1. Geometry of Shape Manifolds
Constraints define a manifold embedded in q0 + L2
Move along manifold by moving in tangent space and projecting back to manifold
Tangent space is infinite dimensional, but normal space is characterized by three constraints defined in f1

2. Tangents and Normals
The derivative of f1 in the direction of f at  is:
Implies df1 is surjective
If f is orthogonal to {1, sin q, cos q}, then df1=0 in the direction of f and hence f is in the tangent space
Surjective because essentially computing Fourier coeffs

3. Projections
Want to find the closest element in C1 to an arbitrary q  q0 + L2
Basic idea: move orthogonal to level sets so projections under f form a straight line in R3
For a point b  R3, we define the level set as:
Let b1=(p,0,0). Then its level set is the preshape space C1

4. Approximate Projections
If points are close to C1, then one can use a faster method
Let dq be the normal vector at q for which f(q+dq)=b1. Can do first order approximation to compute this
Approximate Jacobian as:
Don't justify convergence or why good approx.

5. Iterative algorithm Define the residual (error) vector as Then: where
Iteratively update q + dq q until the error goes to zero
Call this projection operator P
Dtheta is in normal space, parameterized by 1, sin, cos
Betas are then weights on basis vectors
Defining normal space for element not on manifold—this is an approximation to the normal space for the element on the manifold
Convergence?

6. Example Projections Fig. 1: Projections of arbitrary curves into C1
No real intuition as to why these are good
Fig. 1: Projections of arbitrary curves into C1

7. Geodesics
Definition: For a manifold embedded in Euclidean space, a geodesic is a constant speed curve whose acceleration vector is always perpendicular to the manifold
Define the metric between two shapes as the distance along the manifold between the shapes with respect to the L2 inner product
Nice features:
Defined for all closed curves
Interpolants are closed curves
Finds geodesics in a local sense, not necessarily global
For any two points on complete manifold, geodesic exists
Even if start with simple closed, end with simple closed, curves in between may not be simple closed, though they haven’t been able to show—note that this results because they used rotation index 1 curves to define manifold

8. Paths from initial conditions
Assume we have a q in C1 and an f in the tangent space
Approximate geodesic along manifold by moving to q+fDt and projecting that back onto the manifold (Dt is step size)
So q(t+Dt) = P(q(t)+f(t)Dt)

9. Transporting the tangent vector
Now f(t) is not in the tangent space of q(t+Dt)
Two conditions for a geodesic:
The acceleration vector must be perpendicular to the manifold: simply project f into the next tangent space
The curve must move at constant speed: renormalize so ||f(t+1)||=||f(t)||
hk is the orthonormal basis of the normal space
Not clear why this direction has to be chosen for a geodesic
Earlier seems that you can pick any f and start plotting a geodesic in that direction

10. Geodesics on shape spaces
S1 is a quotient space of C1 under actions of S1 by isometries, so finding geodesics in S1 equivalent to finding geodesics in C1 which are orthogonal to S1 orbits
S1 acting by isometries implies that if a geodesic in preshape space is orthogonal to one S1 orbit, it’s orthogonal to all S1 orbits which it meets
So now normal space has one additional component spanned by
The algorithm is the same as detailed earlier except with an expanded normal space
Isometry because all we do is shift theta
Not shown how they derive new basis vector

11. Geodesics between shapes
We know how to generate geodesic paths given q and f
Now we want to construct a geodesic path from q1 to q2
So we need to find all f that lead from q1 to an S1 orbit of q2 in unit time, and then choose the one that leads to the shortest path
Let Y define the geodesic flow, with (q1,0,f)=q1 as the initial condition
We then want Y(q1,1,f)=q2
OK to land anywhere on orbit of theta2

12. Finding the geodesic
Define an error functional which measures how close we are to the target at t=1:
Choose the geodesic as the flow Y which has the smallest initial velocity ||f||
i.e., min ||f|| s.t. H[f]=0
Hard because infinite dimensional search

13. Fourier decomposition
f  L2, so it has a Fourier decomposition
Approximate f with its first m+1 cosine components and its first m sine components:
Let a be the vector containing all of the Fourier coefficients
Now optimization problem is min ||a|| s.t. H[a]=0
Don't explain how they accomplish this (seems like very hard to predict Psi(theta1,1,f)

14. Geodesic paths
Fig. 2: Geodesic paths between two shapes