I have a set of points P that are evenly spaced on a square grid of length 1.
I would like to map these points to the circle of radius 1 as nicely as possible, while preserving relative positions as much as possible. That is, the points in the square grid should map to the points in the unit circle.
How can this be done in Mathematica?
Your question is a little vague, but one possible mapping from a square to a circle is the following:
$$\begin{bmatrix}x^\prime\\y^\prime\end{bmatrix}=\begin{bmatrix}x\sqrt{1-\frac{y^2}{2}}\\y\sqrt{1-\frac{x^2}{2}}\end{bmatrix}$$
This transformation wrapped into a Mathematica function:
transformation[points_] := {#[[1]] Sqrt[1 - (#[[2]]^2/2)],
#[[2]] Sqrt[1 - (#[[1]]^2/2)]} & /@ # & /@ points;
And a little demonstration:
points = Table[{a, b}, {a, -1, 1, 0.2}, {b, -1, 1, 0.2}];
Show[
Graphics[{EdgeForm[Thick], White, Rectangle[{-1, -1}, {1, 1}]}],
ListPlot[Flatten[points, 1], AspectRatio -> 1, PlotStyle -> Red]
]
Show[
Graphics[{Thick, Circle[]}],
ListPlot[Flatten[transformation[points], 1], AspectRatio -> 1, PlotStyle -> Red]
]
Show and ListPlot. Suggest Graphics[{EdgeForm[Thick], FaceForm[None], Rectangle[{-1, -1}, {1, 1}], Red, PointSize[Medium], Point[Flatten[points, 1]]}] and Graphics[{Thick, Circle[], Red, PointSize[Medium], Point[Flatten[transformation@points, 1]]}]
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Commented
Aug 30, 2014 at 5:35
I've used the Schwarz-Christoffel conformal map from the square to the disk plenty of times on this site (search around!); here's how to use it for this question:
pts = Flatten[Table[{x, y}, {x, -1, 1, 1/5}, {y, 1, -1, -1/5}], 1];
{Graphics[{Directive[AbsolutePointSize[3], ColorData[61, 8]], Point[pts]},
Frame -> True, PlotLabel -> "Before"],
Graphics[{Directive[AbsolutePointSize[3], ColorData[61, 8]],
Point[pts /. {x_, y_} :>
With[{ω = N[3 Beta[5/4, 5/4]/2]},
Through[{Re, Im}[JacobiSN[ω (x + I y), 1/2]
JacobiDC[ω (x + I y), 1/2]]]]]},
Frame -> True, PlotLabel -> "After"]} // GraphicsRow
CircleorDisk? $\endgroup$