Seeking formulas for mappings that distort one or more of area, shape, and distance 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.
 A: 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.)
A: If all distances are preserved, area will also be preserved (isometries have determinant $+1$ or $-1$, and therefore, preserve area).  However, $(x,y)\mapsto(2x,y/2)$ preserves area, but alters most distances.
If, as it has been commented, by shape you mean "the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes," then anything that preserves all distances, will also preserve shape. Again, $(x,y)\mapsto(2x,y/2)$ preserves area and alters shape. Furthermore, $(x,y)\mapsto(2x,2y)$ preserves shape, but alters distance and area.
Maps that preserve distance are called isometries. Maps that preserve shape are constant multiples of isometries. Maps that preserve areas have Jacobian Determinant equal to $+1$ or $-1$ everywhere.
Non-Trivial Example of an Area-Preserving Map
The Lambert Cylindrical Projection from the sphere to the cylinder tangent to the equator of the sphere is an area-preserving map.  The cylinder can then be isometrically unrolled to a rectangle in the plane.  The composition of these maps is an area-preserving map from the sphere to a rectangle:
$$
(x,y,z) \mapsto (\tan^{-1}(x/z),y)\tag{1}
$$
The inverse of map $(1)$ takes a rectangle to the sphere:
$$
(x,y) \mapsto (\sqrt{1-y^2}\;\sin(x),y,\sqrt{1-y^2}\;\cos(x))\tag{2}
$$
To get a non-trivial area-preserving map from the plane to the plane, we will map the plane to the sphere using $(2)$, rotate the sphere $\pi/2$ radians around the $z$-axis, then map back to the plane using $(1)$, and finally rotate the plane to return the original orientation at the origin. Here is the the resulting area-preserving map:
$$
(x,y)\mapsto\left(\sqrt{1-y^2}\;\sin(x),\tan^{-1}\left(\frac{y}{\sqrt{1-y^2}}\sec(x)\right)\right)
$$
Other rotations of the sphere yield other area-preserving maps.
