Do you have reference or explanation for how you compute the tangent space jacobians using autodiff?

It is a cool trick that I have never seen before, and I can't quite wrap my head around why it works. I am guessing it has something to do with the definition of the derivative

but you are applying autodiff to only the numerator portion which I thought would be a different value.

• Remove dynamic size Lie groups?
• Add size argument in Default
• Include tests for dynamically sized types
• Define dynamically sized LieGroup interface for std::variant<LieGroup ...>

Split into a fixed number N of points that partition [0, S]

It holds that v(t + \delta) = \sqrt(v(t)^2 + 2 \delta a), where a is constant acceleration on [s(t), s(t+\delta)].

Reverse pass: calculate maximal v(t) for each point

Forward pass: use maximal a subject to constraints from reverse pass, also calculate t's

Can work in v^2 to avoid square roots.

Stretch: higher orders and use interval reachability in reverse pass?

Implement Lie group operators with free functions

Types keep their "native" API

Concept checks that free functions are available

This would be a breaking change, i.e. g.log() becomes log(g) and G::exp(v) becomes exp<G>(v)

1. Fit PiecewiseBezier that minimizes integral over k:th derivative

Given knot points (t_i, x_i), i = 0, ..., n-1, solve

min ∫ \| x^{(k)} (t) \|^2 dt
s.t     x(t) k times continuously differentiable (continuity at knots)
         x(t_i) = x_i, i = 0, ..., n-1

Each interval is defined by a PiecewiseBezier.

• Decision variables are the (k+1) difference vectors of the Bezier components for each interval
• Continuity at knots can be linearly approximated from formulas for Bernstein polynomial derivatives
• Objective function can be linearly approximated to be quadratic in decision variables, can find coefficients from Bernsteil polynomial derivatives and integrals

https://en.wikipedia.org/wiki/Bernstein_polynomial

2. Template curve class on polynomial degree

3. Make order arbitrary in reparameterization

Some types (e.g. AnyManifold) are difficult/impossible to adapt to the Manifold concept due to these requirements:

• Castable
• Default-constructible

Todo: List where those two requirements are necessary.

Solution: Break up into multiple concepts, where the "base" type does not have those requirements.