distance. It is not constructive to make all articles in a related cluster of topics become duplicates of each other by incorporating their content into each other. The article you linked to, 6682:

are different concepts and both are called "metric" in their respective fields. The first one cannot be a squared distance, while the second one is, in fact, the squared distance. You can see this, for example, in the disambiguation page

2806: 5881: 6705:: But we are not talking about the word "metric", we are talking about the phrase "metric space". Is that exact phrase ever used to mean something that cannot be viewed as a special case of the definition at the article 2511:{\displaystyle a_{1}^{2}+2a_{1}b_{1}+b_{1}^{2}+\cdots +a_{n}^{2}+2a_{n}b_{n}+b_{n}^{2}\leq a_{1}^{2}+\cdots +a_{n}^{2}+b_{1}^{2}+\cdots +b_{n}^{2}+2{\sqrt {(a_{1}^{2}+\cdots +a_{n}^{2})(b_{1}^{2}+\cdots +b_{n}^{2})}}.} 1294: 598:

Well. 0.41 sounds like an approximation of sqrt(2). I don't know where the other coefficient comes from or why it's necessary to have 6 decimal place accuracy (while 0.41 only has a two decimal place accuracy).

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audience maturity required to go from one of these concepts to the next? Despite having this repeatedly pointed out to you, none of your comments, including this one, exhibit any understanding of these issues. —

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simple fact has a positive contribution to technology and thus humanity which nowadays suffers from slow, overbloated and resource-hungry software. Knowledge is not only about abstract mathematical concepts.--

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I agree that the coordinate-based view is relatively recent (didn't really dominate until the 20th century, is my impression). Which is further reason to try to flesh out a broader geometrical explanation!

2837: 7839: 4663: 6380: 6274: 3596:{\displaystyle a_{1}b_{1}+\cdots +a_{n}b_{n}\leq |a_{1}b_{1}+\cdots +a_{n}b_{n}|={\sqrt {(a_{1}b_{1}+\cdots +a_{n}b_{n})^{2}}}\leq {\sqrt {(a_{1}^{2}+\cdots +a_{n}^{2})(b_{1}^{2}+\cdots +b_{n}^{2})}}} 7206: 7213:"quadrance" is something new and different. But I suppose we could try again to delete the rational trigonometry article — if it were gone, I would support also removing mention of it from here. — 5124: 4974: 4786: 7715:

The way you keep referring to this article as being only about coordinates makes me think you haven't read the history section of this article. The coordinate-based view is relatively recent. —

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and then the statement is not true: squared Euclidean distance generates the metric of the flat 3-D (Euclidean) space. This is a source of confusion for readers with different backgrounds.

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Perhaps we need a whole new article on distance approximations. Can you give me an example of a place where min and max functions are available but multiplication and division are not ?

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It was accidental. I never delete people's comments deliberately. Partially agree for the two variable points: it is a Cartesian product with twice the dimensions, something like this:

3110: 1887: 5588:{\displaystyle {\frac {\partial }{\partial x_{j}}}f(x)={\frac {\partial }{\partial x_{j}}}\sum _{k=1}^{n}x_{k}^{2}=\sum _{k=1}^{n}{\frac {\partial }{\partial x_{j}}}x_{k}^{2}=2x_{j}.} 3916: 3858: 3800: 6497:(written exactly so) it is understood the squared distance in the respective space. That's why the statement that the squared distance is not a metric sounds strange in this field. 4303:{\displaystyle {\sqrt {(p_{1}-r_{1})^{2}+\cdots +(p_{n}-r_{n})^{2}}}\leq {\sqrt {(p_{1}-q_{1})^{2}+\cdots +(p_{n}-q_{n})^{2}}}+{\sqrt {(q_{1}-r_{1})^{2}+\cdots +(q_{n}-r_{n})^{2}}}} 5206: 5044: 4868: 4707: 4568: 3714: 3670: 3139: 1823: 1746: 1669: 1625: 1562: 7232: 4824: 4024: 3971: 1206: 577: 7113:), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of 504: 441: 6823: 5643: 115: 734:((( max << 8 ) + ( max << 3 ) - ( max << 4 ) - ( max << 1 ) + ( min << 7 ) - ( min << 5 ) + ( min << 3 ) - ( min << 1 )) : --> 5867:{\displaystyle {\frac {\partial }{\partial q}}\left({\frac {\partial (p(q)-q(p))^{2}}{\partial p}}\right)=-2{\frac {\partial p}{\partial q}}{\frac {\partial q}{\partial p}}} 5162: 1159: 1779: 5000: 7703:

concepts (parallel lines, circles, chord-based distance between points on surfaces embedded in Euclidean space, ...) rather than only arithmetized coordinate concepts. –

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in the sense that their distance is symmetric, positive, and obeys the triangle inequality. They have additional properties, of course, as does the Euclidean distance. —

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I think it is worth noting that the Euclidean metric used to be called Pythagorean metric. At least there should be a page title Pythagorean metric that redirects here.

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Do they really call it a "metric space", though? I think that term unambiguously means a set with a symmetric positive distance satisfying the triangle inequality. —

7109:"Although accurate measurements of long distances on the earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see 4897: 4421: 7124:

crow flies"), and well known to cognoscenti that maps of the Earth distort distance and that spherical geometry/trigonometry differ from the Euclidean version.

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However, we have a little twist here: the Cartesian frame in Euclidean space is orthogonal meaning that terms with mixed partial derivatives are zero, that is:

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articles already have a reputation for turning elementary concepts into turgid prose, and imo we should do as much as possible to fight that. Readers first!

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Mentioning "metric space" is wading into another bog because there are many kinds of metric spaces. Specifically, in general relativity the space is a

2698: 738:. This is just like the 2 coefficient min max algorithm presented earlier, but with the coefficients 123/128 and 51/128. I have an article about it at 7496:

2. Various measures of the "distance" between various kinds of shapes are based on Euclidean distance, and have useful applications. For example, the

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In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied.

6856:, p. 134: "Strangely, this formula did not appear in print until 1731, when the French mathematician Alexis Claude Clairaut (1713-1765) published..." 1478:{\displaystyle {\sqrt {(a_{1}+b_{1})^{2}+\cdots +(a_{n}+b_{n})^{2}}}\leq {\sqrt {a_{1}^{2}+\cdots +a_{n}^{2}}}+{\sqrt {b_{1}^{2}+\cdots +b_{n}^{2}}}} 7244: 843:

Also, I moved the approximation sections to the end of the page because it seems important to at least see the 3D version before this is discussed.

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be included or sketched in wikipedia articles, it is usually not desired to have longer technically detailed prrof or textbook derivations/proofs.

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Hm. A google scholar search turns up a lot of sources about geodesics with "geodesic distance" in them. But maybe it should be a disambig page. –

732:. Also it is possible to implement a distance approximation without using either multiplication or division when you have very limited hardware: 7844: 658:=1, it strongly resembles a plain straight line. And those mysterious coefficients are just the description of the optimal interpolation line ( 250: 7508:

in relating circular chord length to arc length had to do with this "chordal" Euclidean distance inherited from the ambient Euclidean space.

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The most common use I see of the phrase "geodesic distance" involves shortest paths in polygons, about which we do not even have an article.

7549:) to give a proper definition of the Frechet distances for curves (there is more than one incompatible way of defining Frechet distance). — 908: 7699:

for coordinates, but ditch the "Euclidean" part'. But when I think of the term "Euclidean distance", I think the idea should also include

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True, but this is already implied by the section that talks about how distances can be compared while skipping the square root operation.

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I tried rewriting the rational trig article to be about the book. If that sticks, I don't think we need to continue mentioning it here. —

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as WP:UNDUE. This seemed clear enough; if the article were trying to catalogue appearances and avatars of squared distance, things like

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I changed it to "does not form a metric space, as it does not satisfy the triangle inequality". Is that adequately unambiguous? —

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If equality holds in (**), then the left side of (**) is non-negative and equality in (***) holds. Therefore, there is some real

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tensors. In any case, in a curved space, it makes difference in which direction you are going. Often, it is shorter to go from

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Those coefficients come from a certain optimal interpolation, which I have calculated some years ago. It goes like this:

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currently redirects to a place where it is, or can be, clearly explained that it is a new name for an old quantity.

7385: 1219: 1043: 7363:...a metric or distance function is a function that gives a distance between each pair of point elements of a set. 739: 848: 6093: 7755: 7720: 7673: 7601: 7554: 7490: 7460: 7452: 7409: 7301: 7218: 7163: 6950: 6883: 6714: 6665: 6613: 6516: 6076: 5717: 5679: 4454: 3297:{\displaystyle (a_{1}b_{1}+\cdots +a_{n}b_{n})\leq (a_{1}^{2}+\cdots +a_{n}^{2})(b_{1}^{2}+\cdots +b_{n}^{2}).} 730:

1007/1024 max(|x|,|y|) + 441/1024 min(|x|,|y|) - if ( max(|x|.|y|)<16min(|x|,|y|), 40/1024 max(|x|,|y|), 0 )

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than directly, whatever "directly" is. The shortest distance between two points in a curved space is along a

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of the lengths of the paths (continuously differentially curves) connecting them. Also, a positive-definite

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I don't relish the thought of the inevitable "keep, because it's long and has footnotes" comments that

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you can see how Euclidean space and Euclidean metric are defined through Riemannian space and metric.

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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.

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gets stuck on spam websites, but the report is clean otherwise. Provided inline citation checks out.

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Both sides of the inequality (*) are non-negative. For this reason, it is equivalent to it's square.

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FWIW, I saw that mention of "rational trigonometry" a while back and thought about cutting it per

7139:, but I do not understand what he thinks is wrong with it or how it is inferior to the preceding. 249:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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is the shortest distance between two points on the Earth surface but it is not a straight line.

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letting the other be variable. If both points are variable the gradient is higher-dimenional. —

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are both ancient concepts, the Pythagorean formula for distance was not published until 1731?

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A fast approximation of 2D distance based on an octagonal boundary can be computed as follows.

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are both ancient concepts, the Pythagorean formula for distance was not published until 1731?

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As you can see in the first section of this talk page, the approximation section looks like

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The place for explaining in detail how different distance concepts relate to each other is

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2. Reworking of the statements on non-Euclidean geometry. The problematic passage states:

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I'm not sure where/how best to integrate a few subjects that seem worth mentioning here.

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One more addition, IIRC, the optimality criterium comes from error-factor formula sqrt(1+

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curve, which is not a straight line. We can see this already on the Earth surface. The

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then regardless of the state of other articles, they should be removed from this one.

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very least, there should be a reference here, or some explanation as to why it works.

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https://en.wikipedia.org/search/?title=Euclidean_distance&oldid=996504346#History

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The following is an archived discussion of the DYK nomination of the article below.

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Is that formula for approximating the distance using only integers wrong? Surely if

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As for "very short": this article falls well into the middle range of lengths of

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Edit warring on rational trigonometry and description of non-Euclidean geometry

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Subsequent comments should be made on the appropriate discussion page (such as

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http://web.oroboro.com:90/rafael/docserv.php/index/programming/article/distance

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Keep or remove "quadrance"/rational trigonometry (split from previous section)

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Maybe the section can be moved to a suitable project on Wikibooks or to the

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already calls it a "function" -- what else does the WP-Math cabal need?

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1. Removal of reference to squared distance being called "quadrance" in

865:, so I guess that it can be safely removed altogether. Afaic, go ahead. 7272:

If quadrance/rat.trig are unimportant to explaining or contextualizing

6536: 7764:

While we're on the subject of history, it's probably worth mentioning

1748:, then equality holds in (*). Prove we the statement in the case that 6785:), unless there is consensus to re-open the discussion at this page. 6684: 904: 886: 761:

You inspired me to start a wikibooks page, please share and improve:

330:) will always be 0, and the approximation will always be 100% error? 7695:

make sense of the material currently here, or really be "off topic".

3736:

can not be negative because the left side of (***) is non-negative.

