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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.

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area and distance are pretty well defined terms; what exactly do you mean by shape? – robjohn Aug 26 '11 at 21:38
I like this definition from Wikipedia ( "the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. – Sol Aug 26 '11 at 21:42
A world atlas (in paper, physical, book format!) often has pictures of map projections as preliminary material, including discussion of what is preserved and what is distorted in each particular projection. – zyx Aug 26 '11 at 21:44
World maps are projections of a sphere onto (typically) a plane. I'm looking to simplify the lesson by starting with a plane-to-plane projection. – Sol Aug 26 '11 at 21:48
You might also want to consider maps that preserve angles, i.e. conformal maps. One can think of them as preserving shape locally but not globally. – Rahul Aug 27 '11 at 16:33
up vote 7 down vote accepted

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.

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Something I'm not clear about. Does (x,y)↦(2x,2y) preserve area or not? Yes, I understand that the area of all objects in the mapping are multiplied by 4 from their premapped state but the areas of the mapped objects don't change relative to one another. – Sol Aug 26 '11 at 22:05
Any isometry on the plane is either a rotation or the composition of a rotation and a reflection, yes? – Rahul Aug 26 '11 at 22:19
@Sol: $(x,y)\mapsto(2x,2y)$ alters both distance and area, but not relative distance or relative area. – robjohn Aug 26 '11 at 22:42
@Rahul: yes. To preserve distance, the Jacobian of the mapping, $J$ must satisfy $J^TJ=I$ at every point. To preserve area, $\det(J^TJ)=1$ at every point. To preserve shape, $J^TJ=kI$ for some constant $k>0$ at every point. – robjohn Aug 26 '11 at 22:53
@robjohn I've marked the question as answered although I still wonder if there are interesting plane to plane projections one create that are not linear transformations but are simple enough for students to grok. For example, I coded in Mathematica a projection of a circle onto a square. That was interesting. – Sol Aug 27 '11 at 0:58

Some simple non-linear area-preserving maps from $\mathbb R^2$ to $\mathbb R^2$ include the non-linear shear map

$$(x,y) \mapsto (x,\; y+f(x))$$ Example of non-linear shear

and its polar coordinate version

$$(r,\theta) \mapsto (r,\; \theta+f(r))$$ Example of non-linear polar shear

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.

enter image description here

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

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Very nice. These are quite simple to state, and the Jacobian of these maps is fairly easy to compute (when converted to polar form for the second map). – robjohn Aug 27 '11 at 14:55
Totally cool!!! (I had to put 2 extra exclamation points in order to reach the minimum number of characters.) – Mike Jones Aug 27 '11 at 22:40
Shearing is an excellent example of a mapping that can get very complex and still preserve area. Thanks! – Sol Aug 28 '11 at 14:18

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