TITLE: Seeking formulas for mappings that distort one or more of area, shape, and distance
QUESTION [7 upvotes]: I'm wanting to create a simple geometry lesson for kids on how different projections distort area, shape and distance. I'd like examples of formulas that map points in the plane to other points in the plane that distort one or more of the properties. I realize that where the center of projection is matters.
I'm particularly interested in examples of projections that leave one property intact while distorting the others.

REPLY [9 votes]: Some simple non-linear area-preserving maps from $\mathbb R^2$ to $\mathbb R^2$ include the non-linear shear map
$$(x,y) \mapsto (x,\; y+f(x))$$

and its polar coordinate version
$$(r,\theta) \mapsto (r,\; \theta+f(r))$$

for any arbitrary function $f: \mathbb R \to \mathbb R$.  Of course, you can also combine these in various ways to create very complicated area-preserving maps.

(Illustrations based on this public-domain image by Scott Bauer / USDA, created using GIMP.)