Talk:Bilinear interpolation

1866:) 'In mathematics, bilinear interpolation is an extension of linear interpolation 'for interpolating functions of two variables. The key idea is to perform linear 'interpolation first in one direction, and then in the other direction. ' 'Suppose that we want to find the value of the unknown function f at 'the point P = (x, y). It is assumed that we know the value of f at the four 'points Q11 = (x1, y1), Q12 = (x1, y2), Q21 = (x2, y1), and Q22 = (x2, y2). ' 'We first do linear interpolation in the x-direction. This yields ' ' f(R1) = (x2 - x)/(x2 - x1) * f(Q11) + (x - x1)/(x2 - x1)*f(Q21) ' where R1 = (x, y1) 'and ' f(R2) = (x2 - x)/(x2 - x1) * f(Q12) + (x - x1)/(x2 - x1)*f(Q22) ' where R2 = (x, y2) ' 'We then proceed by interpolating in the y-direction. ' ' f(P) = (y2 - y)/(y2-y1) * f(R1) + (y - y1)/(y2 - y1) * f(R2) ' where P = (x, y) ' 'Constraints: ' x is a Double and xHeaderRowMinValue <= x <= xHeaderRowMaxValue ' y is a Double and yHeaderColumnMinValue <= y <= yHeaderColumnMaxValue ' rgTable is an Excel range, specified in the normal Excel way ' The range should be a rectangle including the xHeader Row, ' the yHeaderRow, and all of the data table. ' xHeaderRow values increase to the right ' yHeaderColumn values increase downward Const minPositiveDouble As Double = 4.94065645841247E-324 Const maxPositiveDouble As Double = 1.7976931348623E+308 Const Epsilon As Double = 0.0001 'Tolerable error allowed Dim rgXHdr, rgYHdr, rgXYData As Range Dim xHdr(), yHdr(), xyData() As Double 'Array sizes still unknown Dim xHdrMin, xHdrMax, yHdrMin, yHdrMax As Double Dim x1, x2, y1, y2, ixX1, ixX2, ixY1, ixY2, fQ11, fQ12, fQ21, fQ22, fR1, fR2, fP As Double Dim row, col As Range Dim i, j As Integer Dim t As Variant 'Get xHeaderRow, and set up array to hold values Set rgXHdr = rgTable.Offset(0, 1).Resize(1, rgTable.columns.Count - 1) ReDim xHdr(rgXHdr.columns.Count - 1) 'Size the array 'Find min/max limits for xHeaderRow, and read values into array ' Assumes minimum xHeader value is 0, ie no negative numbers xHdrMin = maxPositiveDouble xHdrMax = 0 i = 0 For Each col In rgXHdr.columns If xHdrMax < col.Cells(1, 1).Value Then xHdrMax = col.Cells(1, 1).Value End If If xHdrMin : --> 1877:

xyData(rgXYData.columns.Count - 1, rgXYData.rows.Count - 1) 'Size the 2D array 'Read Excel table values into xyData array i = 0 For Each col In rgXYData.columns j = 0 For Each row In col.rows xyData(i, j) = row.Cells(1, 1).Value j = j + 1 Next i = i + 1 Next 'Find x1, x2, and corresponding indices ixX1, ixX2 in xHdr i = 0 x1 = xHdrMin x2 = xHdrMin ixX1 = 0 ixX2 = 0 For Each t In xHdr If x = t Then x1 = t x2 = t ixX1 = i ixX2 = i Exit For ElseIf x : -->

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fQ22 = xyData(ixX2, ixY2) 'Get f(R1) and f(R2) If x1 = x2 Then fR1 = fQ11 fR2 = fQ12 Else fR1 = ((x2 - x) / (x2 - x1)) * fQ11 + ((x - x1) / (x2 - x1)) * fQ21 fR2 = ((x2 - x) / (x2 - x1)) * fQ12 + ((x - x1) / (x2 - x1)) * fQ22 End If 'Get f(P) If y1 = y2 Then fP = fR1 Else fP = ((y2 - y) / (y2 - y1)) * fR1 + ((y - y1) / (y2 - y1)) * fR2 End If Interp2D = fP End Function

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ReDim yHdr(rgYHdr.rows.Count - 1) 'Size the array 'Find min/max limits for yHeaderColumn, and read values into array ' Assumes minimum yHeader value is 0, ie no negative numbers yHdrMin = maxPositiveDouble yHdrMax = 0 i = 0 For Each row In rgYHdr.rows If yHdrMax < row.Cells(1, 1).Value Then yHdrMax = row.Cells(1, 1).Value End If If yHdrMin : -->

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t Then y1 = t y2 = yHdr(i + 1) ixY1 = i ixY2 = i + 1 End If i = i + 1 Next 'Get fQ11, fQ12, fQ21, fQ22 fQ11 = xyData(ixX1, ixY1) fQ12 = xyData(ixX1, ixY2) fQ21 = xyData(ixX2, ixY1)

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t Then x1 = t x2 = xHdr(i + 1) ixX1 = i ixX2 = i + 1 End If i = i + 1 Next 'Find y1, y2, and corresponding indices ixY1, ixY2 in yHdr i = 0 y1 = yHdrMin y2 = yHdrMin ixY1 = 0

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col.Cells(1, 1).Value Then xHdrMin = col.Cells(1, 1).Value End If xHdr(i) = col.Cells(1, 1).Value i = i + 1 Next 'Get yHeaderColumn, and set up array to hold values Set rgYHdr = rgTable.Offset(1, 0).Resize(rgTable.rows.Count - 1, 1)

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It would seem useful to include software code for many of these excellent mathematical explanations/algorithms. I've included here an Excel user function I wrote yesterday that should work with nearly any 2D table in Excel. The algorithm is based on the algorithm on the main page. There has been very

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The idea of explaining the method using the "capital-H" (or capital-I) shape may help less mathematically skilled readers, but overcomplicates it, IMHO. After reading (and before some thinking) I was left with the impression that three separate linear operations were required, but the meaning of the

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I think it is an excellent formulation for interpretation -- the weights of each corner are proportional to the area of the opposing sub-rectangle. The (x2-x1)(y2-y1) term is the area of the whole rectangle, and, for instance, the (x-x1)(y-y1) is the area of the rectangle opposite Q_22. That would

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yHdrMax Then MsgBox ("Interp2D: x or y value out of range: x= " & x & " y= " & y) End End If 'Get xyData, and set up array to hold values Set rgXYData = rgTable.Offset(1, 1).Resize(rgTable.rows.Count - 1, rgTable.columns.Count - 1) ReDim

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Applying the interpolation is a linear transform of the input variables (Q_**), but the surface between the points is not a plane. Three points specify a plane, and the fourth would require some sort of curvature. If you were interpolating on a triangle by finding the point on the plane defined

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We did a test in a spread sheet - with all plus signs the answers were correct - minus signs would have given an incorrect answer. If the f(p) equation is plugged into the the f(R1) and f(R2) equations then minus signs do not seem to occur. We could be wrong though .. :-) I can provide a link to a

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Compound interpolation operations, whether bilinear or trilinear, are defined by functions that have the same depth as the space in which points are defined, e.g. a compound interpolation operation performed in a 2D grid (i.e. bilinear interpolation), requires an interpolant of degree two, and a

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If the phrase is bothering you, then perhaps you could try reformulating it? By the way, I don't know that this non-linear interpolant actually generates a linear filter that will be used to process the image. Different editors come from different backgrounds, and I don't do image manipulation.