825:

I corrected the formula for the distance in circular coordinates. --

5049:

We can conclude that equality holds in the triangle inequality iff

7334:

Knowledge:Redirects for discussion/Log/2022 May 4#Euclidean metric

7127:

My attempt to state this more correctly is the last paragraph of

7324:

An editor has identified a potential problem with the redirect

6601:

But all of those things are called metric spaces because they

159: 6088:

Euclidean metric (that is, the metric of the Euclidean space)

6052:; however, in general relativity by metric is understood the 6943:

Article is sourced, neutral, and free of copyright problems

7664:

and far above the "combine with other topics" threshold of

6006: 889:? If so, should it be mentioned or at least 'See also'ed? 763:

http://en.wikibooks.org/Algorithms/Distance_approximations

5692:

This is not a gradient of the Euclidean norm, but of the

4658:{\displaystyle q_{i}-p_{i}={\frac {t}{1+t}}(r_{i}-p_{i})} 7128: 6828: 6819: 6375:{\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}} 6269:{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} 6085:

The problem is different. In relativity, the books say:

2521:

After cancelation, we obtain the equivalent inequality

653:^2). Now, when you plot the square-root expression for 83: 7840:

Knowledge Did you know articles that are good articles

947:

not a textbook". While proofs or short derivation may

7380:

I still don't get why folks seem upset with applying

6996: 6936: 6896: 6473: 6392: 6284: 6178: 6096: 5884: 5734: 5607: 5409: 5232: 5170: 5132: 5057: 5008: 4982: 4907: 4879: 4832: 4794: 4719: 4671: 4581: 4532: 4457: 4429: 4403: 4323: 4038: 3979: 3926: 3866: 3808: 3750: 3722: 3678: 3635: 3615: 3315: 3142: 3119: 3113:. Because this quadric inequality holds for all real 3055: 3034: 2840: 2817: 2701: 2692:

To prove this inequality, consider we the inequality

2530: 2193: 1902: 1832: 1787: 1754: 1710: 1677: 1633: 1590: 1570: 1526: 1493: 1297: 1222: 1167: 1129: 1046: 1023: 1003: 983: 547: 513: 453: 390: 7611:

in Cartesian coordinate systems, with a few asides".

7485:

draw the locus of points equidistant to some curve (

245:, a collaborative effort to improve the coverage of 7668:. The formulas are only one of its five sections. — 7003:Hook has been verified by provided inline citation 6703:

