We can measure the distance between Los Angeles and New York.  We can measure the distance between species of plants and animals.  So why not measure the distance between a triangle and a square?

Of course, this comes down to what we chose as our definition of distance.

When we think of each shape as a single point in some space – that space is a Shape Space!  We can endow shape spaces with different topological structures and metrics. The result is an  infinite dimensional geometry.  Though often amusingly lacking, these can be fruitful for studying well-posedness of PDEs and other cool things like information geometry.

A few metrics that we introduce on the shape space of all unit-length unit-speed curves with a single fixed endpoint include:

1.  $\langle{\langle{u,v}\rangle}\rangle = \int_0^1 \langle{u(s),v(s)}\rangle ds + \langle{u(1),v(1)}\rangle$
2. $\langle\langle{u,v}\rangle\rangle = \int_0^1 \lambda(s)\langle{u(s),v(s)}\rangle ds$
3. $\langle\langle{u,v}\rangle\rangle = \int_0^1\langle{u(s), v(s)}\rangle ds + \sum_{j=1}^km_j\langle{u(s_j),v(s_j)}\rangle$

Here $u,v$ are vectors at the point $\gamma$. But $\gamma$ is really a curve so $u,v$ are really vector fields along $\gamma$!

The metrics above attempt to model the motion of a whip with arbitrary weights attached. We were able to prove some theoretical results.  Along the way, we made some pretty great videos too! (See below.)

Closing the loop, we get shapes rather than strings. We hope that these designated weights can aid in preserving and creating geometric features useful in animation, shape recognition, and image processing.

Collaborators: Steve Preston and Anna Broido