sym.unit3 module¶
- class Unit3(vec)[source]¶
Bases:
objectAutogenerated Python implementation of
symforce.geo.unit3.Unit3.Direction in R^3 represented as a unit vector on the S^2 sphere manifold.
Storage is three dimensional, and tangent space is two dimensional. Due to the nature of the manifold, the unit X vector is handled as a singularity.
The implementation of the retract and local_coordinates functions are based on Appendix B.2 :
[Hertzberg 2013] Integrating Generic Sensor Fusion Algorithms with Sound State Representations through Encapsulation of Manifolds
The retract operation performs a perturbation to the desired unit X vector, which is then rotated to desired position along the actual stored unit vector through a Householder-reflection + relection across the XZ plane.
x.retract(delta) = x [+] delta = Rx * Exp(delta), where Exp(delta) = [cos(||delta||), sinc(||delta||) * delta], and Rx = (I - 2 vv^T / (v^Tv))X, v = x - e_x != 0, X is a matrix negating 2nd vector component
= diag(1, -1, -1) , x = e_x
See: unit3_visualization.ipynb for a visualization of the Unit3 manifold.
- Parameters:
vec (T.Union[T.Sequence[float], numpy.ndarray]) –
- basis(epsilon)[source]¶
Returns a
Matrix32with the basis vectors of the tangent space (in R^3) at the current Unit3 direction.
- static random_from_uniform_samples(u1, u2, epsilon)[source]¶
Generate a random
Unit3direction from two variables uniformly sampled in [0, 1].
- static from_vector(a, epsilon)[source]¶
Return a
Unit3that points along the direction of vectoraawill be normalized.
- static from_unit_vector(a)[source]¶
Return a
Unit3that points along the direction of vectoraais expected to be a unit vector.