Geometry
Friday, September 10th, 2004What exactly is geometry? I have yet to find a satisfactory definition. All the ones I’ve seen say it is something like the study of invariants under certain groups of transformations. Like Euclidean geometry is the study of invariants under the transformations of rigid motions. That I understand, but what about differential geometry, for instance? In that case, what would the relevant invariants and transformations be? Maybe arc length, curvature, torsion? And the things I don’t know anything about yet, like the various surface curvatures (intrinsic and extrinsic)? It isn’t really so hard to imagine the invariants, but what is the transformation group?
Anyhow, here’s a question I just came up with: Usually, we view rigid motions as reflections, translations, and rotations in Euclidean spaces. But, we can generalize the notion, by saying a map between two homeomorphic metric spaces is a rigid motion if it preserves distances. The question is simply to show that, when applied to Euclidean spaces, this definition of a rigid motion is equivalent to the standard one.
where 
and 
are smooth functions and 
is a simple closed curve.
. Namely, let 
. Then let 
and 
. Then by Green’s theorem, 
, so 
, where 
is the period of 
.
, this gives the simple expression 
!
and find an expression for 
in terms of 
, 
. Solve this for an expression for 
as 
to 
: 
. Then notice that the area of the ellipse is twice that of 
, or 
. Recall that the antiderivative of 
is 
— or more likely, give up. Then plug and chug: ![A = b \left[x \sqrt{ 1 - \left(\frac{x}{a}\right)^2} + a\arcsin\left(\frac{x}{a}\right)\right]_{-a}^a = ab \left( \arcsin(1) - \arcsin(-1) \right) = \pi ab](/cz/latexrender/pictures/decad88576e119c0ac66307aa091bae8.png "A = b \left[x \sqrt{ 1 - \left(\frac{x}{a}\right)^2} + a\arcsin\left(\frac{x}{a}\right)\right]_{-a}^a = ab \left( \arcsin(1) - \arcsin(-1) \right) = \pi ab")