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row.Cells(1, 1).Value Then yHdrMin = row.Cells(1, 1).Value End If yHdr(i) = row.Cells(1, 1).Value i = i + 1 Next 'Verify that x & y are in proper range, ' after allowing for Epsilon differences in values If x <

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hold to higher dimensions: In trilinear, the weights of each component are proportional to the size of the opposing volume. Back down to the univariate linear, case, the weight of each extreme is proportional to the distance opposite the extreme. Conceptually, it seems equivalent to the

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compound interpolation operation in a 3D grid (i.e. trilinear interpolation) requires an interpolant of degree three. Performing one linear interpolation on each dimension of the space you are working in is how the interpolant is defined for these 'grid interpolation operations'.

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I don't know why this phrase is bothering me... Perhaps because I also want to include a dicussion on the fact that this non-linear interpolant actually generates a linear filter that will be used to process the image (as you all know). Wat do you think?... --

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However, explaining bilinear interpolation as successive 1d linear interpolations also has its advantages. It explains the name (you do first an interpolation in the x-direction and then one in the y-direction) and it points to an easy way to do an error

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Regardless of the depth (i.e. number of dimensions) of the space in which points are defined, linear interpolation operation is a one dimensional operation; essentially you generate a third point between two known points, all three being on a

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I found the formula with b_xx coefficients would be very useful, but I couldn’t figure out how is the derivation done with only the final formula provided. It might be a lot better to add more information and explanation of that formula. :-)

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I like your idea of coupling the areas with the weights very much. It is easy to see it as a generalization of the one-dimensional case, where the weights are determined by the distances. I agree that this will be less complicated than the

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Does the alternate algorithm (assuming the final functional form, and solving the matrix to find the weights) also trivially generalise to individual (convex) non-rectilinear quadrilaterals? (And what about convex quadrilateral meshes?)

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Here's the formula I use for bilinear interpolation of a quadrilateral region bounded by curved edges. The position is parametrized by , each going 0 to 1, with 4 corner positions A,B,C,D, and 4 edge curves e(u), f(v), g(v), and h(u).

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On using this page to check some interpolations we noticed that some minus signs should be plus signs. These have been changed. Otherwise this page is excellent. How about something similar in the tri-linear equations page?

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Could someone please explain what f(Q11), f(Q12), f(Q21), and f(Q22) actually are? Everything else is explicitly defined, but I have no idea what the function for these points is calculating. In other words, f(Q??) = ???

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I would describe the process as a weighted average of the four nearest points, with the weights being determined by the areas of the four rectangles divided by their sum. This sum is the area of the rectangle QQQQ.

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Trilinear interpolation is a compound interpolation operation that consists of seven linear interpolations (two bilinear interpolations followed by a linear interpolation) that are done in three dimensional

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Bilinear and trilinear interpolations are compound interpolation operations, that consist of multiple linear interpolations along the grid lines, performed on 2D and 3D grids respectively. They are linear

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ixY2 = 0 For Each t In yHdr If y = t Then y1 = t y2 = t ixY1 = i ixY2 = i Exit For ElseIf y : -->

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Like you said, the result of interpolation is x+y indeed, and the form that you said was not definitive for the situation is the very form you used to find the result of interpolation actually.

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Of course, I could also be wrong. Please post the link, or email the code to me directly (you can find my email address by going to my user page and following the link to my home page). --

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There are three kinds of interpolation operations depending on the form of the interpolant used: piecewise constant interpolation, linear interpolation and non-linear interpolation.

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You either don't have the original function that was used to generate the data, or you have it but it is too complex and thus inefficient to evaluate with the resources at hand

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are determined by their relative distances to y on the x-axis. However, note that the weights are inversely proportional to the distances on the x-axis, i.e. the weight of y

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So, what do you think of the following outline: First give the formula and your area-based explanation and then say that the formula can also be derived via the H-shape. --

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Also, the 'four nearest points' idea works this way only if the points are on a regular grid. If the points are irregular, one violates the first assumption on the page.

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Public Function Interp2D(ByVal x As Double, ByVal y As Double, ByVal rgTable As Range) As Double '**** Interp2D BILINEAR INTERPOLATION FUNCTION **** '(From

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Bilinear interpolation is a compound interpolation operation that consists of three consecutive interpolation operations that are done on two dimensional (2D) grids.

3597: 2098: 198: 672:{\displaystyle f(x,y)\approx {\frac {f(Q_{11})}{(x_{1}-x_{2})(y_{1}-y_{2})}}(x-x_{2})(y-y_{2})-{\frac {f(Q_{21})}{(x_{1}-x_{2})(y_{1}-y_{2})}}(x-x_{1})(y-y_{2})} 3304:, you assume that the two known data points and the new data point between the two are connected by a line having a zero slope, and thus the interpolant is y=b 2796: 1507: 1301: 3587: 222: 3577: 1768:

The text mentions the fact that "the interpolant is not linear", but have you ever seen a linear interpolant?... Linear in relation to what, after all??

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Related to code writing based on the given example value data, as it allows for seemingly right, but still wrong, code to output the right value.

3058:). In case the example+image are ever changed/updated. Suggest to not use a example value which is right in the middle. Like 14.5 in this case. ( 3509:

Which edge cases? Sampling at the edges of the image? I'd say that there's no convention for this because of the ad hoc nature of this problem.

1834:. The name does not suggest linearity just because it contains the latter's adjective form. unregistered.user 5:38, 30 September 2009 (UTC) 318: 3221:

The article is well written I think. Examples and figures are pretty cool. The figure illustrating the geometric visualisation is fantastic.

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page with a C code listing showing all plus signs - but I'm not sure if such links are allowed on these pages. I won't change the page.

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Even if you don't like my alternative interpretation, it'd be easier to read if 1/( (x2-x1)(y2-y1) ) were in a sigle term at the start.

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and closely compare the example images. Basically, bicubic is great if you like 135° angles and terrible if you don't like edge halos.