both are called "metric" in their respective fields

6489: 6450: 6374: 6268: 6160: 6012: 5866: 5637: 5587: 5389: 5200: 5156: 5118: 5038: 4994: 4968: 4891: 4862: 4818: 4780: 4701: 4657: 4562: 4518: 4441: 4415: 4386: 4302: 4018: 3965: 3910: 3852: 3794: 3728: 3708: 3664: 3621: 3595: 3296: 3125: 3103: 3040: 3016: 2823: 2800: 2682: 2510: 2173: 1880: 1817: 1773: 1740: 1696: 1663: 1619: 1576: 1556: 1512: 1477: 1280: 1200: 1153: 1112: 1029: 1009: 989: 571: 531: 498: 435: 5119:{\displaystyle q_{i}-p_{i}=\lambda (r_{i}-p_{i})} 4969:{\displaystyle q_{i}-p_{i}=\lambda (r_{i}-p_{i})} 4781:{\displaystyle q_{i}-p_{i}=\lambda (r_{i}-p_{i})} 6581:Riemannian metric (or Riemannian metric tensor) 3104:{\displaystyle b_{1}^{2}+\cdots +b_{n}^{2}: --> 1881:{\displaystyle b_{1}^{2}+\cdots +b_{n}^{2}: --> 7131:. This has been reverted as "tendentious" by 1281:{\displaystyle a_{1},b_{1},\dots ,a_{n},b_{n}} 1113:{\displaystyle d(p,r)\leqslant d(p,q)+d(q,r),} 48:If it no longer meets these criteria, you can 6854:The Pythagorean Theorem: A 4,000-Year History 3133:, the discriminant is less or equal to zero. 314:are the same (eg, at a 45-degree angle) then 8: 7584:distance. We have a more general article on 7080:The following edits were recently reverted. 7058:returns a length of 8939 characters. Earwig 6787:No further edits should be made to this page 5255: 5248: 5195: 5177: 5033: 5015: 4857: 4839: 4696: 4678: 4557: 4539: 3703: 3685: 1812: 1794: 1735: 1717: 1658: 1640: 1551: 1533: 1208:. To prove it, we need the following Lemma. 7618:points the wrong place; I meant to link to 6161:{\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}} 5648: 5216:If we present the Euclidean distance as a 4519:{\displaystyle p_{i}-q_{i}=t(q_{i}-r_{i})} 191: 62: 15: 7054:As a GA, the article is long enough, and 6674:Your last sentence is exactly the point. 6481: 6472: 6451:{\displaystyle ds^{2}=g_{ik}dx^{i}dx^{k}} 6442: 6429: 6413: 6400: 6391: 6366: 6353: 6340: 6324: 6308: 6292: 6283: 6260: 6244: 6228: 6212: 6199: 6186: 6177: 6152: 6136: 6120: 6104: 6095: 5998: 5987: 5962: 5946: 5937: 5932: 5919: 5906: 5900: 5889: 5883: 5844: 5824: 5794: 5754: 5735: 5733: 5606: 5576: 5560: 5555: 5542: 5529: 5523: 5512: 5499: 5494: 5484: 5473: 5460: 5447: 5423: 5410: 5408: 5381: 5376: 5366: 5355: 5342: 5328: 5324: 5313: 5308: 5298: 5287: 5263: 5258: 5231: 5169: 5131: 5107: 5094: 5075: 5062: 5056: 5007: 4981: 4957: 4944: 4925: 4912: 4906: 4878: 4831: 4793: 4769: 4756: 4737: 4724: 4718: 4670: 4646: 4633: 4608: 4599: 4586: 4580: 4531: 4507: 4494: 4475: 4462: 4456: 4428: 4402: 4322: 4292: 4282: 4269: 4247: 4237: 4224: 4215: 4204: 4194: 4181: 4159: 4149: 4136: 4127: 4116: 4106: 4093: 4071: 4061: 4048: 4039: 4037: 4010: 3997: 3984: 3978: 3957: 3944: 3931: 3925: 3899: 3880: 3865: 3841: 3822: 3807: 3783: 3764: 3749: 3721: 3677: 3656: 3640: 3634: 3614: 3582: 3577: 3558: 3553: 3537: 3532: 3513: 3508: 3499: 3488: 3478: 3468: 3449: 3439: 3430: 3422: 3416: 3406: 3387: 3377: 3368: 3359: 3349: 3330: 3320: 3314: 3282: 3277: 3258: 3253: 3237: 3232: 3213: 3208: 3189: 3179: 3160: 3150: 3141: 3118: 3089: 3084: 3065: 3060: 3054: 3033: 2999: 2994: 2975: 2970: 2951: 2941: 2922: 2912: 2890: 2877: 2872: 2853: 2848: 2839: 2816: 2783: 2773: 2757: 2735: 2725: 2709: 2700: 2666: 2661: 2642: 2637: 2621: 2616: 2597: 2592: 2583: 2574: 2564: 2545: 2535: 2529: 2494: 2489: 2470: 2465: 2449: 2444: 2425: 2420: 2411: 2399: 2394: 2375: 2370: 2357: 2352: 2333: 2328: 2315: 2310: 2297: 2287: 2271: 2266: 2247: 2242: 2229: 2219: 2203: 2198: 2192: 2157: 2152: 2133: 2128: 2112: 2107: 2088: 2083: 2074: 2062: 2057: 2038: 2033: 2020: 2015: 1996: 1991: 1978: 1968: 1955: 1933: 1923: 1910: 1901: 1866: 1861: 1842: 1837: 1831: 1786: 1759: 1753: 1709: 1682: 1676: 1632: 1611: 1595: 1589: 1569: 1525: 1498: 1492: 1467: 1462: 1443: 1438: 1432: 1421: 1416: 1397: 1392: 1386: 1375: 1365: 1352: 1330: 1320: 1307: 1298: 1296: 1272: 1259: 1240: 1227: 1221: 1166: 1128: 1045: 1022: 1002: 982: 546: 512: 491: 485: 472: 463: 452: 428: 422: 409: 400: 389: 4387:{\displaystyle d(p,r)\leq d(p,q)+d(q,r)} 585:where do 0.41 and 0.941246 come from? -- 334:Yea, it was wrong and has been changed. 7235:was in 2013, so maybe it's time again. 5697:sentence is somehow out-of-place here. 678:). The last step is a realization that 193: 7432: 6903:Article is new enough and long enough 6702: 5212:Gradient of squared Euclidean distance 3911:{\displaystyle r=(r_{1},\dots ,r_{n})} 3853:{\displaystyle q=(q_{1},\dots ,q_{n})} 3795:{\displaystyle p=(p_{1},\dots ,p_{n})} 1123:with equality if and only if there is 7453:wilfully continuing to miss the point 3606:and therefore inequality (**) holds. 7: 7580:Again, this is a focused article on 942:removed section: triangle inequality 641:is greater or equal to 1). Distance 239:This article is within the scope of 165: 163: 7742:Only if that fleshing-out is about 7477:Sub-subjects that seem missing here 6878:Improved to Good Article status by 5201:{\displaystyle i\in \{1,\dots ,n\}} 5039:{\displaystyle i\in \{1,\dots ,n\}} 4863:{\displaystyle i\in \{1,\dots ,n\}} 4702:{\displaystyle i\in \{1,\dots ,n\}} 4563:{\displaystyle i\in \{1,\dots ,n\}} 3709:{\displaystyle i\in \{1,\dots ,n\}} 3665:{\displaystyle a_{i}=\lambda b_{i}} 1818:{\displaystyle i\in \{1,\dots ,n\}} 1741:{\displaystyle i\in \{1,\dots ,n\}} 1664:{\displaystyle i\in \{1,\dots ,n\}} 1620:{\displaystyle a_{i}=\lambda b_{i}} 1557:{\displaystyle i\in \{1,\dots ,n\}} 885:Does this have anything to do with 182:It is of interest to the following 141:Did you know ... that although the 7850:High-priority mathematics articles 7806:overview article for distance. -- 7588:and a separate focused article on 6680:metric tensor (general relativity) 6054:metric tensor (general relativity) 5912: 5908: 5855: 5847: 5835: 5827: 5802: 5757: 5741: 5737: 5608: 5535: 5531: 5453: 5449: 5416: 5412: 14: 4819:{\displaystyle \lambda \in [0,1)} 4029:Applying previous lemma, we have 4019:{\displaystyle b_{i}=q_{i}-r_{i}} 3966:{\displaystyle a_{i}=p_{i}-q_{i}} 1201:{\displaystyle q-p=\lambda (r-p)} 572:{\displaystyle 0.41dx+0.941246dy} 259:Knowledge:WikiProject Mathematics 41:. If you can improve it further, 7332:. This discussion will occur at 7319: 7049: 7022: 7014: 6985: 6965: 6954: 6925: 6914: 6384:general metric of a curved space 499:{\displaystyle dy=|p_{y}-q_{y}|} 436:{\displaystyle dx=|p_{x}-q_{x}|} 262:Template:WikiProject Mathematics 226: 216: 195: 164: 105: 19: 7489:) or other point set. Also cf. 6770:Please do not modify this page. 5638:{\displaystyle \nabla f(x)=2x.} 279:This article has been rated as 6579:of inner products is called a 5791: 5787: 5781: 5772: 5766: 5760: 5620: 5614: 5441: 5435: 5242: 5236: 5151: 5139: 5113: 5087: 4963: 4937: 4813: 4801: 4775: 4749: 4652: 4626: 4513: 4487: 4381: 4369: 4360: 4348: 4339: 4327: 4289: 4262: 4244: 4217: 4201: 4174: 4156: 4129: 4113: 4086: 4068: 4041: 3905: 3873: 3847: 3815: 3789: 3757: 3588: 3546: 3543: 3501: 3485: 3432: 3423: 3369: 3288: 3246: 3243: 3201: 3195: 3143: 3028:This is quadric inequality by 3005: 2963: 2957: 2905: 2883: 2841: 2780: 2750: 2732: 2702: 2672: 2630: 2627: 2585: 2500: 2458: 2455: 2413: 2163: 2121: 2118: 2076: 1975: 1948: 1930: 1903: 1372: 1345: 1327: 1300: 1195: 1183: 1148: 1136: 1104: 1092: 1083: 1071: 1062: 1050: 775:09:20, 23 September 2012 (UTC) 492: 464: 429: 401: 339:16:40, 21 September 2005 (UTC) 29:has been listed as one of the 1: 7845:GA-Class mathematics articles 7816:22:36, 18 November 2023 (UTC) 7782:01:03, 19 November 2023 (UTC) 7760:22:32, 18 November 2023 (UTC) 7738:22:19, 18 November 2023 (UTC) 7725:21:59, 18 November 2023 (UTC) 7711:21:46, 18 November 2023 (UTC) 7690:21:42, 18 November 2023 (UTC) 7678:21:11, 18 November 2023 (UTC) 7646:20:54, 18 November 2023 (UTC) 7606:20:50, 18 November 2023 (UTC) 7576:20:49, 18 November 2023 (UTC) 7559:20:42, 18 November 2023 (UTC) 7534:20:07, 18 November 2023 (UTC) 7312:"Euclidean metric" listed at 7306:18:22, 30 December 2020 (UTC) 7290:19:28, 30 December 2020 (UTC) 7260:19:28, 30 December 2020 (UTC) 7245:19:25, 28 December 2020 (UTC) 7231:would get in an AfD, but the 7223:06:48, 28 December 2020 (UTC) 7207:05:56, 28 December 2020 (UTC) 7188:20:22, 30 December 2020 (UTC) 7168:07:17, 27 December 2020 (UTC) 7149:07:13, 27 December 2020 (UTC) 7071:08:42, 13 December 2020 (UTC) 6888:07:43, 11 December 2020 (UTC) 6808:04:48, 14 December 2020 (UTC) 6734:09:30, 11 November 2020 (UTC) 6719:07:57, 11 November 2020 (UTC) 6697:07:12, 11 November 2020 (UTC) 6670:22:06, 10 November 2020 (UTC) 6653:22:00, 10 November 2020 (UTC) 6618:20:39, 10 November 2020 (UTC) 6597:20:08, 10 November 2020 (UTC) 6521:18:38, 10 November 2020 (UTC) 6507:18:26, 10 November 2020 (UTC) 6081:17:30, 10 November 2020 (UTC) 6066:10:21, 10 November 2020 (UTC) 6031:06:50, 10 November 2020 (UTC) 917:10:03, 13 November 2008 (UTC) 875:13:55, 20 December 2007 (UTC) 853:01:16, 20 December 2007 (UTC) 696:09:54, 22 February 2006 (UTC) 604:19:57, 21 February 2006 (UTC) 590:12:59, 21 February 2006 (UTC) 253:and see a list of open tasks. 