Apply non-linear transformation to graphics object

Question

I would like to define some simple geometric primitives (e.g. points and lines) and see what happens when I transform them by a non-linear transformation (in my case $(x,y)\mapsto (\sqrt x,\sqrt y)$). The result should be an image showing $\{(\sqrt x, \sqrt y)|(x,y)\in X\}$, where $X$ is some set I defined before, like union of points and lines.

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I tried playing with TransformedRegion. I can apply it to a single line, but for example {Line[p1,p2],Line[p2,p3]} is not a correctly specified region. Also, putting TransformedRegion[Line[p1,p2],F] (for some defined function F) to Graphics does not work, since "TransformedRegion is not a Graphics primitive or directive".

Answer 1

There are a couple things to discuss here.

TransformedRegion output

The output of TransformedRegion is a region object. In general, to view a region object, you need to use Region or DiscretizeRegion on it. The exception is if TransformedRegion returns a primitive region, e.g., Point, Line, etc., in which case you can just use Graphics to view it. For example, here is a trivial transformation where TransformedRegion is able to return a primitive region:

reg = Line[{{1, 0}, {5, 1}}];
new = TransformedRegion[reg, RotationTransform[Pi/4]]

Line[{{{1/Sqrt[2], 1/Sqrt[2]}, {2 Sqrt[2], 3 Sqrt[2]}}}]

You can use Graphics to view this primitive:

Graphics[new]

Here is your square root transformation, where TransformedRegion is unable to create a primitive region:

new = TransformedRegion[reg, Sqrt]

TransformedRegion[Line[{{1, 0}, {5, 1}}], Sqrt]

For this kind of output, you need to use Region or DiscretizeRegion:

Region[new]

TransformedRegion input

The input to TransformedRegion should be a region object. This is why Line[p1, p2] works and {Line[p1, p2], Line[p2, p3]} does not work. If you want to transform the union of the two lines, use RegionUnion:

reg = RegionUnion[
    Line[{{1, 0}, {3, 1}}],
    Line[{{0, 2}, {3, 0}}]
];
new = TransformedRegion[reg, Sqrt];
Region[new, PlotRange -> {{0,Sqrt[3]},{0,Sqrt[2]}}]