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Removed assertion that bicubic interpolation was superior to bilinear for various things including "edge halos" which is untrue. See

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As long as you are not using the original function, the data you will generate will be estimated/approximated/interpolated data

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Of course it's not linear, it's bilinear. Bilinear means it's linear in terms of either its arguments. Rather like the use in

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Contrary to what the name suggests, the bilinear interpolant is not linear; rather, it is a product of two linear functions:

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a good starting point to understand interpolation. And, it is a horrible starting point to understand computer graphics ;]

3318:, you assume that the three points are connected by a curve (varying slope), and thus the interpolant is a polynomial of 3199:

from the page, and I need to know how to perform Bilinear Interpolation on a calculator, but the page isn't helping me.

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xHdrMax And x <= xHdrMax + Epsilon Then x = xHdrMax End If If y < yHdrMin And y : -->

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This article does not cite any references or sources. (difficult when I'm trying to find references for my coursework)

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yHdrMax And y <= yHdrMax + Epsilon Then y = yHdrMax End If If x < xHdrMin Or x : -->

3311:, you assume that the three points are connected by a line having a non-zero slope, and thus the interpolant is y=mx+b 2554: 2459: 3514: 2297: 1786:, which is a linear function. So "the interpolant is not linear" means nonlinear with respect to the arguments of 3440: 3240: 3051:

Also, this example has an inverted y axis which leads to difficulties when trying to apply the formulas above.

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the result of interpolation is f(x,y) = x + y, which cannot be written in the form of (a1 x + a2)(a3 y + a4).

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

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

2038: 1982: 83: 3469: 3244: 3020: 3016: 2083: 1913: 1227:{\displaystyle -{\frac {f(Q_{12})}{(x_{1}-x_{2})(y_{1}-y_{2})}}(x_{1}-x_{2})(y_{2}-y_{1})=f(Q_{12}).} 1992:

Response to Pot- The geneal expression z=b1+b2*x+b3*y+b4*x*y does not factor into an expression like

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The palm tree image is very rough to start with. Can someone do this with a better original image?

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on the grid lines whereas they are non-linear between the grid lines. They can be referred to as

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I think that this article explains the topic in not the best way and I propose a major rewrite.

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Notable omission in the article is what if any conventions exist for dealing with edge cases.

3554: 3494: 3172: 3130: 3067: 2275:{\displaystyle (a_{1}x+a_{2})(a_{3}y+a_{4})=a_{2}a_{4}+a_{1}a_{4}x+a_{2}a_{3}y+a_{1}a_{3}xy\,} 1803: 1715: 1276: 1246: 351: 301: 194: 228: 3449: 3122: 3091: 1927:

How would one put the page back to what it was before there were a lot of failed parsings?

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the expression z=(a1+a2x)(a3+a4y) does not since a1 and a3 factor out leaving 3 constants

1909: 1894: 3371:-x), because the weighted average should be closer to the value whose weight is larger. 2986:{\displaystyle f(x,y)\approx f(0,0)\,(1-x)(1-y)+f(1,0)\,x(1-y)+f(0,1)\,(1-x)y+f(1,1)xy.} 1697:{\displaystyle f(x,y)\approx f(0,0)\,(1-x)(1-y)+f(1,0)\,x(1-y)+f(0,1)\,(1-x)y+f(1,1)xy.} 1491:{\displaystyle f(x,y)\approx f(0,0)\,(x-1)(y-1)-f(1,0)\,x(y-1)-f(0,1)\,(x-1)y+f(1,1)xy.} 3566: 3248: 2056: 1947: 1831: 1773: 3168: 3126: 3118: 3063: 1873:= yHdrMin - Epsilon Then y = yHdrMin End If If y : --> 1871:= xHdrMin - Epsilon Then x = xHdrMin End If If x : --> 283: 3558: 3539: 3518: 3502: 3473: 3453: 3426: 3214: 3176: 3134: 3106: 3071: 3045: 3028: 2087: 2060: 2042: 1986: 1951: 1917: 1898: 1823: 1811: 1776: 1723: 1284: 1269: 1254: 359: 332: 309: 286: 1744:) represent the weights on the pixels, and this way the alternate weights ( (1 − 1710:

You're probably right. I modified the formulas to avoid negative quantities like

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Maybe it's just me, but I find this way easier because in this case the (1 −

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I think there is an error in the matrix formulation. Product of 2x2 and 1x2?

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Consider 2D Cartesian space: Linear interpolation can be interpreted as a

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The matrix formula for unit square interpolation has x and y transposed.

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The function that you will use to generate new data is called interpolant

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For example in the case of f(0,0) = 0, f(0,1) = f(1,0) = 1, f(1, 1) = 2,

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Error in example picture: Bilinear interpolation in grayscale values.

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Yes, you're right. Thanks for the notice. The error is now fixed. --

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the 4 corners are z1=z(0,0), z2=z(1,0), z3=z(1,1), z4=z(0,1) and

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Ok, I figured it out. It is the value at the points Q11..Q22.--

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in the banner shell. Please resolve this conflict if possible.

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This article has been given a rating which conflicts with the

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Explanations or Derivation of the second Alternative formula

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In either case, you need a new function to generate new data

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I agree. The formula (a1 x + a2)(a3 y + a4) is incorrect.

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Four equalities above, one for each constant, come from;

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by the triangle, then you'd have a linear interpolant.

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z=A(1+Bu)(1+Ct)+D with u,t the normalized coordinates

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No, these in fact are four independent constants --