7626:should probably redirect to 5722:23:10, 9 November 2020 (UTC) 5707:22:47, 9 November 2020 (UTC) 5684:22:34, 9 November 2020 (UTC) 5667:22:29, 9 November 2020 (UTC) 5157:{\displaystyle \lambda \in } 3740:Proof of triangle inequality 1671:. It is easy to see that if 1154:{\displaystyle \lambda \in } 780:Comparing against a distance 356:20:07, 21 October 2005 (UTC) 7033: 6783:Knowledge talk:Did you know 6775:this nomination's talk page 1774:{\displaystyle b_{i}\neq 0} 1564:or there is a non-negative 937:02:03, 29 August 2015 (UTC) 541:, approximated distance is 351:Sure, go ahead and add it. 139:The text of the entry was: 7866: 7386:Squared Euclidean distance 5223:norm, the squared norm is 4995:{\displaystyle \lambda =1} 3306:Now, we can conclude that 2811:which holds for all reals 968:23:53, 19 March 2019 (UTC) 377:fast 2d calculation values 372:23:15, 14 March 2008 (UTC) 7835:Mathematics good articles 7547:Good Article criterion 3b 4397:where equality holds iff 2831:. It can be rewritten as 899:17:46, 4 April 2008 (UTC) 830:12:25, 23 July 2007 (UTC) 800:05:37, 29 July 2006 (UTC) 790:06:18, 21 July 2006 (UTC) 278: 211: 190: 65: 61: 33:Mathematics good articles 7491:Signed distance function 7314:Redirects for discussion 3729:{\displaystyle \lambda } 3622:{\displaystyle \lambda } 3126:{\displaystyle \lambda } 3041:{\displaystyle \lambda } 2824:{\displaystyle \lambda } 1577:{\displaystyle \lambda } 821:Two-dimensional distance 816:11:27, 14 May 2008 (UTC) 754:16:10, 18 May 2006 (UTC) 285:project's priority scale 118:appeared on Knowledge's 7830:Knowledge good articles 7636:distance (graph theory) 7511:4. Squared distance in 7465:00:52, 5 May 2022 (UTC) 7446:00:33, 5 May 2022 (UTC) 7428:22:56, 4 May 2022 (UTC) 7414:22:44, 4 May 2022 (UTC) 7398:22:35, 4 May 2022 (UTC) 7384:to "Euclidean metric"? 7346:17:05, 4 May 2022 (UTC) 7328:and has thus listed it 6779:the article's talk page 6760:Did you know nomination 4573:which is equivalent to 4442:{\displaystyle t\geq 0} 2184:which is equivalent to 1697:{\displaystyle b_{i}=0} 1513:{\displaystyle b_{i}=0} 242:WikiProject Mathematics 7513:pseudo-Euclidean space 7378: 7365: 7115:non-Euclidean geometry 7100:related to the topic. 6491: 6490:{\displaystyle ds^{2}} 6467:Everywhere, by metric 6452: 6376: 6270: 6162: 6014: 5905: 5868: 5639: 5589: 5528: 5489: 5400:Then the gradient is 5391: 5371: 5303: 5202: 5158: 5120: 5040: 4996: 4970: 4893: 4864: 4820: 4782: 4703: 4659: 4564: 4520: 4443: 4417: 4388: 4304: 4020: 3967: 3912: 3854: 3796: 3730: 3710: 3666: 3623: 3597: 3298: 3127: 3106: 3042: 3018: 2825: 2802: 2684: 2512: 2175: 1883: 1819: 1775: 1742: 1698: 1665: 1621: 1578: 1558: 1514: 1479: 1282: 1202: 1155: 1114: 1031: 1011: 991: 573: 534: 532:{\displaystyle dy: --> 500: 437: 172:This article is rated 116:fact from this article 7373: 7361: 7229:rational trigonometry 7085:rational trigonometry 6886:). Self-nominated at 6492: 6453: 6377: 6271: 6163: 6015: 5885: 5869: 5640: 5590: 5508: 5469: 5392: 5351: 5283: 5203: 5159: 5121: 5041: 4997: 4971: 4894: 4865: 4821: 4783: 4704: 4660: 4565: 4521: 4444: 4418: 4389: 4305: 4021: 3968: 3913: 3855: 3797: 3731: 3711: 3667: 3624: 3598: 3299: 3128: 3107: 3043: 3019: 2826: 2803: 2685: 2513: 2176: 1884: 1820: 1776: 1743: 1699: 1666: 1622: 1579: 1559: 1515: 1480: 1283: 1203: 1156: 1115: 1032: 1012: 992: 574: 535: 501: 438: 39:good article criteria 7357:Metric (mathematics) 6974:copyright violations 6676:Metric (mathematics) 6641:orthodromic distance 6471: 6390: 6282: 6176: 6094: 6050:metric (mathematics) 5882: 5732: 5605: 5407: 5230: 5168: 5130: 5055: 5006: 4980: 4905: 4877: 4830: 4792: 4717: 4669: 4579: 4530: 4455: 4427: 4401: 4321: 4036: 3977: 3924: 3864: 3806: 3748: 3720: 3676: 3633: 3613: 3313: 3140: 3117: 3053: 3032: 2838: 2815: 2699: 2528: 2191: 1900: 1830: 1785: 1752: 1708: 1675: 1631: 1588: 1568: 1524: 1491: 1295: 1220: 1165: 1127: 1044: 1021: 1001: 981: 545: 511: 451: 388: 265:mathematics articles 91:Good article nominee 7178:(Moved from above. 7154:It is ridiculously 6846:Pythagorean theorem 6530:Riemannian manifold 5942: 5565: 5504: 5386: 5318: 5268: 4892:{\displaystyle q=r} 4416:{\displaystyle q=r} 3587: 3563: 3542: 3518: 3287: 3263: 3242: 3218: 3094: 3070: 3004: 2980: 2882: 2858: 2671: 2647: 2626: 2602: 2499: 2475: 2454: 2430: 2404: 2380: 2362: 2338: 2320: 2276: 2252: 2208: 2162: 2138: 2117: 2093: 2067: 2043: 2025: 2001: 1871: 1847: 1472: 1448: 1426: 1402: 973:Triangle inequality 903:Yes, it's the same 714:^2)/(0.41+0.941246* 149:Pythagorean theorem 7744:Euclidean distance 7614:You're right that 7351:"Euclidean metric" 7274:Euclidean distance 7111:history of geodesy 7097:Bregman divergence 6982:close paraphrasing 6841:Euclidean distance 6838:... that although 6551:is defined on the 6487: 6448: 6372: 6266: 6158: 6010: 6005: 5928: 5864: 5635: 5585: 5551: 5490: 5387: 5372: 5304: 5254: 5198: 5154: 5116: 5036: 4992: 4966: 4889: 4860: 4816: 4778: 4699: 4655: 4560: 4516: 4439: 4413: 4384: 4300: 4016: 3963: 3918:. Use we notation 3908: 3850: 3792: 3726: 3706: 3662: 3619: 3593: 3573: 3549: 3528: 3504: 3294: 3273: 3249: 3228: 3204: 3123: 3101: 3080: 3056: 3038: 3014: 2990: 2966: 2868: 2844: 2821: 2798: 2680: 2657: 2633: 2612: 2588: 2508: 2485: 2461: 2440: 2416: 2390: 2366: 2348: 2324: 2306: 2262: 2238: 2194: 2171: 2148: 2124: 2103: 2079: 2053: 2029: 2011: 1987: 1878: 1857: 1833: 1815: 1771: 1738: 1694: 1661: 1617: 1574: 1554: 1510: 1487:with equality iff 1475: 1458: 1434: 1412: 1388: 1278: 1198: 1151: 1110: 1027: 1007: 987: 922:Incomplete article 569: 529: 496: 433: 234:Mathematics portal 178:content assessment 144:Euclidean distance 66:Article milestones 27:Euclidean distance 7624:geodesic distance 7616:geodesic distance 7594:geodesic distance 7565:geodesic distance 7044: 7043: 7032: 7031: 6995: 6994: 6951:Adequate sourcing 6935: 6934: 6891: 6858: 6001: 5965: 5926: 5862: 5842: 5809: 5748: 5669: 5653:comment added by 5549: 5467: 5430: 4899:is equivalent to 4624: 4298: 4210: 4122: 3591: 3494: 2675: 2503: 2166: 1781:for at least one 1473: 1427: 1381: 1030:{\displaystyle r} 1010:{\displaystyle q} 990:{\displaystyle p} 863:original research 645:is then equal to 299: 298: 295: 294: 291: 290: 158: 157: 133:December 26, 2020 100: 99: 57: 7857: 7632:intrinsic metric 7500:between curves. 7498:Fréchet distance 7382:MOS:BOLDREDIRECT 7326:Euclidean metric 7323: 7138: 7053: 7034: 7026: 7025: 7018: 7017: 6997: 6989: 6988: 6969: 6968: 6958: 6957: 6937: 6929: 6928: 6918: 6917: 6897: 6877: 6849: 6794:The result was: 6772: 6496: 6494: 6493: 6488: 6486: 6485: 6457: 6455: 6454: 6449: 6447: 6446: 6434: 6433: 6421: 6420: 6405: 6404: 6381: 6379: 6378: 6373: 6371: 6370: 6358: 6357: 6345: 6344: 6329: 6328: 6313: 6312: 6297: 6296: 6276:or alternatively 6275: 6273: 6272: 6267: 6265: 6264: 6249: 6248: 6233: 6232: 6217: 6216: 6204: 6203: 6191: 6190: 6170:Minkowski metric 6167: 6165: 6164: 6159: 6157: 6156: 6141: 6140: 6125: 6124: 6109: 6108: 6019: 6017: 6016: 6011: 6009: 6008: 6002: 5999: 5992: 5991: 5966: 5963: 5941: 5936: 5927: 5925: 5924: 5923: 5907: 5904: 5899: 5873: 5871: 5870: 5865: 5863: 5861: 5853: 5845: 5843: 5841: 5833: 5825: 5814: 5810: 5808: 5800: 5799: 5798: 5755: 5749: 5747: 5736: 5644: 5642: 5641: 5636: 5594: 5592: 5591: 5586: 5581: 5580: 5564: 5559: 5550: 5548: 5547: 5546: 5530: 5527: 5522: 5503: 5498: 5488: 5483: 5468: 5466: 5465: 5464: 5448: 5431: 5429: 5428: 5427: 5411: 5396: 5394: 5393: 5388: 5385: 5380: 5370: 5365: 5347: 5346: 5341: 5337: 5336: 5332: 5323: 5319: 5317: 5312: 5302: 5297: 5267: 5262: 5207: 5205: 5204: 5199: 5163: 5161: 5160: 5155: 5125: 5123: 5122: 5117: 5112: 5111: 5099: 5098: 5080: 5079: 5067: 5066: 5045: 5043: 5042: 5037: 5001: 4999: 4998: 4993: 4975: 4973: 4972: 4967: 4962: 4961: 4949: 4948: 4930: 4929: 4917: 4916: 4898: 4896: 4895: 4890: 4869: 4867: 4866: 4861: 4825: 4823: 4822: 4817: 4787: 4785: 4784: 4779: 4774: 4773: 4761: 4760: 4742: 4741: 4729: 4728: 4708: 4706: 4705: 4700: 4664: 4662: 4661: 4656: 4651: 4650: 4638: 4637: 4625: 4623: 4609: 4604: 4603: 4591: 4590: 4569: 4567: 4566: 4561: 4525: 4523: 4522: 4517: 4512: 4511: 4499: 4498: 4480: 4479: 4467: 4466: 4448: 4446: 4445: 4440: 4422: 4420: 4419: 4414: 4393: 4391: 4390: 4385: 4309: 4307: 4306: 4301: 4299: 4297: 4296: 4287: 4286: 4274: 4273: 4252: 4251: 4242: 4241: 4229: 4228: 4216: 4211: 4209: 4208: 4199: 4198: 4186: 4185: 4164: 4163: 4154: 4153: 4141: 4140: 4128: 4123: 4121: 4120: 4111: 4110: 4098: 4097: 4076: 4075: 4066: 4065: 4053: 4052: 4040: 4025: 4023: 4022: 4017: 4015: 4014: 4002: 4001: 3989: 3988: 3972: 3970: 3969: 3964: 3962: 3961: 3949: 3948: 3936: 3935: 3917: 3915: 3914: 3909: 3904: 3903: 3885: 3884: 3859: 3857: 3856: 3851: 3846: 3845: 3827: 3826: 3801: 3799: 3798: 3793: 3788: 3787: 3769: 3768: 3735: 3733: 3732: 