Associated curves

Given a path $\gamma(t)$ in the plane, we can construct many
-associated curves-.

More generally given a parametrized manifold, we can construct similar
things.

Note that some of these are -intrinsic-, IE, not depending on the
parametrization.

Often (for curves at least) we take unit speed parametrizations.

That is, given a curve, we also have its
-tangent- and -normal- at a point,
so we get $(t,k) \mapsto \gamma(t) + k\dot \gamma(t)$
...and likewise for normal
(instead of adding, in general we need to take the exponential map)

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parallel curve:
shift by fixed normal amount.

Evolute:
- centers of curvature
"envelope of normals"?
  (these are unique)
If $C'$ is the evolute of $C$, then we say $C$ is -an-
involute of $C'$ -- but so is every parallel curve

Inverse curve (w/r/t a circle)
- recall inverse w/r/t a circle is a mobius transform,
aka fractional linear,
IE, element of $\PSL_2 \bC$, aka, linear transform of $\CP^1$
(hence this preserves algebraicness)

caustics:
if light comes from a point (including infinity),
then can reflect (catacaustic) or refract (diacaustic)
and get the points where lines intersect.

EG: for parabola (headlights); for circle (silly)
  for hyperbola (backwards...: defracts nicely
  from center, right?)

(pedal curves: don't really get)

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There's also the -inverse- problem:
when is a system "holonomic".
EG, when is a set of lines/planes the tangent lines (normal lines)
to an immersed curve/surface/manifold.
I think this is AKA integrable.

Hmm...similarly, can ask if a -vector field- is the derivatives of a
function, AKA, is it -integrable-?