1802:Please do add something about this application. -- 2985: 2777: 2637: 2542: 2447: 2366: 2274: 1696: 1490: 1226: 1036: 962: 671: 3583:C-Class articles with conflicting quality ratings 3441:The myth of infinite detail: Bilinear vs. Bicubic 2288:Alternatively, the interpolant can be written as: 1998:If z=A(1+Bu)(1+Ct)+D then in terms of b1,b2,b3,b4 1859:little testing so take this as a starting point. 3331:operation, where the values to be averaged are y 2638:{\displaystyle b_{3}=a_{2}a_{3}=f(0,1)-f(0,0)\,} 2543:{\displaystyle b_{2}=a_{1}a_{4}=f(1,0)-f(0,0)\,} 1295:Wouldn't it be much more intuitive to replace 3275:You have a set of data but you need more data 3008:The correct value should be 146.1, not 139.5 1864:http://en.wikipedia.org/Bilinear_interpolation 2367:{\displaystyle b_{1}+b_{2}x+b_{3}y+b_{4}xy\,} 207:Knowledge (XXG):WikiProject Computer graphics 8: 19: 3484: 152: 47: 3593:Mid-importance computer graphics articles 3373:This is linear interpolation in 2D space. 2798: 2680: 2670: 2657: 2651: 2585: 2575: 2562: 2556: 2490: 2480: 2467: 2461: 2448:{\displaystyle b_{1}=a_{2}a_{4}=f(0,0)\,} 2416: 2406: 2393: 2387: 2350: 2334: 2318: 2305: 2299: 2258: 2248: 2232: 2222: 2206: 2196: 2183: 2173: 2157: 2141: 2125: 2109: 2100: 1509: 1303: 1212: 1190: 1177: 1161: 1148: 1129: 1116: 1100: 1087: 1069: 1056: 1051: 1025: 1012: 985: 948: 926: 901: 888: 872: 859: 841: 828: 816: 794: 769: 756: 740: 727: 709: 696: 691: 689: 660: 638: 613: 600: 584: 571: 553: 540: 528: 506: 481: 468: 452: 439: 421: 408: 385: 3138: 1241:It would be great if you could edit the 2933: 2892: 2839: 2773: 2633: 2538: 2443: 2362: 2270: 1995:z=(a1+a2*x)(a3+a4*y)unless b1*b4=b2*b3. 1644: 1603: 1550: 1438: 1397: 1344: 154: 115:Knowledge (XXG):WikiProject Mathematics 49: 3598:WikiProject Computer graphics articles 1238:So the minus sign seems correct to me. 378:Are you sure? The formula used to be: 210:Template:WikiProject Computer graphics 1854:Excel Bilinear Interpolation Function 319:Barycentric_coordinates_(mathematics) 7: 1875:xHdrMax Or y < yHdrMin Or y : --> 187:This article is within the scope of 95:This article is within the scope of 3121:while in fact it can also apply to 1044:, the right-hand side evaluates to 1037:{\displaystyle (x,y)=(x_{1},y_{2})} 38:It is of interest to the following 3588:C-Class computer graphics articles 1931:error in linear product expression 229:project-independent quality rating 14: 3578:Low-priority mathematics articles 3359:), and conversely the weight of y 272:intermediates was not explained. 3573:Start-Class mathematics articles 3302:piecewise constant interpolation 3182: 3154: 3145: 1782:The interpolant is the function 180: 156: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 3235:You said you didn't understand 1966:A=(z2-z1)(z4-z1)/(z3+z1-z4-z2) 241:This article has been rated as 135:This article has been rated as 3459:Non-rectilinear quadrilaterals 3271:A problem definition would be: 2971: 2959: 2947: 2935: 2930: 2918: 2909: 2897: 2889: 2877: 2868: 2856: 2853: 2841: 2836: 2824: 2815: 2803: 2767: 2755: 2746: 2734: 2725: 2713: 2704: 2692: 2630: 2618: 2609: 2597: 2535: 2523: 2514: 2502: 2440: 2428: 2163: 2134: 2131: 2102: 2031:b4=z(0,0)-z(1,0)-z(0,1)+z(1,1) 1918:20:23, 19 September 2008 (UTC) 1824:01:13, 25 September 2006 (UTC) 1682: 1670: 1658: 1646: 1641: 1629: 1620: 1608: 1600: 1588: 1579: 1567: 1564: 1552: 1547: 1535: 1526: 1514: 1476: 1464: 1452: 1440: 1435: 1423: 1414: 1402: 1394: 1382: 1373: 1361: 1358: 1346: 1341: 1329: 1320: 1308: 1218: 1205: 1196: 1170: 1167: 1141: 1135: 1109: 1106: 1080: 1075: 1062: 1031: 1005: 999: 987: 954: 935: 932: 913: 907: 881: 878: 852: 847: 834: 822: 803: 800: 781: 775: 749: 746: 720: 715: 702: 666: 647: 644: 625: 619: 593: 590: 564: 559: 546: 534: 515: 512: 493: 487: 461: 458: 432: 427: 414: 402: 390: 333:19:11, 24 September 2006 (UTC) 1: 3427:05:56, 13 November 2013 (UTC) 3392:grid interpolation operations 3177:03:56, 16 December 2011 (UTC) 3046:13:11, 18 November 2010 (UTC) 3029:12:41, 18 November 2010 (UTC) 1245:page along similar lines. -- 201:and see a list of open tasks. 190:WikiProject Computer graphics 109:and see a list of open tasks. 