3727: 3715: 3713: 3712: 3707: 3671: 3669: 3668: 3663: 3661: 3660: 3645: 3644: 3628: 3626: 3625: 3620: 3602: 3600: 3599: 3594: 3592: 3586: 3581: 3562: 3557: 3541: 3536: 3517: 3512: 3500: 3495: 3493: 3492: 3483: 3482: 3473: 3472: 3454: 3453: 3444: 3443: 3431: 3426: 3421: 3420: 3411: 3410: 3392: 3391: 3382: 3381: 3372: 3364: 3363: 3354: 3353: 3335: 3334: 3325: 3324: 3303: 3301: 3300: 3295: 3286: 3281: 3262: 3257: 3241: 3236: 3217: 3212: 3194: 3193: 3184: 3183: 3165: 3164: 3155: 3154: 3132: 3130: 3129: 3124: 3112: 3109: 3108: 3102: 3093: 3088: 3069: 3064: 3047: 3045: 3044: 3039: 3023: 3021: 3020: 3015: 3003: 2998: 2979: 2974: 2956: 2955: 2946: 2945: 2927: 2926: 2917: 2916: 2895: 2894: 2881: 2876: 2857: 2852: 2830: 2828: 2827: 2822: 2807: 2805: 2804: 2799: 2788: 2787: 2778: 2777: 2762: 2761: 2740: 2739: 2730: 2729: 2714: 2713: 2689: 2687: 2686: 2681: 2676: 2670: 2665: 2646: 2641: 2625: 2620: 2601: 2596: 2584: 2579: 2578: 2569: 2568: 2550: 2549: 2540: 2539: 2517: 2515: 2514: 2509: 2504: 2498: 2493: 2474: 2469: 2453: 2448: 2429: 2424: 2412: 2403: 2398: 2379: 2374: 2361: 2356: 2337: 2332: 2319: 2314: 2302: 2301: 2292: 2291: 2275: 2270: 2251: 2246: 2234: 2233: 2224: 2223: 2207: 2202: 2180: 2178: 2177: 2172: 2167: 2161: 2156: 2137: 2132: 2116: 2111: 2092: 2087: 2075: 2066: 2061: 2042: 2037: 2024: 2019: 2000: 1995: 1983: 1982: 1973: 1972: 1960: 1959: 1938: 1937: 1928: 1927: 1915: 1914: 1889: 1886: 1885: 1879: 1870: 1865: 1846: 1841: 1824: 1822: 1821: 1816: 1780: 1778: 1777: 1772: 1764: 1763: 1747: 1745: 1744: 1739: 1703: 1701: 1700: 1695: 1687: 1686: 1670: 1668: 1667: 1662: 1626: 1624: 1623: 1618: 1616: 1615: 1600: 1599: 1583: 1581: 1580: 1575: 1563: 1561: 1560: 1555: 1519: 1517: 1516: 1511: 1503: 1502: 1484: 1482: 1481: 1476: 1474: 1471: 1466: 1447: 1442: 1433: 1428: 1425: 1420: 1401: 1396: 1387: 1382: 1380: 1379: 1370: 1369: 1357: 1356: 1335: 1334: 1325: 1324: 1312: 1311: 1299: 1287: 1285: 1284: 1279: 1277: 1276: 1264: 1263: 1245: 1244: 1232: 1231: 1207: 1205: 1204: 1199: 1160: 1158: 1157: 1152: 1119: 1117: 1116: 1111: 1036: 1034: 1033: 1028: 1016: 1014: 1013: 1008: 996: 994: 993: 988: 578: 576: 575: 570: 540: 537: 536: 530: 505: 503: 502: 497: 495: 490: 489: 477: 476: 467: 442: 440: 439: 434: 432: 427: 426: 414: 413: 404: 267: 266: 263: 260: 257: 236: 231: 230: 220: 213: 212: 207: 199: 192: 175: 169: 168: 167: 160: 109: 86: 84:December 8, 2020 63: 46: 23: 16: 7865: 7864: 7860: 7859: 7858: 7856: 7855: 7854: 7820: 7819: 7526:any thoughts? – 7479: 7353: 7317: 7176: 7132: 7089:quadratic forms 7078: 7073: 7023: 7015: 6986: 6966: 6955: 6926: 6915: 6835: 6833: 6829:Article history 6768: 6762: 6578: 6562: 6550: 6477: 6469: 6468: 6438: 6425: 6409: 6396: 6388: 6387: 6362: 6349: 6336: 6320: 6304: 6288: 6280: 6279: 6256: 6240: 6224: 6208: 6195: 6182: 6174: 6173: 6148: 6132: 6116: 6100: 6092: 6091: 6045: 6004: 6003: 5996: 5983: 5977: 5976: 5960: 5947: 5915: 5911: 5880: 5879: 5854: 5846: 5834: 5826: 5801: 5790: 5756: 5750: 5740: 5730: 5729: 5603: 5602: 5572: 5538: 5534: 5456: 5452: 5419: 5415: 5405: 5404: 5282: 5278: 5277: 5273: 5272: 5228: 5227: 5222: 5214: 5166: 5165: 5128: 5127: 5103: 5090: 5071: 5058: 5053: 5052: 5004: 5003: 4978: 4977: 4953: 4940: 4921: 4908: 4903: 4902: 4875: 4874: 4828: 4827: 4790: 4789: 4765: 4752: 4733: 4720: 4715: 4714: 4667: 4666: 4642: 4629: 4613: 4595: 4582: 4577: 4576: 4528: 4527: 4503: 4490: 4471: 4458: 4453: 4452: 4425: 4424: 4399: 4398: 4319: 4318: 4288: 4278: 4265: 4243: 4233: 4220: 4200: 4190: 4177: 4155: 4145: 4132: 4112: 4102: 4089: 4067: 4057: 4044: 4034: 4033: 4006: 3993: 3980: 3975: 3974: 3953: 3940: 3927: 3922: 3921: 3895: 3876: 3862: 3861: 3837: 3818: 3804: 3803: 3779: 3760: 3746: 3745: 3742: 3718: 3717: 3674: 3673: 3652: 3636: 3631: 3630: 3611: 3610: 3484: 3474: 3464: 3445: 3435: 3412: 3402: 3383: 3373: 3355: 3345: 3326: 3316: 3311: 3310: 3185: 3175: 3156: 3146: 3138: 3137: 3115: 3114: 3050: 3049: 3030: 3029: 2947: 2937: 2918: 2908: 2886: 2836: 2835: 2813: 2812: 2779: 2769: 2753: 2731: 2721: 2705: 2697: 2696: 2570: 2560: 2541: 2531: 2526: 2525: 2293: 2283: 2225: 2215: 2189: 2188: 1974: 1964: 1951: 1929: 1919: 1906: 1898: 1897: 1827: 1826: 1783: 1782: 1755: 1750: 1749: 1706: 1705: 1678: 1673: 1672: 1629: 1628: 1607: 1591: 1586: 1585: 1566: 1565: 1522: 1521: 1494: 1489: 1488: 1371: 1361: 1348: 1326: 1316: 1303: 1293: 1292: 1268: 1255: 1236: 1223: 1218: 1217: 1214: 1163: 1162: 1125: 1124: 1042: 1041: 1019: 1018: 999: 998: 979: 978: 977:For any points 975: 944: 924: 883: 845:Triathematician 837: 823: 782: 674:* (0.41 + 0.94* 543: 542: 508: 507: 481: 468: 449: 448: 418: 405: 386: 385: 379: 304: 264: 261: 258: 255: 254: 232: 225: 205: 176:on Knowledge's 173: 154: 153: 137: 82: 12: 11: 5: 7863: 7861: 7853: 7852: 7847: 7842: 7837: 7832: 7822: 7821: 7803: 7802: 7801: 7800: 7799: 7798: 7797: 7796: 7795: 7794: 7793: 7792: 7791: 7790: 7789: 7788: 7787: 7786: 7785: 7784: 7752:David Eppstein 7717:David Eppstein 7696: 7692: 7670:David Eppstein 7662:MOS:LEADLENGTH 7658: 7651: 7612: 7598:David Eppstein 7551:David Eppstein 7524:David Eppstein 7487:Parallel curve 7478: 7475: 7474: 7473: 7472: 7471: 7470: 7469: 7468: 7467: 7457:David Eppstein 7406:David Eppstein 7352: 7349: 7330:for discussion 7316: 7310: 7309: 7308: 7298:David Eppstein 7294: 7293: 7292: 7267: 7266: 7265: 7264: 7263: 7262: 7215:David Eppstein 7175: 7172: 7171: 7170: 7160:David Eppstein 7135:David Eppstein 7121: 7120: 7119: 7118: 7077: 7074: 7042: 7041: 7030: 7029: 7028: 7027: 7019: 7005: 7004: 6993: 6992: 6991: 6990: 6970: 6959: 6945: 6944: 6933: 6932: 6931: 6930: 6919: 6905: 6904: 6894: 6880:David Eppstein 6875: 6874: 6873: 6872: 6870:Teresa Rodrigo 6860: 6859: 6832: 6831: 6826: 6816: 6814: 6810: 6792: 6791: 6763: 6761: 6758: 6757: 6756: 6755: 6754: 6753: 6752: 6751: 6750: 6749: 6748: 6747: 6746: 6745: 6744: 6743: 6742: 6741: 6740: 6739: 6738: 6737: 6736: 6711:David Eppstein 6662:David Eppstein 6610:David Eppstein 6574: 6566:at each point 6558: 6546: 6513:David Eppstein 6484: 6480: 6476: 6462: 6461: 6460: 6459: 6445: 6441: 6437: 6432: 6428: 6424: 6419: 6416: 6412: 6408: 6403: 6399: 6395: 6385: 6382: 6369: 6365: 6361: 6356: 6352: 6348: 6343: 6339: 6335: 6332: 6327: 6323: 6319: 6316: 6311: 6307: 6303: 6300: 6295: 6291: 6287: 6277: 6263: 6259: 6255: 6252: 6247: 6243: 6239: 6236: 6231: 6227: 6223: 6220: 6215: 6211: 6207: 6202: 6198: 6194: 6189: 6185: 6181: 6171: 6168: 6155: 6151: 6147: 6144: 6139: 6135: 6131: 6128: 6123: 6119: 6115: 6112: 6107: 6103: 6099: 6089: 6086: 6073:David Eppstein 6044: 6041: 6040: 6039: 6038: 6037: 6036: 6035: 6034: 6033: 6020: 6007: 5997: 5995: 5990: 5986: 5982: 5979: 5978: 5975: 5972: 5969: 5961: 5959: 5956: 5953: 5952: 5950: 5945: 5940: 5935: 5931: 5922: 5918: 5914: 5910: 5903: 5898: 5895: 5892: 5888: 5877: 5874: 5860: 5857: 5852: 5849: 5840: 5837: 5832: 5829: 5823: 5820: 5817: 5813: 5807: 5804: 5797: 5793: 5789: 5786: 5783: 5780: 5777: 5774: 5771: 5768: 5765: 5762: 5759: 5753: 5746: 5743: 5739: 5727: 5714:David Eppstein 5687: 5686: 5676:David Eppstein 5646: 5645: 5634: 5631: 5628: 5625: 5622: 5619: 5616: 5613: 5610: 5596: 5595: 5584: 5579: 5575: 5571: 5568: 5563: 5558: 5554: 5545: 5541: 5537: 5533: 5526: 5521: 5518: 5515: 5511: 5507: 5502: 5497: 5493: 5487: 5482: 5479: 5476: 5472: 5463: 5459: 5455: 5451: 5446: 5443: 5440: 5437: 5434: 5426: 5422: 5418: 5414: 5398: 5397: 5384: 5379: 5375: 5369: 5364: 5361: 5358: 5354: 5350: 5345: 5340: 5335: 5331: 5327: 5322: 5316: 5311: 5307: 5301: 5296: 5293: 5290: 5286: 5281: 5276: 5271: 5266: 5261: 5257: 5253: 5250: 5247: 5244: 5241: 5238: 5235: 5220: 5213: 5210: 5197: 5194: 5191: 5188: 5185: 5182: 5179: 5176: 5173: 5153: 5150: 5147: 5144: 5141: 5138: 5135: 5115: 5110: 5106: 5102: 5097: 5093: 5089: 5086: 5083: 5078: 5074: 5070: 5065: 5061: 5035: 5032: 5029: 5026: 5023: 5020: 5017: 5014: 5011: 4991: 4988: 4985: 4965: 4960: 4956: 4952: 4947: 4943: 4939: 4936: 4933: 4928: 4924: 4920: 4915: 4911: 4888: 4885: 4882: 4859: 4856: 4853: 4850: 4847: 4844: 4841: 4838: 4835: 4815: 4812: 4809: 4806: 4803: 4800: 4797: 4777: 4772: 4768: 4764: 4759: 4755: 4751: 4748: 4745: 4740: 4736: 4732: 4727: 4723: 4698: 4695: 4692: 4689: 4686: 4683: 4680: 4677: 4674: 4654: 4649: 4645: 4641: 4636: 4632: 4628: 4622: 4619: 4616: 4612: 4607: 4602: 4598: 4594: 4589: 4585: 4559: 4556: 4553: 4550: 4547: 4544: 4541: 4538: 4535: 4526:holds for all 4515: 4510: 4506: 4502: 4497: 4493: 4489: 4486: 4483: 4478: 4474: 4470: 4465: 4461: 4438: 4435: 4432: 4412: 4409: 4406: 4395: 4394: 4383: 4380: 4377: 4374: 4371: 4368: 4365: 4362: 4359: 4356: 4353: 4350: 4347: 4344: 4341: 4338: 4335: 4332: 4329: 4326: 4312: 4311: 4295: 4291: 4285: 4281: 4277: 4272: 4268: 4264: 4261: 4258: 4255: 4250: 4246: 4240: 4236: 4232: 4227: 4223: 4219: 4214: 4207: 4203: 4197: 4193: 4189: 4184: 4180: 4176: 4173: 4170: 4167: 4162: 4158: 4152: 4148: 4144: 4139: 4135: 4131: 4126: 4119: 4115: 4109: 4105: 4101: 4096: 4092: 4088: 4085: 4082: 