# Sphere

$$\operatorname{Sphere}(n, r)$$ is the sphere in $$\mathbb{R}^n$$ with radius $$r$$:

$\operatorname{Sphere}(n,r) = \{x \in \mathbb{R}^n\:\mid\:\lVert x \rVert = r\}$

Warning

In mathematics, $$\mathbb{S}^n$$ represents the $$n$$-dimensional sphere. With this notation, $$\operatorname{Sphere}(n, 1.) = \mathbb{S}^{n-1}$$.

Sphere as a map from the tangent space onto the sphere using the exponential map.

Parameters
• size (torch.size) – Size of the tensor to be parametrized

• radius (float) – Optional. Radius of the sphere. A positive number. Default: 1.

sample()[source]

Returns a uniformly sampled vector on the sphere.

in_manifold(x, eps=1e-05)[source]

Checks that a vector is on the sphere.

For tensors with more than 2 dimensions the first dimensions are treated as batch dimensions.

Parameters
• X (torch.Tensor) – The vector to be checked.

• eps (float) – Optional. Threshold at which the norm is considered to be equal to 1. Default: 1e-5

Sphere as the orthogonal projection from $$\mathbb{R}^n$$ to $$\mathbb{S}^{n-1}$$, that is, $$x \mapsto \frac{x}{\lVert x \rVert}$$.

Parameters
• size (torch.size) – Size of the tensor to be parametrized

• radius (float) – Optional. Radius of the sphere. A positive number. Default: 1.

sample()[source]

Returns a uniformly sampled vector on the sphere.

in_manifold(x, eps=1e-05)[source]

Checks that a vector is on the sphere.

For tensors with more than 2 dimensions the first dimensions are treated as batch dimensions.

Parameters
• X (torch.Tensor) – The vector to be checked.

• eps (float) – Optional. Threshold at which the norm is considered to be equal to 1. Default: 1e-5

geotorch.sphere.uniform_init_sphere_(x, r=1.0)[source]

Samples a point uniformly on the sphere into the tensor x. If x has $$d > 1$$ dimensions, the first $$d-1$$ dimensions are treated as batch dimensions.