3519:18:18, 6 December 2019 (UTC) 3503:07:26, 1 December 2019 (UTC) 3474:23:39, 9 February 2017 (UTC) 3454:23:32, 27 October 2015 (UTC) 3160:NOT bilinearly interpolated 3135:02:56, 3 December 2011 (UTC) 321:interpolation often used in 310:13:27, 19 January 2006 (UTC) 287:02:24, 19 January 2006 (UTC) 3239:. I suggest you start with 2088:12:32, 16 August 2012 (UTC) 2016:z=(b2*b3/b4)*+(b1-b2*b3/b4) 1812:14:54, 25 August 2006 (UTC) 1777:14:18, 25 August 2006 (UTC) 342:Error in matrix formulation 3614: 3226:Bilinear interpolation is 1957:the correct expression is 1899:22:35, 23 April 2008 (UTC) 360:05:53, 20 April 2006 (UTC) 247:project's importance scale 213:computer graphics articles 3559:20:08, 19 July 2021 (UTC) 3215:16:41, 16 July 2013 (UTC) 3191:Please simplify this page 3072:05:38, 27 June 2015 (UTC) 2061:14:12, 12 July 2010 (UTC) 2043:21:07, 11 July 2010 (UTC) 1987:14:51, 3 April 2010 (UTC) 1952:00:59, 4 April 2010 (UTC) 240: 226: 175: 134: 67: 46: 3316:non-linear interpolation 3107:02:00, 30 May 2011 (UTC) 3092:01:58, 30 May 2011 (UTC) 1724:04:50, 22 May 2006 (UTC) 1285:03:43, 18 May 2006 (UTC) 1270:21:38, 17 May 2006 (UTC) 1255:04:09, 17 May 2006 (UTC) 141:project's priority scale 3540:20:04, 7 May 2020 (UTC) 3479:treatment of edge cases 3363:should be taken as (x-x 3076: 1291:Better form of equation 1243:trilinear interpolation 323:Finite_element_analysis 98:WikiProject Mathematics 3151:Bilinear interpolated 2987: 2779: 2639: 2544: 2449: 2368: 2276: 1698: 1492: 1228: 1038: 964: 673: 369:Error in Big equations 28:This article is rated 3351:should be taken as (x 3036:I agree. Good catch. 2988: 2780: 2640: 2545: 2450: 2369: 2277: 1935:you need 4 constants 1904:Bad Image for Example 1699: 1493: 1229: 1039: 965: 674: 32:on Knowledge (XXG)'s 3434:bilinear vs. bicubic 3309:linear interpolation 3245:linear interpolation 3243:, and continue with 3195:I do not understand 2797: 2650: 2555: 2460: 2386: 2298: 2099: 1508: 1302: 1050: 984: 688: 384: 121:mathematics articles 3355:-x) instead of (x-x 3258:is synonymous with 3141: 3056:Hobby_Coder_Nitpick 1870:xHdrMin And x : --> 3329:weighted averaging 3241:weighted averaging 3139: 3113:Curvilinear grids? 2983: 2934: 2893: 2840: 2775: 2774: 2635: 2634: 2540: 2539: 2445: 2444: 2364: 2363: 2272: 2271: 1694: 1645: 1604: 1551: 1488: 1439: 1398: 1345: 1224: 1034: 960: 669: 90:Mathematics portal 34:content assessment 3505: 3489:comment added by 3417:comment added by 3277:in the same range 3205:comment added by 3164: 3163: 3140:curvilinear grid 3123:curvilinear grids 3032: 3015:comment added by 1840:comment added by 1139: 911: 779: 623: 491: 261: 260: 257: 256: 253: 252: 204:Computer graphics 195:computer graphics 164:Computer graphics 151: 150: 147: 146: 3605: 3511:BernardoSulzbach 3429: 3217: 3186: 3158: 3149: 3142: 3031: 3009: 2992: 2990: 2989: 2984: 2784: 2782: 2781: 2776: 2685: 2684: 2675: 2674: 2662: 2661: 2644: 2642: 2641: 2636: 2590: 2589: 2580: 2579: 2567: 2566: 2549: 2547: 2546: 2541: 2495: 2494: 2485: 2484: 2472: 2471: 2454: 2452: 2451: 2446: 2421: 2420: 2411: 2410: 2398: 2397: 2373: 2371: 2370: 2365: 2355: 2354: 2339: 2338: 2323: 2322: 2310: 2309: 2281: 2279: 2278: 2273: 2263: 2262: 2253: 2252: 2237: 2236: 2227: 2226: 2211: 2210: 2201: 2200: 2188: 2187: 2178: 2177: 2162: 2161: 2146: 2145: 2130: 2129: 2114: 2113: 2028:b3=z(0,1)-z(0,0) 2025:b2=z(1,0)-z(0,0) 1849: 1703: 1701: 1700: 1695: 1497: 1495: 1494: 1489: 1233: 1231: 1230: 1225: 1217: 1216: 1195: 1194: 1182: 1181: 1166: 1165: 1153: 1152: 1140: 1138: 1134: 1133: 1121: 1120: 1105: 1104: 1092: 1091: 1078: 1074: 1073: 1057: 1043: 1041: 1040: 1035: 1030: 1029: 1017: 1016: 969: 967: 966: 961: 953: 952: 931: 930: 912: 910: 906: 905: 893: 892: 877: 876: 864: 863: 850: 846: 845: 829: 821: 820: 799: 798: 780: 778: 774: 773: 761: 760: 745: 744: 732: 731: 718: 714: 713: 697: 692: 678: 676: 675: 670: 665: 664: 643: 642: 624: 622: 618: 617: 605: 604: 589: 588: 576: 575: 562: 558: 557: 541: 533: 532: 511: 510: 492: 490: 486: 485: 473: 472: 457: 456: 444: 443: 430: 426: 425: 409: 215: 214: 211: 208: 205: 184: 177: 176: 171: 168: 160: 153: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3613: 3612: 3608: 3607: 3606: 3604: 3603: 3602: 3563: 3562: 3547: 3527: 3481: 3461: 3436: 3412: 3370: 3367:) instead of (x 3366: 3362: 3358: 3354: 3350: 3346: 3342: 3338: 3334: 3200: 3193: 3115: 3079: 3077:I don't get it. 3038:—Ben FrantzDale 3010: 3006: 2795: 2794: 2676: 2666: 2653: 2648: 2647: 2581: 2571: 2558: 2553: 2552: 2486: 2476: 2463: 2458: 2457: 2412: 2402: 2389: 2384: 2383: 2346: 2330: 2314: 2301: 2296: 2295: 2254: 2244: 2228: 2218: 2202: 2192: 2179: 2169: 2153: 2137: 2121: 2105: 2097: 2096: 1933: 1925: 1906: 1887: 1882: 1856: 1835: 1763: 1506: 1505: 1300: 1299: 1293: 1208: 1186: 1173: 1157: 1144: 1125: 1112: 1096: 1083: 1079: 1065: 1058: 1048: 1047: 1021: 1008: 982: 981: 944: 922: 897: 884: 868: 855: 851: 837: 830: 812: 790: 765: 752: 736: 723: 719: 705: 698: 686: 685: 656: 634: 609: 596: 580: 567: 563: 549: 542: 524: 502: 477: 464: 448: 435: 431: 417: 410: 382: 381: 371: 344: 266: 212: 209: 206: 203: 202: 169: 166: 120: 117: 114: 111: 110: 88: 81: 61: 29: 12: 11: 5: 3611: 3609: 3601: 3600: 3595: 3590: 3585: 3580: 3575: 3565: 3564: 3546: 3543: 3526: 3523: 3522: 3521: 3480: 3477: 3460: 