4079: 4074: 4070: 4064: 4060: 4056: 4051: 4047: 4043: 4013: 4009: 4005: 4000: 3996: 3992: 3987: 3983: 3960: 3956: 3952: 3947: 3943: 3939: 3934: 3930: 3907: 3902: 3898: 3894: 3891: 3888: 3883: 3879: 3875: 3872: 3869: 3849: 3844: 3840: 3836: 3833: 3830: 3825: 3821: 3817: 3814: 3811: 3791: 3786: 3782: 3778: 3775: 3772: 3767: 3763: 3759: 3756: 3753: 3741: 3738: 3725: 3705: 3702: 3699: 3696: 3693: 3690: 3687: 3684: 3681: 3659: 3655: 3651: 3648: 3643: 3639: 3618: 3604: 3603: 3590: 3585: 3580: 3576: 3572: 3569: 3566: 3561: 3556: 3552: 3548: 3545: 3540: 3535: 3531: 3527: 3524: 3521: 3516: 3511: 3507: 3503: 3498: 3491: 3487: 3481: 3477: 3471: 3467: 3463: 3460: 3457: 3452: 3448: 3442: 3438: 3434: 3429: 3425: 3419: 3415: 3409: 3405: 3401: 3398: 3395: 3390: 3386: 3380: 3376: 3371: 3367: 3362: 3358: 3352: 3348: 3344: 3341: 3338: 3333: 3329: 3323: 3319: 3293: 3290: 3285: 3280: 3276: 3272: 3269: 3266: 3261: 3256: 3252: 3248: 3245: 3240: 3235: 3231: 3227: 3224: 3221: 3216: 3211: 3207: 3203: 3200: 3197: 3192: 3188: 3182: 3178: 3174: 3171: 3168: 3163: 3159: 3153: 3149: 3145: 3122: 3100: 3097: 3092: 3087: 3083: 3079: 3076: 3073: 3068: 3063: 3059: 3037: 3026: 3025: 3013: 3010: 3007: 3002: 2997: 2993: 2989: 2986: 2983: 2978: 2973: 2969: 2965: 2962: 2959: 2954: 2950: 2944: 2940: 2936: 2933: 2930: 2925: 2921: 2915: 2911: 2907: 2904: 2901: 2898: 2893: 2889: 2885: 2880: 2875: 2871: 2867: 2864: 2861: 2856: 2851: 2847: 2843: 2820: 2809: 2808: 2797: 2794: 2791: 2786: 2782: 2776: 2772: 2768: 2765: 2760: 2756: 2752: 2749: 2746: 2743: 2738: 2734: 2728: 2724: 2720: 2717: 2712: 2708: 2704: 2679: 2674: 2669: 2664: 2660: 2656: 2653: 2650: 2645: 2640: 2636: 2632: 2629: 2624: 2619: 2615: 2611: 2608: 2605: 2600: 2595: 2591: 2587: 2582: 2577: 2573: 2567: 2563: 2559: 2556: 2553: 2548: 2544: 2538: 2534: 2519: 2518: 2507: 2502: 2497: 2492: 2488: 2484: 2481: 2478: 2473: 2468: 2464: 2460: 2457: 2452: 2447: 2443: 2439: 2436: 2433: 2428: 2423: 2419: 2415: 2410: 2407: 2402: 2397: 2393: 2389: 2386: 2383: 2378: 2373: 2369: 2365: 2360: 2355: 2351: 2347: 2344: 2341: 2336: 2331: 2327: 2323: 2318: 2313: 2309: 2305: 2300: 2296: 2290: 2286: 2282: 2279: 2274: 2269: 2265: 2261: 2258: 2255: 2250: 2245: 2241: 2237: 2232: 2228: 2222: 2218: 2214: 2211: 2206: 2201: 2197: 2182: 2181: 2170: 2165: 2160: 2155: 2151: 2147: 2144: 2141: 2136: 2131: 2127: 2123: 2120: 2115: 2110: 2106: 2102: 2099: 2096: 2091: 2086: 2082: 2078: 2073: 2070: 2065: 2060: 2056: 2052: 2049: 2046: 2041: 2036: 2032: 2028: 2023: 2018: 2014: 2010: 2007: 2004: 1999: 1994: 1990: 1986: 1981: 1977: 1971: 1967: 1963: 1958: 1954: 1950: 1947: 1944: 1941: 1936: 1932: 1926: 1922: 1918: 1913: 1909: 1905: 1877: 1874: 1869: 1864: 1860: 1856: 1853: 1850: 1845: 1840: 1836: 1814: 1811: 1808: 1805: 1802: 1799: 1796: 1793: 1790: 1770: 1767: 1762: 1758: 1737: 1734: 1731: 1728: 1725: 1722: 1719: 1716: 1713: 1704:holds for all 1693: 1690: 1685: 1681: 1660: 1657: 1654: 1651: 1648: 1645: 1642: 1639: 1636: 1614: 1610: 1606: 1603: 1598: 1594: 1573: 1553: 1550: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1509: 1506: 1501: 1497: 1470: 1465: 1461: 1457: 1454: 1451: 1446: 1441: 1437: 1431: 1424: 1419: 1415: 1411: 1408: 1405: 1400: 1395: 1391: 1385: 1378: 1374: 1368: 1364: 1360: 1355: 1351: 1347: 1344: 1341: 1338: 1333: 1329: 1323: 1319: 1315: 1310: 1306: 1302: 1275: 1271: 1267: 1262: 1258: 1254: 1251: 1248: 1243: 1239: 1235: 1230: 1226: 1216:For any reals 1213: 1210: 1197: 1194: 1191: 1188: 1185: 1182: 1179: 1176: 1173: 1170: 1150: 1147: 1144: 1141: 1138: 1135: 1132: 1121: 1120: 1109: 1106: 1103: 1100: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 1049: 1026: 1006: 986: 974: 971: 943: 940: 923: 920: 909:129.132.152.15 882: 879: 878: 877: 836: 835:Approximations 833: 822: 819: 803: 802: 781: 778: 759: 758: 757: 756: 744: 743: 724: 723: 722: 721: 720: 719: 703: 702: 701: 700: 699: 698: 662:= 0.41 + 0.94* 609: 608: 607: 606: 593: 592: 568: 565: 562: 559: 556: 553: 550: 528: 525: 522: 519: 516: 494: 488: 484: 480: 475: 471: 466: 462: 459: 456: 445:absolute value 431: 425: 421: 417: 412: 408: 403: 399: 396: 393: 378: 375: 359: 358: 342: 341: 303: 300: 297: 296: 293: 292: 289: 288: 277: 271: 270: 268: 251:the discussion 238: 237: 221: 209: 208: 200: 188: 187: 181: 170: 156: 155: 138: 113: 112: 110: 102: 101: 98: 97: 94: 87: 79: 78: 75: 72: 68: 67: 59: 58: 24: 13: 10: 9: 6: 4: 3: 2: 7862: 7851: 7848: 7846: 7843: 7841: 7838: 7836: 7833: 7831: 7828: 7827: 7825: 7818: 7817: 7813: 7809: 7783: 7780: 7777: 7773: 7772: 7767: 7763: 7762: 7761: 7757: 7753: 7749: 7745: 7741: 7740: 7739: 7736: 7733: 7728: 7727: 7726: 7722: 7718: 7714: 7713: 7712: 7709: 7706: 7702: 7697: 7693: 7691: 7688: 7685: 7681: 7680: 7679: 7675: 7671: 7667: 7663: 7659: 7656: 7652: 7649: 7648: 7647: 7644: 7641: 7637: 7633: 7629: 7625: 7621: 7617: 7613: 7609: 7608: 7607: 7603: 7599: 7595: 7591: 7587: 7583: 7579: 7578: 7577: 7574: 7571: 7566: 7562: 7561: 7560: 7556: 7552: 7548: 7543: 7538: 7537: 7536: 7535: 7532: 7529: 7525: 7520: 7518: 7514: 7509: 7507: 7501: 7499: 7494: 7492: 7488: 7482: 7476: 7466: 7462: 7458: 7454: 7449: 7448: 7447: 7443: 7439: 7434: 7431: 7430: 7429: 7425: 7421: 7417: 7416: 7415: 7411: 7407: 7402: 7401: 7400: 7399: 7395: 7391: 7387: 7383: 7377: 7372: 7370: 7364: 7360: 7358: 7350: 7348: 7347: 7343: 7339: 7335: 7331: 7327: 7322: 7315: 7311: 7307: 7303: 7299: 7295: 7291: 7287: 7283: 7279: 7275: 7271: 7270: 7269: 7268: 7261: 7257: 7253: 7248: 7247: 7246: 7242: 7238: 7234: 7230: 7226: 7225: 7224: 7220: 7216: 7211: 7210: 7209: 7208: 7204: 7200: 7196: 7191: 7189: 7185: 7181: 7173: 7169: 7165: 7161: 7157: 7153: 7152: 7151: 7150: 7146: 7142: 7136: 7130: 7125: 7116: 7112: 7108: 7107: 7106: 7105: 7104: 7101: 7098: 7094: 7093:Hilbert space 7090: 7086: 7081: 7075: 7072: 7068: 7064: 7060: 7057: 7052: 7047: 7039: 7036: 7035: 7021:Interesting: 7020: 7012: 7009: 7008: 7007: 7006: 7002: 6999: 6998: 6983: 6979: 6975: 6971: 6963: 6960: 6952: 6949: 6948: 6947: 6946: 6942: 6939: 6938: 6923: 6920: 6912: 6909: 6908: 6907: 6906: 6902: 6899: 6898: 6895: 6892: 6889: 6885: 6881: 6871: 6867: 6864: 6863: 6862: 6861: 6857: 6855: 6847: 6843: 6842: 6837: 6836: 6830: 6827: 6825: 6821: 6818: 6817: 6813: 6811: 6809: 6805: 6801: 6797: 6790: 6788: 6784: 6780: 6776: 6771: 6765: 6764: 6759: 6735: 6731: 6727: 6722: 6721: 6720: 6716: 6712: 6708: 6704: 6700: 6699: 6698: 6694: 6690: 6686: 6681: 6677: 6673: 6672: 6671: 6667: 6663: 6658: 6657: 6656: 6655: 6654: 6650: 6646: 6642: 6638: 6634: 6630: 6626: 6621: 6620: 6619: 6615: 6611: 6607: 6606:metric spaces 6604: 6600: 6599: 6598: 6594: 6590: 6586: 6582: 6577: 6573: 6570:. The family 6569: 6565: 6561: 6557: 6554: 6553:tangent space 6549: 6545: 6542: 6541:inner product 6538: 6534: 6531: 6528: 6524: 6523: 6522: 6518: 6514: 6510: 6509: 6508: 6504: 6500: 6482: 6478: 6474: 6466: 6465: 6464: 6463: 6443: 6439: 6435: 6430: 6426: 6422: 6417: 6414: 6410: 6406: 6401: 6397: 6393: 6386: 6383: 6367: 6363: 6359: 6354: 6350: 6346: 6341: 6337: 6333: 6330: 6325: 6321: 6317: 6314: 6309: 6305: 6301: 6298: 6293: 6289: 6285: 6278: 6261: 6257: 6253: 6250: 6245: 6241: 6237: 6234: 6229: 6225: 6221: 6218: 6213: 6209: 6205: 6200: 6196: 6192: 6187: 6183: 6179: 6172: 6169: 6153: 6149: 6145: 6142: 6137: 6133: 6129: 6126: 6121: 6117: 6113: 6110: 6105: 6101: 6097: 6090: 6087: 6084: 6083: 6082: 6078: 6074: 6070: 6069: 6068: 6067: 6063: 6059: 6055: 6051: 6042: 6032: 6028: 6024: 6021: 5993: 5988: 5984: 5980: 5973: 5970: 5967: 5957: 5954: 5948: 5943: 5938: 5933: 5929: 5920: 5916: 5901: 5896: 5893: 5890: 5886: 5878: 5875: 5858: 5850: 5838: 5830: 5821: 5818: 5815: 5811: 5805: 5795: 5784: 5778: 5775: 5769: 5763: 5751: 5744: 5728: 5725: 5724: 5723: 5719: 5715: 5710: 5709: 5708: 5704: 5700: 5695: 5691: 5690: 5689: 5688: 5685: 5681: 5677: 5672: 5671: 5670: 5668: 5664: 5660: 5656: 5652: 5632: 5629: 5626: 5623: 5617: 5611: 5601: 5600: 5599: 5582: 5577: 5573: 5569: 5566: 5561: 5556: 5552: 5543: 5539: 5524: 5519: 5516: 5513: 5509: 5505: 5500: 5495: 5491: 5485: 5480: 5477: 5474: 5470: 5461: 5457: 5444: 5438: 5432: 5424: 5420: 5403: 5402: 5401: 5382: 5377: 5373: 5367: 5362: 5359: 5356: 5352: 5348: 5343: 5338: 5333: 5329: 5325: 5320: 5314: 5309: 5305: 5299: 5294: 5291: 5288: 5284: 5279: 5274: 5269: 5264: 5259: 5251: 5245: 5239: 5233: 5226: 5225: 5224: 5219: 5211: 5209: 5192: 5189: 5186: 5183: 5180: 5174: 5171: 5148: 5145: 5142: 5136: 5133: 5108: 5104: 5100: 5095: 5091: 5084: 5081: 5076: 5072: 5068: 5063: 5059: 5050: 5047: 5030: 5027: 5024: 5021: 5018: 5012: 5009: 4989: 4986: 4983: 4958: 4954: 4950: 4945: 4941: 4934: 4931: 4926: 4922: 4918: 4913: 4909: 4900: 4886: 4883: 4880: 4871: 4854: 4851: 4848: 4845: 4842: 4836: 4833: 4810: 4807: 4804: 4798: 4795: 4770: 4766: 4762: 4757: 4753: 4746: 4743: 4738: 4734: 4730: 4725: 4721: 4712: 4709: 4693: 4690: 4687: 4684: 4681: 4675: 4672: 4647: 4643: 4639: 4634: 4630: 4620: 4617: 4614: 4610: 4605: 4600: 4596: 4592: 4587: 4583: 4574: 4571: 4554: 