This pull request addresses issues #32 and #33 by projecting nudges onto the tangent space of the solution variety using a Euclidean-invariant inner product, which I'm calling the *uniform* inner product.
### Definition of the uniform inner product
For spheres and planes, the uniform inner product is defined on the tangent space of the hyperboloid $\langle v, v \rangle = 1$. For points, it's defined on the tangent space of the paraboloid $\langle v, v \rangle = 0,\; \langle v, I_\infty \rangle = 1$.
The tangent space of an assembly can be expressed as the direct sum of the tangent spaces of the elements. We extend the uniform inner product to assemblies by declaring the tangent spaces of different elements to be orthogonal.
#### For spheres and planes
If $v = [x, y, z, b, c]^\top$ is on the hyperboloid $\langle v, v \rangle = 1$, the vectors
$$\left[ \begin{array}{c} 2b \\ \cdot \\ \cdot \\ \cdot \\ x \end{array} \right],\;\left[ \begin{array}{c} \cdot \\ 2b \\ \cdot \\ \cdot \\ y \end{array} \right],\;\left[ \begin{array}{c} \cdot \\ \cdot \\ 2b \\ \cdot \\ z \end{array} \right],\;\left[ \begin{array}{l} 2bx \\ 2by \\ 2bz \\ 2b^2 \\ 2bc + 1 \end{array} \right]$$
form a basis for the tangent space of hyperboloid at $v$. We declare this basis to be orthonormal with respect to the uniform inner product.
The first three vectors in the basis are unit-speed translations along the coordinate axes. The last vector moves the surface at unit speed along its normal field. For spheres, this increases the radius at unit rate. For planes, this translates the plane parallel to itself at unit speed. This description makes it clear that the uniform inner product is invariant under Euclidean motions.
#### For points
If $v = [x, y, z, b, c]^\top$ is on the paraboloid $\langle v, v \rangle = 0,\; \langle v, I_\infty \rangle = 1$, the vectors
$$\left[ \begin{array}{c} 2b \\ \cdot \\ \cdot \\ \cdot \\ x \end{array} \right],\;\left[ \begin{array}{c} \cdot \\ 2b \\ \cdot \\ \cdot \\ y \end{array} \right],\;\left[ \begin{array}{c} \cdot \\ \cdot \\ 2b \\ \cdot \\ z \end{array} \right]$$
form a basis for the tangent space of paraboloid at $v$. We declare this basis to be orthonormal with respect to the uniform inner product.
The meanings of the basis vectors, and the argument that the uniform inner product is Euclidean-invariant, are the same as for spheres and planes. In the engine, we pad the basis with $[0, 0, 0, 0, 1]^\top$ to keep the number of uniform coordinates consistent across element types.
### Confirmation of intended behavior
Two new tests confirm that we've corrected the misbehaviors described in issues #32 and #33.
Issue | Test
---|---
#32 | `proj_equivar_test`
#33 | `tangent_test_kaleidocycle`
Co-authored-by: Aaron Fenyes <aaron.fenyes@fareycircles.ooo>
Reviewed-on: #34
Co-authored-by: Vectornaut <vectornaut@nobody@nowhere.net>
Co-committed-by: Vectornaut <vectornaut@nobody@nowhere.net>
Vectornaut
817a446fad