3457: 3435: 3432: 3431: 3430: 3407: 3406: 3401: 3400: 3396: 3395: 3382: 3381: 3376: 3375: 3368: 3364: 3360: 3356: 3352: 3348: 3344: 3340: 3336: 3332: 3324: 3323: 3312: 3305: 3298: 3295: 3292: 3289: 3285: 3284: 3283: 3282: 3279: 3267: 3266: 3252: 3251: 3232: 3231: 3223: 3222: 3192: 3189: 3188: 3187: 3162: 3161: 3159: 3152: 3150: 3114: 3111: 3110: 3109: 3099:61.194.119.130 3084:61.194.119.130 3078: 3075: 3049: 3048: 3005: 3002: 3001: 3000: 2996: 2995: 2994: 2993: 2982: 2979: 2976: 2973: 2970: 2967: 2964: 2961: 2958: 2955: 2952: 2949: 2946: 2943: 2940: 2937: 2932: 2929: 2926: 2923: 2920: 2917: 2914: 2911: 2908: 2905: 2902: 2899: 2896: 2891: 2888: 2885: 2882: 2879: 2876: 2873: 2870: 2867: 2864: 2861: 2858: 2855: 2852: 2849: 2846: 2843: 2838: 2835: 2832: 2829: 2826: 2823: 2820: 2817: 2814: 2811: 2808: 2805: 2802: 2788: 2787: 2786: 2785: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2751: 2748: 2745: 2742: 2739: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2697: 2694: 2691: 2688: 2683: 2679: 2673: 2669: 2665: 2660: 2656: 2645: 2632: 2629: 2626: 2623: 2620: 2617: 2614: 2611: 2608: 2605: 2602: 2599: 2596: 2593: 2588: 2584: 2578: 2574: 2570: 2565: 2561: 2550: 2537: 2534: 2531: 2528: 2525: 2522: 2519: 2516: 2513: 2510: 2507: 2504: 2501: 2498: 2493: 2489: 2483: 2479: 2475: 2470: 2466: 2455: 2442: 2439: 2436: 2433: 2430: 2427: 2424: 2419: 2415: 2409: 2405: 2401: 2396: 2392: 2377: 2376: 2375: 2374: 2361: 2358: 2353: 2349: 2345: 2342: 2337: 2333: 2329: 2326: 2321: 2317: 2313: 2308: 2304: 2290: 2289: 2285: 2284: 2283: 2282: 2269: 2266: 2261: 2257: 2251: 2247: 2243: 2240: 2235: 2231: 2225: 2221: 2217: 2214: 2209: 2205: 2199: 2195: 2191: 2186: 2182: 2176: 2172: 2168: 2165: 2160: 2156: 2152: 2149: 2144: 2140: 2136: 2133: 2128: 2124: 2120: 2117: 2112: 2108: 2104: 2068: 2066: 2065: 2064: 2063: 2046: 2045: 2032: 2029: 2026: 2023: 2020: 2017: 2014: 2011: 2008: 2005: 2002: 1999: 1996: 1993: 1955: 1954: 1932: 1929: 1924: 1921: 1905: 1902: 1886: 1883: 1861: 1855: 1852: 1851: 1850: 1842:137.189.90.241 1827: 1826: 1815: 1814: 1799: 1762: 1759: 1758: 1757: 1756: 1755: 1754: 1753: 1752:) add up to 1. 1729: 1728: 1727: 1726: 1705: 1704: 1693: 1690: 1687: 1684: 1681: 1678: 1675: 1672: 1669: 1666: 1663: 1660: 1657: 1654: 1651: 1648: 1643: 1640: 1637: 1634: 1631: 1628: 1625: 1622: 1619: 1616: 1613: 1610: 1607: 1602: 1599: 1596: 1593: 1590: 1587: 1584: 1581: 1578: 1575: 1572: 1569: 1566: 1563: 1560: 1557: 1554: 1549: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1522: 1519: 1516: 1513: 1499: 1498: 1487: 1484: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1451: 1448: 1445: 1442: 1437: 1434: 1431: 1428: 1425: 1422: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1396: 1393: 1390: 1387: 1384: 1381: 1378: 1375: 1372: 1369: 1366: 1363: 1360: 1357: 1354: 1351: 1348: 1343: 1340: 1337: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1313: 1310: 1307: 1292: 1289: 1288: 1287: 1267:safetycritical 1260: 1258: 1257: 1239: 1236: 1235: 1234: 1223: 1220: 1215: 1211: 1207: 1204: 1201: 1198: 1193: 1189: 1185: 1180: 1176: 1172: 1169: 1164: 1160: 1156: 1151: 1147: 1143: 1137: 1132: 1128: 1124: 1119: 1115: 1111: 1108: 1103: 1099: 1095: 1090: 1086: 1082: 1077: 1072: 1068: 1064: 1061: 1055: 1033: 1028: 1024: 1020: 1015: 1011: 1007: 1004: 1001: 998: 995: 992: 989: 978: 977: 976: 975: 974: 973: 972: 971: 970: 959: 956: 951: 947: 943: 940: 937: 934: 929: 925: 921: 918: 915: 909: 904: 900: 896: 891: 887: 883: 880: 875: 871: 867: 862: 858: 854: 849: 844: 840: 836: 833: 827: 824: 819: 815: 811: 808: 805: 802: 797: 793: 789: 786: 783: 777: 772: 768: 764: 759: 755: 751: 748: 743: 739: 735: 730: 726: 722: 717: 712: 708: 704: 701: 695: 668: 663: 659: 655: 652: 649: 646: 641: 637: 633: 630: 627: 621: 616: 612: 608: 603: 599: 595: 592: 587: 583: 579: 574: 570: 566: 561: 556: 552: 548: 545: 539: 536: 531: 527: 523: 520: 517: 514: 509: 505: 501: 498: 495: 489: 484: 480: 476: 471: 467: 463: 460: 455: 451: 447: 442: 438: 434: 429: 424: 420: 416: 413: 407: 404: 401: 398: 395: 392: 389: 370: 367: 365: 363: 362: 343: 340: 338: 336: 335: 326: 313: 312: 298: 294: 265: 262: 259: 258: 255: 254: 251: 250: 243:Mid-importance 239: 233: 232: 225: 219: 218: 216: 199:the discussion 185: 173: 172: 170:Mid‑importance 161: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3610: 3599: 3596: 3594: 3591: 3589: 3586: 3584: 3581: 3579: 3576: 3574: 3571: 3570: 3568: 3561: 3560: 3556: 3552: 3545:Formula error 3544: 3542: 3541: 3537: 3533: 3524: 3520: 3516: 3512: 3508: 3507: 3506: 3504: 3500: 3496: 3492: 3488: 3478: 3476: 3475: 3471: 3467: 3458: 3456: 3455: 3451: 3447: 3443: 3442: 3433: 3428: 3424: 3420: 3416: 3409: 3408: 3403: 3402: 3398: 3397: 3393: 3389: 3384: 3383: 3378: 3377: 3374: 