Torus

The torus $𝕋^d ≅ [-π,π)^d$ is modeled as an AbstractPowerManifold of the (real-valued) Circle and uses ArrayPowerRepresentation. Points on the torus are hence row vectors, $x ∈ ℝ^{d}$.

Example

The following code can be used to make a three-dimensional torus $𝕋^3$ and compute a tangent vector:

using Manifolds
M = Torus(3)
p = [0.5, 0.0, 0.0]
q = [0.0, 0.5, 1.0]
X = log(M, p, q)

3-element Array{Float64,1}:
 -0.5
  0.5
  1.0

Types and functions

Most functions are directly implemented for an AbstractPowerManifold with ArrayPowerRepresentation except the following special cases:

Manifolds.Torus — Type

Torus{N} <: AbstractPowerManifold

The n-dimensional torus is the $n$-dimensional product of the Circle.

The Circle is stored internally within M.manifold, such that all functions of AbstractPowerManifold can be used directly.

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ManifoldsBase.check_manifold_point — Method

check_manifold_point(M::Torus{n},p)

Checks whether p is a valid point on the Torus M, i.e. each of its entries is a valid point on the Circle and the length of x is n.

source

ManifoldsBase.check_tangent_vector — Method

check_tangent_vector(M::Torus{n}, p, X; check_base_point = true, kwargs...)

Checks whether X is a valid tangent vector to p on the Torus M. This means, that p is valid, that X is of correct dimension and elementwise a tangent vector to the elements of p on the Circle. The optional parameter check_base_point indicates, whether to call check_manifold_point for p.

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