4551: 4548: 4545: 4542: 4536: 4533: 4508: 4504: 4500: 4495: 4491: 4484: 4481: 4476: 4472: 4468: 4463: 4459: 4450: 4436: 4433: 4430: 4410: 4407: 4404: 4378: 4375: 4372: 4366: 4363: 4357: 4354: 4351: 4345: 4342: 4336: 4333: 4330: 4324: 4317: 4316: 4315: 4293: 4283: 4279: 4275: 4270: 4266: 4259: 4256: 4253: 4248: 4238: 4234: 4230: 4225: 4221: 4212: 4205: 4195: 4191: 4187: 4182: 4178: 4171: 4168: 4165: 4160: 4150: 4146: 4142: 4137: 4133: 4124: 4117: 4107: 4103: 4099: 4094: 4090: 4083: 4080: 4077: 4072: 4062: 4058: 4054: 4049: 4045: 4032: 4031: 4030: 4027: 4011: 4007: 4003: 3998: 3994: 3990: 3985: 3981: 3958: 3954: 3950: 3945: 3941: 3937: 3932: 3928: 3919: 3900: 3896: 3892: 3889: 3886: 3881: 3877: 3870: 3867: 3842: 3838: 3834: 3831: 3828: 3823: 3819: 3812: 3809: 3784: 3780: 3776: 3773: 3770: 3765: 3761: 3754: 3751: 3739: 3737: 3723: 3700: 3697: 3694: 3691: 3688: 3682: 3679: 3657: 3653: 3649: 3646: 3641: 3637: 3616: 3607: 3583: 3578: 3574: 3570: 3567: 3564: 3559: 3554: 3550: 3538: 3533: 3529: 3525: 3522: 3519: 3514: 3509: 3505: 3496: 3489: 3479: 3475: 3469: 3465: 3461: 3458: 3455: 3450: 3446: 3440: 3436: 3427: 3417: 3413: 3407: 3403: 3399: 3396: 3393: 3388: 3384: 3378: 3374: 3365: 3360: 3356: 3350: 3346: 3342: 3339: 3336: 3331: 3327: 3321: 3317: 3309: 3308: 3307: 3304: 3291: 3283: 3278: 3274: 3270: 3267: 3264: 3259: 3254: 3250: 3238: 3233: 3229: 3225: 3222: 3219: 3214: 3209: 3205: 3198: 3190: 3186: 3180: 3176: 3172: 3169: 3166: 3161: 3157: 3151: 3147: 3134: 3120: 3098: 3095: 3090: 3085: 3081: 3077: 3074: 3071: 3066: 3061: 3057: 3035: 3011: 3008: 3000: 2995: 2991: 2987: 2984: 2981: 2976: 2971: 2967: 2960: 2952: 2948: 2942: 2938: 2934: 2931: 2928: 2923: 2919: 2913: 2909: 2902: 2899: 2896: 2891: 2887: 2878: 2873: 2869: 2865: 2862: 2859: 2854: 2849: 2845: 2834: 2833: 2832: 2818: 2795: 2792: 2789: 2784: 2774: 2770: 2766: 2763: 2758: 2754: 2747: 2744: 2741: 2736: 2726: 2722: 2718: 2715: 2710: 2706: 2695: 2694: 2693: 2690: 2677: 2667: 2662: 2658: 2654: 2651: 2648: 2643: 2638: 2634: 2622: 2617: 2613: 2609: 2606: 2603: 2598: 2593: 2589: 2580: 2575: 2571: 2565: 2561: 2557: 2554: 2551: 2546: 2542: 2536: 2532: 2522: 2505: 2495: 2490: 2486: 2482: 2479: 2476: 2471: 2466: 2462: 2450: 2445: 2441: 2437: 2434: 2431: 2426: 2421: 2417: 2408: 2405: 2400: 2395: 2391: 2387: 2384: 2381: 2376: 2371: 2367: 2363: 2358: 2353: 2349: 2345: 2342: 2339: 2334: 2329: 2325: 2321: 2316: 2311: 2307: 2303: 2298: 2294: 2288: 2284: 2280: 2277: 2272: 2267: 2263: 2259: 2256: 2253: 2248: 2243: 2239: 2235: 2230: 2226: 2220: 2216: 2212: 2209: 2204: 2199: 2195: 2187: 2186: 2185: 2168: 2158: 2153: 2149: 2145: 2142: 2139: 2134: 2129: 2125: 2113: 2108: 2104: 2100: 2097: 2094: 2089: 2084: 2080: 2071: 2068: 2063: 2058: 2054: 2050: 2047: 2044: 2039: 2034: 2030: 2026: 2021: 2016: 2012: 2008: 2005: 2002: 1997: 1992: 1988: 1984: 1979: 1969: 1965: 1961: 1956: 1952: 1945: 1942: 1939: 1934: 1924: 1920: 1916: 1911: 1907: 1896: 1895: 1894: 1891: 1875: 1872: 1867: 1862: 1858: 1854: 1851: 1848: 1843: 1838: 1834: 1809: 1806: 1803: 1800: 1797: 1791: 1788: 1768: 1765: 1760: 1756: 1732: 1729: 1726: 1723: 1720: 1714: 1711: 1691: 1688: 1683: 1679: 1655: 1652: 1649: 1646: 1643: 1637: 1634: 1612: 1608: 1604: 1601: 1596: 1592: 1571: 1548: 1545: 1542: 1539: 1536: 1530: 1527: 1507: 1504: 1499: 1495: 1485: 1468: 1463: 1459: 1455: 1452: 1449: 1444: 1439: 1435: 1429: 1422: 1417: 1413: 1409: 1406: 1403: 1398: 1393: 1389: 1383: 1376: 1366: 1362: 1358: 1353: 1349: 1342: 1339: 1336: 1331: 1321: 1317: 1313: 1308: 1304: 1289: 1273: 1269: 1265: 1260: 1256: 1252: 1249: 1246: 1241: 1237: 1233: 1228: 1224: 1211: 1209: 1192: 1189: 1186: 1180: 1177: 1174: 1171: 1168: 1145: 1142: 1139: 1133: 1130: 1107: 1101: 1098: 1095: 1089: 1086: 1080: 1077: 1074: 1068: 1065: 1059: 1056: 1053: 1047: 1040: 1039: 1038: 1024: 1004: 984: 972: 970: 969: 965: 961: 957: 952: 950: 941: 939: 938: 934: 930: 921: 919: 918: 914: 910: 906: 901: 900: 896: 892: 888: 880: 876: 872: 868: 864: 861: 857: 856: 855: 854: 850: 846: 841: 834: 832: 831: 828: 820: 818: 817: 813: 809: 801: 798: 794: 793: 792: 791: 788: 779: 777: 776: 772: 768: 764: 755: 752: 748: 747: 746: 745: 741: 737: 731: 726: 725: 717: 713: 709: 708: 707: 706: 705: 704: 697: 694: 689: 685: 681: 677: 673: 669: 665: 661: 656: 652: 648: 644: 640: 636: 632: 628: 624: 619: 615: 614: 613: 612: 611: 610: 605: 602: 597: 596: 595: 594: 591: 588: 584: 583: 582: 581: 566: 563: 560: 557: 554: 551: 548: 526: 523: 520: 517: 514: 486: 482: 478: 473: 469: 460: 457: 454: 446: 423: 419: 415: 410: 406: 397: 394: 391: 383: 376: 374: 373: 369: 365: 357: 354: 350: 349: 348: 347: 340: 337: 333: 332: 331: 329: 325: 321: 317: 313: 309: 301: 286: 282: 281:High-priority 276: 273: 272: 269: 252: 248: 244: 243: 235: 229: 224: 222: 219: 215: 214: 210: 206:High‑priority 204: 201: 198: 194: 189: 185: 179: 171: 162: 161: 152: 150: 146: 145: 135: 134: 129: 127: 126:Did you know? 121: 117: 111: 108: 104: 103: 95: 93: 92: 88: 85: 81: 80: 76: 73: 70: 69: 64: 60: 55: 53: 52: 44: 40: 36: 35: 34: 28: 25: 22: 18: 17: 7804: 7770: 7748:WP:TECHNICAL 7743: 7700: 7581: 7521: 7510: 7506:trigonometry 7502: 7495: 7483: 7480: 7379: 7374: 7369:Metric space 7366: 7362: 7354: 7318: 7282:73.89.25.252 7252:73.89.25.252 7192: 7180:73.89.25.252 7177: 7156:WP:TECHNICAL 7141:73.89.25.252 7126: 7122: 7102: 7082: 7079: 7045: 7037: 7000: 6940: 6900: 6893: 6876: 6865: 6853: 6839: 6812: 6795: 6793: 6786: 6778: 6774: 6769: 6766: 6707:metric space 6632: 6628: 6624: 6602: 6575: 6571: 6567: 6563: 6559: 6555: 6547: 6543: 6532: 6046: 5693: 5649:— Preceding 5647: 5597: 5399: 5217: 5215: 5164:and for all 5051: 5048: 5002:and for all 4901: 4872: 4826:and for all 4713: 4710: 4575: 4572: 4451: 4423:or there is 4396: 4313: 4028: 3920: 3743: 3716:. The value 3608: 3605: 3305: 3135: 3027: 2810: 2691: 2523: 2520: 2183: 1892: 1486: 1290: 1215: 1122: 976: 953: 948: 945: 929:Blackbombchu 925: 902: 884: 842: 838: 824: 804: 783: 760: 733: 729: 715: 711: 687: 683: 679: 675: 671: 667: 663: 659: 654: 650: 646: 642: 638: 634: 630: 626: 625:, therefore 622: 617: 580: 381: 380: 360: 343: 327: 323: 319: 315: 311: 307: 305: 280: 240: 184:WikiProjects 142: 140: 131: 123: 89: 49: 47: 43:please do so 31: 30: 26: 7701:geometrical 7666:WP:SIZERULE 7634:instead of 7622:. The link 7420:fgnievinski 7390:fgnievinski 7338:fgnievinski 7056:WP:DYKcheck 6922:Long enough 5598:It follows 949:on occasion 691:it :-)). -- 682:is in fact 649:* sqrt(1 + 539:dx}" /: --> 256:Mathematics 247:mathematics 203:Mathematics 7824:Categories 7542:Arc length 7237:XOR'easter 7199:XOR'easter 6978:plagiarism 6911:New enough 4449:such that 3629:such that 3111:0}" /: --> 1888:0}" /: --> 1584:such that 1161:such that 958:project.-- 927:rotation? 827:EarthJoker 364:80.2.17.86 130:column on 37:under the 7776:jacobolus 7732:jacobolus 7705:jacobolus 7684:jacobolus 7640:jacobolus 7590:geodesics 7582:Euclidean 7570:jacobolus 7528:jacobolus 7517:spacetime 7278:Quadrance 6527:connected 5126:for some 4788:for some 956:ProofWiki 860:unsourced 666:). Then, 510:dx}": --> 120:Main Page 7655:distance 7628:geodesic 7620:geodesic 7586:distance 7367:Article 7355:Article 7233:last one 7063:Joofjoof 7040:: Done. 6972:Free of 6901:General: 6866:Reviewed 6850:Source: 6844:and the 6796:promoted 6726:Lantonov 6689:Lantonov 6645:Lantonov 6637:geodesic 6631:through 6589:Lantonov 6499:Lantonov 6058:Lantonov 6023:Lantonov 5699:Lantonov 5663:contribs 5655:Lantonov 5651:unsigned 4873:Because 4665:for all 3672:for all 3052:0}": --> 3048:because 1829:0}": --> 1627:for all 1520:for all 891:Bitwiseb 742:--Rafael 561:0.941246 322:- 2×min( 302:Comments 174:GA-class 147:and the 51:reassess 7771:Metrica 7046:Overall 6962:Neutral 6941:Policy: 6820:Comment 6537:infimum 6043:Metric? 5694:squared 1825:. Then 718:). --BB 637:(where 283:on the 122:in the 74:Process 7808:Kmhkmh 7563:Maybe 7438:Ovinus 7371:says: 7359:says: 7195:WP:DUE 6980:, and 6852:Maor, 6685:metric 6458:, etc. 3136:(***): 1288:holds 1037:holds 960:Kmhkmh 905:Euclid 887:Euclid 881:Euclid 797:StuRat 751:StuRat 601:StuRat 587:Abdull 447:) and 353:StuRat 336:StuRat 180:scale. 96:Listed 77:Result 7766:Heron 7011:Cited 7001:Hook: 6800:Amkgp 6000:else 3096:: --> 2524:(**) 1873:: --> 1212:Lemma 787:Myria 735:: --> 657:: --> 620:: --> 521:: --> 506:. If 7812:talk 7756:talk 7721:talk 7674:talk 7602:talk 7555:talk 7461:talk 7442:talk 7424:talk 7410:talk 7394:talk 7342:talk 7302:talk 7286:talk 7256:talk 7241:talk 7219:talk 7203:talk 7184:talk 7164:talk 7145:talk 7095:and 7067:talk 6884:talk 6824:view 6804:talk 6730:talk 6715:talk 6693:talk 6678:and 6666:talk 6649:talk 6614:talk 6593:talk 6585:Here 6517:talk 6503:talk 6077:talk 6062:talk 6027:talk 5718:talk 5703:talk 5680:talk 5659:talk 4976:for 3860:and 3744:Let 1291:(*) 1017:and 964:talk 933:talk 913:talk 895:talk 871:talk 867:DVdm 849:talk 812:talk 771:talk 767:mcld 549:0.41 384:Let 368:talk 310:and 275:High 71:Date 7779:(t) 7768:'s 7735:(t) 7708:(t) 7687:(t) 7643:(t) 7638:. – 7630:or 7573:(t) 7531:(t) 7190:) 7038:QPQ 6822:or 6798:by 6781:or 6709:? — 6701:Re 6627:to 6603:are 5964:if 4711:or 4314:or 736:8 ) 670:=~ 533:dx} 346:127 7826:: 7814:) 7758:) 7723:) 7676:) 7604:) 7557:) 7519:. 