3330: 3326: 3325: 3321: 3317: 3313: 3310: 3306: 3303: 3299: 3296: 3293: 3290: 3287: 3286: 3280: 3278: 3274: 3273: 3272: 3269: 3268: 3265: 3264:approximation 3261: 3257: 3256:Interpolation 3254: 3253: 3250: 3249:interpolation 3246: 3242: 3238: 3234: 3233: 3229: 3225: 3224: 3220: 3219: 3218: 3216: 3212: 3208: 3207:86.133.88.157 3204: 3198: 3190: 3185: 3181: 3180: 3179: 3178: 3174: 3170: 3157: 3153: 3148: 3144: 3143: 3137: 3136: 3132: 3128: 3124: 3120: 3119:regular grids 3112: 3108: 3104: 3100: 3096: 3095: 3094: 3093: 3089: 3085: 3074: 3073: 3069: 3065: 3061: 3057: 3052: 3047: 3043: 3039: 3035: 3034: 3033: 3030: 3026: 3022: 3018: 3014: 3003: 2998: 2997: 2980: 2977: 2974: 2968: 2965: 2962: 2956: 2953: 2950: 2944: 2941: 2938: 2927: 2924: 2921: 2915: 2912: 2906: 2903: 2900: 2894: 2886: 2883: 2880: 2874: 2871: 2865: 2862: 2859: 2850: 2847: 2844: 2833: 2830: 2827: 2821: 2818: 2812: 2809: 2806: 2800: 2793: 2792: 2790: 2789: 2770: 2764: 2761: 2758: 2752: 2749: 2743: 2740: 2737: 2731: 2728: 2722: 2719: 2716: 2710: 2707: 2701: 2698: 2695: 2689: 2686: 2681: 2677: 2671: 2667: 2663: 2658: 2654: 2646: 2627: 2624: 2621: 2615: 2612: 2606: 2603: 2600: 2594: 2591: 2586: 2582: 2576: 2572: 2568: 2563: 2559: 2551: 2532: 2529: 2526: 2520: 2517: 2511: 2508: 2505: 2499: 2496: 2491: 2487: 2481: 2477: 2473: 2468: 2464: 2456: 2437: 2434: 2431: 2425: 2422: 2417: 2413: 2407: 2403: 2399: 2394: 2390: 2382: 2381: 2379: 2378: 2359: 2356: 2351: 2347: 2343: 2340: 2335: 2331: 2327: 2324: 2319: 2315: 2311: 2306: 2302: 2294: 2293: 2292: 2291: 2287: 2286: 2267: 2264: 2259: 2255: 2249: 2245: 2241: 2238: 2233: 2229: 2223: 2219: 2215: 2212: 2207: 2203: 2197: 2193: 2189: 2184: 2180: 2174: 2170: 2166: 2158: 2154: 2150: 2147: 2142: 2138: 2126: 2122: 2118: 2115: 2110: 2106: 2095: 2094: 2092: 2091: 2090: 2089: 2085: 2081: 2076: 2073: 2070: 2062: 2058: 2054: 2050: 2049: 2048: 2047: 2044: 2040: 2036: 2033: 2030: 2027: 2024: 2021: 2018: 2015: 2012: 2010:D=b1-b2*b3/b4 2009: 2006: 2003: 2000: 1997: 1994: 1991: 1990: 1989: 1988: 1984: 1980: 1976: 1973: 1970: 1967: 1964: 1961: 1958: 1953: 1949: 1945: 1941: 1940: 1939: 1936: 1930: 1928: 1922: 1920: 1919: 1915: 1911: 1903: 1901: 1900: 1896: 1892: 1884: 1865: 1860: 1853: 1847: 1843: 1839: 1833: 1832:Bilinear form 1829: 1828: 1825: 1822: 1817: 1816: 1813: 1809: 1805: 1800: 1797: 1793: 1789: 1785: 1781: 1780: 1779: 1778: 1775: 1769: 1766: 1760: 1751: 1747: 1743: 1739: 1735: 1734: 1733: 1732: 1731: 1730: 1725: 1721: 1717: 1713: 1709: 1708: 1707: 1706: 1691: 1688: 1685: 1679: 1676: 1673: 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757: 753: 741: 737: 733: 728: 724: 710: 706: 699: 693: 684: 683: 682: 681: 680: 679: 661: 657: 653: 650: 639: 635: 631: 628: 614: 610: 606: 601: 597: 585: 581: 577: 572: 568: 554: 550: 543: 537: 529: 525: 521: 518: 507: 503: 499: 496: 482: 478: 474: 469: 465: 453: 449: 445: 440: 436: 422: 418: 411: 405: 399: 396: 393: 387: 380: 379: 377: 376: 375: 368: 366: 361: 357: 353: 349: 348: 347: 341: 339: 334: 331: 327: 324: 320: 315: 314: 311: 307: 303: 299: 295: 291: 290: 289: 288: 285: 280: 277: 273: 269: 263: 248: 244: 238: 235: 234: 230: 224: 221: 220: 217: 200: 196: 192: 191: 186: 183: 179: 178: 174: 165: 162: 159: 155: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3548: 3532:1.36.219.240 3528: 3485:— Preceding 3482: 3462: 3439: 3437: 3419:85.110.50.54 3413:— Preceding 3391: 3387: 3372: 3328: 3315: 3308: 3301: 3276: 3270: 3263: 3259: 3255: 3236: 3227: 3201:— Preceding 3196: 3194: 3165: 3116: 3080: 3059: 3055: 3053: 3050: 3007: 2077: 2074: 2071: 2067: 1977: 1974: 1972:C=(z4-z1)/A 1971: 1969:B=(z2-z1)/A 1968: 1965: 1962: 1959: 1956: 1937: 1934: 1926: 1907: 1888: 1885:No citations 1857: 1804:Jitse Niesen 1795: 1791: 1790:, which are 1787: 1783: 1770: 1767: 1765:Hi there... 1764: 1749: 1745: 1741: 1737: 1716:Jitse Niesen 1711: 1500: 1294: 1277:Jitse Niesen 1263: 1261:(continued) 1259: 1247:Jitse Niesen 980:Substituing 372: 364: 352:Jitse Niesen 345: 337: 302:Jitse Niesen 281: 278: 274: 270: 267: 242: 188: 137:Low-priority 136: 96: 62:Low‑priority 40:WikiProjects 3322:two or more 3011:—Preceding 1836:—Preceding 1761:Linearities 1740:) and (1 − 112:Mathematics 103:mathematics 59:Mathematics 30:Start-class 3567:Categories 3466:Cesiumfrog 3260:estimation 3017:Skogkatten 2001:A=b2*b3/b4 1910:daviddoria 282:thoughts? 2022:b1=z(0,0) 297:analysis. 3499:contribs 3487:unsigned 3415:unsigned 3237:anything 3203:unsigned 3197:anything 3169:Tom Ruen 3127:Tom Ruen 3025:contribs 3013:unsigned 1838:unsigned 293:H-shape. 3551:Shg4421 3491:Norlesh 3064:MvGulik 2380:where, 2035:Jdf6042 2007:C=b4/b2 2004:B=b4/b3 1979:Jdf6042 1975:D=z1-A 1774:NIC1138 284:TomViza 264:Rewrite 245:on the 167:C‑class 139:on the 3446:Foogus 3405:grids. 