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Given the parametric equations used in the article, I think the arc length should be 16a not 8a. The 8a length relates to the parametric equations: x = a cos t (1 - cos t) y= a sin t (1 - cos t) which describe a smaller cardioid that the parametric equations shown in this article. Also, the formula
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The equations express the rectangular coordinates of the curve, x and y, in terms of a parameter theta. That is, for each value of theta, compute x(theta) and y(theta), and plot the point (x,y) measured x units to the right of the y axis, and y units above the x axis (other direction for negatives).
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Plums are almost always an oval shape. Peaches and apples are much closer to the shape than plums. Apples though are usually round and match the shape perfectly, especially so if they are the sweeter ones that supermarkets sell and people like to eat. Apples are the only decent fit when the cardioid
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The other way to understand the images is to ignore x and y, and just look at rho (radius) and theta. At each angle theta measured from the positive x axis, compute the radius rho and put a point that far from the origin. At 180 degrees, the formula 1+cos(theta) gives zero, so you get a point at
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I believe that the definition of the parameter t in the
Equations section may be incorrect. I think t should the (radian) angle from the x-axis to the line from the origin going through the centre of the rolling circle. I could easily derive the given parametric equations for x and y using this
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Whaddaya mean? A squashed square is a great explanation to start out with. It is something easily recognizable from kindergarten. You must begin somewhere recognizable and advance once the audience comprehends. I love all the images: Coffee cup, Spirograph(TM), Apple with grid, Apple with
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I went ahead and did the merge. I took some parts of the page in very summarized and altered form to add to the main article without disturbing the prose style. The old page is still there with the history so it can be restored if someone objects
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is worth including in the main article. The results proved there seem simple enough that they could easily be stated without proof, and adding any portion of those derivations here would clutter up the article.
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so that it can be merged into the main article. In addition, the article name 'Cardioid/Proofs' does not conform to
Knowledge naming conventions. Therefore I propose that the other article be merged with this
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Your a corresponds to the radius to the centre of the circle rolling round, i.e. the diameter of the small circle rather than its radius. The equations are the same otherwise except x is offset a bit.
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Yes, it's the angle. It doesn't matter whether it's in degrees or radians, since you still end up with the same cosine (e.g., cos(90°) is the same thing as cos(pi/2).
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microphone stem. That moving
Spirograph was great, plus being near the top! It instantly lets reader know. A picture's worth 1000 words. They can rest after that.
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definition, whereas I got nowhere using the given definition: "t is the angle at the origin from the horizontal axis to the ray to a point on the cardioid."
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Also, the equation for the area enclosed by a cardioid appears to be incorrect. Instead of A = 6*pi*r^2, isn't it A = 3*pi*r^2/2? Or am I off my rocker?
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The set of all such points makes the curve shown in the image. So, x and y are just numbers to describe the location of a point on a curve.
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Tomatoes don't tend to have a stalk recessed into the fruit. Cherries do but they aren't always round and people think of them in pairs.
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the origin, which is the cusp. By displacing theta, or using sin instead of cosine, the whole picture just rotates around that cusp.
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be a cardioid, and it may be a nephroid. It depends on the angle of the light rays relative to the bottom of the cup. Cheers,
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I tend to agree for the most part and I'll go through the proofs article with a critical eye before doing a merge.--
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is three dimensions because there is no groove in them (peaches and plums have a groove from top to bottom.)
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In the equations, for example; x = a(2cos(t)- cos(2t)) Is t the angle? Is it in degrees or radians?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I'm no geometer, but it looks like the cross-section of a globe tomato to me.
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If you want to know r=4*sin((sin(O/2))) graphs a better "love heart" shape
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I do not understand how the images can be understood - what are x and y? --
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Absolutely not. Read the article. The two shapes are not the same.
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It should be possible to simplify and summarize the material in
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These are not equivalent to those shown in the article.
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Huh?? Why move a chunk of page content off to a subpage?
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IMHO it should be merged with/into Heart (Symbol) --
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210:Knowledge:WikiProject Mathematics/Proofs#Cardioid
772:Knowledge level-5 vital articles in Mathematics
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289:section describes x and y in terms of theta. --
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402:The caustic seen at the bottom of a coffee cup
733:Geometry of the Cardioid by Arseniy V Akopyan
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120:and see a list of open tasks.
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123:Mathematics
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