7493:. 7463:) 7444:) 7426:) 7412:) 7396:) 7344:) 7304:) 7288:) 7258:) 7243:) 7221:) 7205:) 7186:) 7166:) 7147:) 7117:". 7091:, 7069:) 7048:: 7013:: 6984:: 6976:, 6964:: 6953:: 6924:: 6913:: 6868:: 6815:( 6806:) 6777:, 6732:) 6717:) 6695:) 6668:) 6651:) 6616:) 6595:) 6583:. 6519:) 6505:) 6347:− 6251:− 6235:− 6219:− 6079:) 6064:) 6029:) 5971:≠ 5913:∂ 5909:∂ 5887:∑ 5856:∂ 5848:∂ 5836:∂ 5828:∂ 5819:− 5803:∂ 5776:− 5758:∂ 5742:∂ 5738:∂ 5720:) 5705:) 5682:) 5665:) 5661:• 5609:∇ 5536:∂ 5532:∂ 5510:∑ 5471:∑ 5454:∂ 5450:∂ 5417:∂ 5413:∂ 5353:∑ 5285:∑ 5256:‖ 5249:‖ 5208:. 5187:… 5175:∈ 5137:∈ 5134:λ 5101:− 5085:λ 5069:− 5046:. 5025:… 5013:∈ 4984:λ 4951:− 4935:λ 4919:− 4870:. 4849:… 4837:∈ 4799:∈ 4796:λ 4763:− 4747:λ 4731:− 4688:… 4676:∈ 4640:− 4593:− 4570:, 4549:… 4537:∈ 4501:− 4469:− 4434:≥ 4343:≤ 4276:− 4257:⋯ 4231:− 4188:− 4169:⋯ 4143:− 4125:≤ 4100:− 4081:⋯ 4055:− 4026:. 4004:− 3973:, 3951:− 3890:… 3832:… 3802:, 3774:… 3724:λ 3695:… 3683:∈ 3650:λ 3617:λ 3568:⋯ 3523:⋯ 3497:≤ 3459:⋯ 3397:⋯ 3366:≤ 3340:⋯ 3268:⋯ 3223:⋯ 3199:≤ 3170:⋯ 3121:λ 3105:0} 3075:⋯ 3036:λ 3009:≥ 2985:⋯ 2932:⋯ 2903:λ 2897:− 2888:λ 2863:⋯ 2819:λ 2790:≥ 2767:λ 2764:− 2745:⋯ 2719:λ 2716:− 2652:⋯ 2607:⋯ 2581:≤ 2555:⋯ 2480:⋯ 2435:⋯ 2385:⋯ 2343:⋯ 2322:≤ 2257:⋯ 2143:⋯ 2098:⋯ 2048:⋯ 2006:⋯ 1985:≤ 1943:⋯ 1890:. 1882:0} 1852:⋯ 1804:… 1792:∈ 1766:≠ 1727:… 1715:∈ 1650:… 1638:∈ 1605:λ 1572:λ 1543:… 1531:∈ 1453:⋯ 1407:⋯ 1384:≤ 1340:⋯ 1250:… 1190:− 1181:λ 1172:− 1134:∈ 1131:λ 1066:⩽ 997:, 966:) 935:) 915:) 907:. 897:) 873:) 851:) 814:) 808:PE 773:) 765:-- 693:BB 688:dx 684:dy 672:dx 647:dx 635:dx 629:= 627:dy 623:dx 621:= 618:dy 479:− 416:− 370:) 328:dy 324:dx 320:dy 318:+ 316:dx 312:dy 308:dx 114:A 54:it 45:. 7810:( 7754:( 7730:– 7719:( 7672:( 7657:. 7600:( 7568:– 7553:( 7522:@ 7459:( 7440:( 7422:( 7408:( 7392:( 7340:( 7300:( 7284:( 7254:( 7239:( 7217:( 7201:( 7182:( 7162:( 7143:( 7137:: 7133:@ 7065:( 6890:. 6882:( 6834:) 6802:( 6789:. 6728:( 6713:( 6691:( 6664:( 6660:— 6647:( 6633:q 6629:r 6625:p 6612:( 6591:( 6576:p 6572:g 6568:p 6564:M 6560:p 6556:T 6548:p 6544:g 6533:M 6515:( 6501:( 6483:2 6479:s 6475:d 6444:k 6440:x 6436:d 6431:i 6427:x 6423:d 6418:k 6415:i 6411:g 6407:= 6402:2 6398:s 6394:d 6368:2 6364:t 6360:d 6355:2 6351:c 6342:2 6338:z 6334:d 6331:+ 6326:2 6322:y 6318:d 6315:+ 6310:2 6306:x 6302:d 6299:= 6294:2 6290:s 6286:d 6262:2 6258:z 6254:d 6246:2 6242:y 6238:d 6230:2 6226:x 6222:d 6214:2 6210:t 6206:d 6201:2 6197:c 6193:= 6188:2 6184:s 6180:d 6154:2 6150:z 6146:d 6143:+ 6138:2 6134:y 6130:d 6127:+ 6122:2 6118:x 6114:d 6111:= 6106:2 6102:s 6098:d 6075:( 6060:( 6025:( 5994:, 5989:j 5985:x 5981:2 5974:k 5968:j 5958:, 5955:0 5949:{ 5944:= 5939:2 5934:k 5930:x 5921:j 5917:x 5902:n 5897:1 5894:= 5891:k 5859:p 5851:q 5839:q 5831:p 5822:2 5816:= 5812:) 5806:p 5796:2 5792:) 5788:) 5785:p 5782:( 5779:q 5773:) 5770:q 5767:( 5764:p 5761:( 5752:( 5745:q 5716:( 5701:( 5678:( 5657:( 5633:. 5630:x 5627:2 5624:= 5621:) 5618:x 5615:( 5612:f 5583:. 5578:j 5574:x 5570:2 5567:= 5562:2 5557:k 5553:x 5544:j 5540:x 5525:n 5520:1 5517:= 5514:k 5506:= 5501:2 5496:k 5492:x 5486:n 5481:1 5478:= 5475:k 5462:j 5458:x 5445:= 5442:) 5439:x 5436:( 5433:f 5425:j 5421:x 5383:2 5378:k 5374:x 5368:n 5363:1 5360:= 5357:k 5349:= 5344:2 5339:) 5334:2 5330:/ 5326:1 5321:) 5315:2 5310:k 5306:x 5300:n 5295:1 5292:= 5289:k 5280:( 5275:( 5270:= 5265:2 5260:2 5252:x 5246:= 5243:) 5240:x 5237:( 5234:f 5221:2 5218:l 5196:} 5193:n 5190:, 5184:, 5181:1 5178:{ 5172:i 5152:] 5149:1 5146:, 5143:0 5140:[ 5114:) 5109:i 5105:p 5096:i 5092:r 5088:( 5082:= 5077:i 5073:p 5064:i 5060:q 5034:} 5031:n 5028:, 5022:, 5019:1 5016:{ 5010:i 4990:1 4987:= 4964:) 4959:i 4955:p 4946:i 4942:r 4938:( 4932:= 4927:i 4923:p 4914:i 4910:q 4887:r 4884:= 4881:q 4858:} 4855:n 4852:, 4846:, 4843:1 4840:{ 4834:i 4814:) 4811:1 4808:, 4805:0 4802:[ 4776:) 4771:i 4767:p 4758:i 4754:r 4750:( 4744:= 4739:i 4735:p 4726:i 4722:q 4697:} 4694:n 4691:, 4685:, 4682:1 4679:{ 4673:i 4653:) 4648:i 4644:p 4635:i 4631:r 4627:( 4621:t 4618:+ 4615:1 4611:t 4606:= 4601:i 4597:p 4588:i 4584:q 4558:} 4555:n 4552:, 4546:, 4543:1 4540:{ 4534:i 4514:) 4509:i 4505:r 4496:i 4492:q 4488:( 4485:t 4482:= 4477:i 4473:q 4464:i 4460:p 4437:0 4431:t 4411:r 4408:= 4405:q 4382:) 4379:r 4376:, 4373:q 4370:( 4367:d 4364:+ 4361:) 4358:q 4355:, 4352:p 4349:( 4346:d 4340:) 4337:r 4334:, 4331:p 4328:( 4325:d 4310:, 4294:2 4290:) 4284:n 4280:r 4271:n 4267:q 4263:( 4260:+ 4254:+ 4249:2 4245:) 4239:1 4235:r 4226:1 4222:q 4218:( 4213:+ 4206:2 4202:) 4196:n 4192:q 4183:n 4179:p 4175:( 4172:+ 4166:+ 4161:2 4157:) 4151:1 4147:q 4138:1 4134:p 4130:( 4118:2 4114:) 4108:n 4104:r 4095:n 4091:p 4087:( 4084:+ 4078:+ 4073:2 4069:) 4063:1 4059:r 4050:1 4046:p 4042:( 4012:i 4008:r 3999:i 3995:q 3991:= 3986:i 3982:b 3959:i 3955:q 3946:i 3942:p 3938:= 3933:i 3929:a 3906:) 3901:n 3897:r 3893:, 3887:, 3882:1 3878:r 3874:( 3871:= 3868:r 3848:) 3843:n 3839:q 3835:, 3829:, 3824:1 3820:q 3816:( 3813:= 3810:q 3790:) 3785:n 3781:p 3777:, 3771:, 3766:1 3762:p 3758:( 3755:= 3752:p 3704:} 3701:n 3698:, 3692:, 3689:1 3686:{ 3680:i 3658:i 3654:b 3647:= 3642:i 3638:a 3589:) 3584:2 3579:n 3575:b 3571:+ 3565:+ 3560:2 3555:1 3551:b 3547:( 3544:) 3539:2 3534:n 3530:a 3526:+ 3520:+ 3515:2 3510:1 3506:a 3502:( 3490:2 3486:) 3480:n 3476:b 3470:n 3466:a 3462:+ 3456:+ 3451:1 3447:b 3441:1 3437:a 3433:( 3428:= 3424:| 3418:n 3414:b 3408:n 3404:a 3400:+ 3394:+ 3389:1 3385:b 3379:1 3375:a 3370:| 3361:n 3357:b 3351:n 3347:a 3343:+ 3337:+ 3332:1 3328:b 3322:1 3318:a 3292:. 3289:) 3284:2 3279:n 3275:b 3271:+ 3265:+ 3260:2 3255:1 3251:b 3247:( 3244:) 3239:2 3234:n 3230:a 3226:+ 3220:+ 3215:2 3210:1 3206:a 3202:( 3196:) 3191:n 3187:b 3181:n 3177:a 3173:+ 3167:+ 3162:1 3158:b 3152:1 3148:a 3144:( 3099:0 3091:2 3086:n 3082:b 3078:+ 3072:+ 3067:2 3062:1 3058:b 3024:. 3012:0 3006:) 3001:2 2996:n 2992:a 2988:+ 2982:+ 2977:2 2972:1 2968:a 2964:( 2961:+ 2958:) 2953:n 2949:b 2943:n 2939:a 2935:+ 2929:+ 2924:1 2920:b 2914:1 2910:a 2906:( 2900:2 2892:2 2884:) 2879:2 2874:n 2870:b 2866:+ 2860:+ 2855:2 2850:1 2846:b 2842:( 2796:, 2793:0 2785:2 2781:) 2775:n 2771:b 2759:n 2755:a 2751:( 2748:+ 2742:+ 2737:2 2733:) 2727:1 2723:b 2711:1 2707:a 2703:( 2678:. 2673:) 2668:2 2663:n 2659:b 2655:+ 2649:+ 2644:2 2639:1 2635:b 2631:( 2628:) 2623:2 2618:n 2614:a 2610:+ 2604:+ 2599:2 2594:1 2590:a 2586:( 2576:n 2572:b 2566:n 2562:a 2558:+ 2552:+ 2547:1 2543:b 2537:1 2533:a 2506:. 2501:) 2496:2 2491:n 2487:b 2483:+ 2477:+ 2472:2 2467:1 2463:b 2459:( 2456:) 2451:2 2446:n 2442:a 2438:+ 2432:+ 2427:2 2422:1 2418:a 2414:( 2409:2 2406:+ 2401:2 2396:n 2392:b 2388:+ 2382:+ 2377:2 2372:1 2368:b 2364:+ 2359:2 2354:n 2350:a 2346:+ 2340:+ 2335:2 2330:1 2326:a 2317:2 2312:n 2308:b 2304:+ 2299:n 2295:b 2289:n 2285:a 2281:2 2278:+ 2273:2 2268:n 2264:a 2260:+ 2254:+ 2249:2 2244:1 2240:b 2236:+ 2231:1 2227:b 2221:1 2217:a 2213:2 2210:+ 2205:2 2200:1 2196:a 2169:, 2164:) 2159:2 2154:n 2150:b 2146:+ 2140:+ 2135:2 2130:1 2126:b 2122:( 2119:) 2114:2 2109:n 2105:a 2101:+ 2095:+ 2090:2 2085:1 2081:a 2077:( 2072:2 2069:+ 2064:2 2059:n 2055:b 2051:+ 2045:+ 2040:2 2035:1 2031:b 2027:+ 2022:2 2017:n 2013:a 2009:+ 2003:+ 1998:2 1993:1 1989:a 1980:2 1976:) 1970:n 1966:b 1962:+ 1957:n 1953:a 1949:( 1946:+ 1940:+ 1935:2 1931:) 1925:1 1921:b 1917:+ 1912:1 1908:a 1904:( 1876:0 1868:2 1863:n 1859:b 1855:+ 1849:+ 1844:2 1839:1 1835:b 1813:} 1810:n 1807:, 1801:, 1798:1 1795:{ 1789:i 1769:0 1761:i 1757:b 1736:} 1733:n 1730:, 1724:, 1721:1 1718:{ 1712:i 1692:0 1689:= 1684:i 1680:b 1659:} 1656:n 1653:, 1647:, 1644:1 1641:{ 1635:i 1613:i 1609:b 1602:= 1597:i 1593:a 1552:} 1549:n 1546:, 1540:, 1537:1 1534:{ 1528:i 1508:0 1505:= 1500:i 1496:b 1469:2 1464:n 1460:b 1456:+ 1450:+ 1445:2 1440:1 1436:b 1430:+ 1423:2 1418:n 1414:a 1410:+ 1404:+ 1399:2 1394:1 1390:a 1377:2 1373:) 1367:n 1363:b 1359:+ 1354:n 1350:a 1346:( 1343:+ 1337:+ 1332:2 1328:) 1322:1 1318:b 1314:+ 1309:1 1305:a 1301:( 1274:n 1270:b 1266:, 1261:n 1257:a 1253:, 1247:, 1242:1 1238:b 1234:, 1229:1 1225:a 1196:) 1193:p 1187:r 1184:( 1178:= 1175:p 1169:q 1149:] 1146:1 1143:, 1140:0 1137:[ 1108:, 1105:) 1102:r 1099:, 1096:q 1093:( 1090:d 1087:+ 1084:) 1081:q 1078:, 1075:p 1072:( 1069:d 1063:) 1060:r 1057:, 1054:p 1051:( 1048:d 1025:r 1005:q 985:p 962:( 931:( 911:( 893:( 869:( 847:( 810:( 769:( 716:a 712:a 686:/ 680:a 676:a 668:d 664:a 660:b 655:a 651:a 643:d 639:a 633:* 631:a 579:. 567:y 564:d 558:+ 555:x 552:d 527:x 524:d 518:y 515:d 493:| 487:y 483:q 474:y 470:p 465:| 461:= 458:y 455:d 443:( 430:| 424:x 420:q 411:x 407:p 402:| 398:= 395:x 392:d 366:( 326:, 287:. 186:: 136:. 128:" 124:" 56:.

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