3320:degree 3054:Minor( 2051:So? -- 1923:Revert 1748:) and 36:scale. 3380:line. 3343:and y 3335:and y 2080:Lzn11 2013:thus, 1891:Ms331 1821:Drf5n 1501:with 330:Drf5n 3555:talk 3536:talk 3515:talk 3495:talk 3470:talk 3450:talk 3423:talk 3388:only 3262:and 3247:and 3211:talk 3173:talk 3131:talk 3103:talk 3088:talk 3068:talk 3062:) -- 3042:talk 3021:talk 2084:talk 2057:talk 2039:talk 2019:with 1983:talk 1948:talk 1914:talk 1895:talk 1846:talk 1808:talk 1794:and 1720:talk 1281:talk 1251:talk 356:talk 306:talk 3314:In 3307:In 3300:In 3228:not 2053:Pot 1944:Pot 237:Mid 131:Low 3569:: 3557:) 3538:) 3517:) 3501:) 3497:• 3472:) 3452:) 3425:) 3213:) 3175:) 3133:) 3105:) 3090:) 3082:-- 3070:) 3044:) 3027:) 3023:• 2942:− 2904:− 2863:− 2848:− 2819:≈ 2729:− 2708:− 2613:− 2518:− 2086:) 2078:-- 2059:) 2041:) 1985:) 1950:) 1916:) 1897:) 1848:) 1810:) 1722:) 1653:− 1615:− 1574:− 1559:− 1530:≈ 1447:− 1418:− 1409:− 1377:− 1368:− 1353:− 1324:≈ 1283:) 1253:) 1214:12 1184:− 1155:− 1123:− 1094:− 1071:12 1054:− 942:− 920:− 895:− 866:− 843:22 810:− 788:− 763:− 734:− 711:12 694:− 654:− 632:− 607:− 578:− 555:21 538:− 522:− 500:− 475:− 446:− 423:11 406:≈ 358:) 308:) 3553:( 3534:( 3513:( 3493:( 3468:( 3448:( 3421:( 3394:. 3369:1 3365:0 3361:1 3357:0 3353:1 3349:0 3345:1 3341:0 3337:1 3333:0 3209:( 3171:( 3129:( 3101:( 3086:( 3066:( 3040:( 3019:( 2981:. 2978:y 2975:x 2972:) 2969:1 2966:, 2963:1 2960:( 2957:f 2954:+ 2951:y 2948:) 2945:x 2939:1 2936:( 2931:) 2928:1 2925:, 2922:0 2919:( 2916:f 2913:+ 2910:) 2907:y 2901:1 2898:( 2895:x 2890:) 2887:0 2884:, 2881:1 2878:( 2875:f 2872:+ 2869:) 2866:y 2860:1 2857:( 2854:) 2851:x 2845:1 2842:( 2837:) 2834:0 2831:, 2828:0 2825:( 2822:f 2816:) 2813:y 2810:, 2807:x 2804:( 2801:f 2771:. 2768:) 2765:1 2762:, 2759:1 2756:( 2753:f 2750:+ 2747:) 2744:1 2741:, 2738:0 2735:( 2732:f 2726:) 2723:0 2720:, 2717:1 2714:( 2711:f 2705:) 2702:0 2699:, 2696:0 2693:( 2690:f 2687:= 2682:3 2678:a 2672:1 2668:a 2664:= 2659:4 2655:b 2631:) 2628:0 2625:, 2622:0 2619:( 2616:f 2610:) 2607:1 2604:, 2601:0 2598:( 2595:f 2592:= 2587:3 2583:a 2577:2 2573:a 2569:= 2564:3 2560:b 2536:) 2533:0 2530:, 2527:0 2524:( 2521:f 2515:) 2512:0 2509:, 2506:1 2503:( 2500:f 2497:= 2492:4 2488:a 2482:1 2478:a 2474:= 2469:2 2465:b 2441:) 2438:0 2435:, 2432:0 2429:( 2426:f 2423:= 2418:4 2414:a 2408:2 2404:a 2400:= 2395:1 2391:b 2360:y 2357:x 2352:4 2348:b 2344:+ 2341:y 2336:3 2332:b 2328:+ 2325:x 2320:2 2316:b 2312:+ 2307:1 2303:b 2268:y 2265:x 2260:3 2256:a 2250:1 2246:a 2242:+ 2239:y 2234:3 2230:a 2224:2 2220:a 2216:+ 2213:x 2208:4 2204:a 2198:1 2194:a 2190:+ 2185:4 2181:a 2175:2 2171:a 2167:= 2164:) 2159:4 2155:a 2151:+ 2148:y 2143:3 2139:a 2135:( 2132:) 2127:2 2123:a 2119:+ 2116:x 2111:1 2107:a 2103:( 2082:( 2055:( 2037:( 1981:( 1946:( 1912:( 1893:( 1844:( 1806:( 1798:. 1796:y 1792:x 1788:f 1784:f 1750:x 1746:x 1742:y 1738:x 1718:( 1712:x 1692:. 1689:y 1686:x 1683:) 1680:1 1677:, 1674:1 1671:( 1668:f 1665:+ 1662:y 1659:) 1656:x 1650:1 1647:( 1642:) 1639:1 1636:, 1633:0 1630:( 1627:f 1624:+ 1621:) 1618:y 1612:1 1609:( 1606:x 1601:) 1598:0 1595:, 1592:1 1589:( 1586:f 1583:+ 1580:) 1577:y 1571:1 1568:( 1565:) 1562:x 1556:1 1553:( 1548:) 1545:0 1542:, 1539:0 1536:( 1533:f 1527:) 1524:y 1521:, 1518:x 1515:( 1512:f 1486:. 1483:y 1480:x 1477:) 1474:1 1471:, 1468:1 1465:( 1462:f 1459:+ 1456:y 1453:) 1450:1 1444:x 1441:( 1436:) 1433:1 1430:, 1427:0 1424:( 1421:f 1415:) 1412:1 1406:y 1403:( 1400:x 1395:) 1392:0 1389:, 1386:1 1383:( 1380:f 1374:) 1371:1 1365:y 1362:( 1359:) 1356:1 1350:x 1347:( 1342:) 1339:0 1336:, 1333:0 1330:( 1327:f 1321:) 1318:y 1315:, 1312:x 1309:( 1306:f 1279:( 1249:( 1222:. 1219:) 1210:Q 1206:( 1203:f 1200:= 1197:) 1192:1 1188:y 1179:2 1175:y 1171:( 1168:) 1163:2 1159:x 1150:1 1146:x 1142:( 1136:) 1131:2 1127:y 1118:1 1114:y 1110:( 1107:) 1102:2 1098:x 1089:1 1085:x 1081:( 1076:) 1067:Q 1063:( 1060:f 1032:) 1027:2 1023:y 1019:, 1014:1 1010:x 1006:( 1003:= 1000:) 997:y 994:, 991:x 988:( 958:. 955:) 950:1 946:y 939:y 936:( 933:) 928:1 924:x 917:x 914:( 908:) 903:2 899:y 890:1 886:y 882:( 879:) 874:2 870:x 861:1 857:x 853:( 848:) 839:Q 835:( 832:f 826:+ 823:) 818:1 814:y 807:y 804:( 801:) 796:2 792:x 785:x 782:( 776:) 771:2 767:y 758:1 754:y 750:( 747:) 742:2 738:x 729:1 725:x 721:( 716:) 707:Q 703:( 700:f 667:) 662:2 658:y 651:y 648:( 645:) 640:1 636:x 629:x 626:( 620:) 615:2 611:y 602:1 598:y 594:( 591:) 586:2 582:x 573:1 569:x 565:( 560:) 551:Q 547:( 544:f 535:) 530:2 526:y 519:y 516:( 513:) 508:2 504:x 497:x 494:( 488:) 483:2 479:y 470:1 466:y 462:( 459:) 454:2 450:x 441:1 437:x 433:( 428:) 419:Q 415:( 412:f 403:) 400:y 397:, 394:x 391:( 388:f 354:( 325:. 304:( 249:. 223:C 143:. 42::

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