Abstract Class for Special Orthogonal Groups in Matrix Representation

Class for special orthogonal groups in matrix representation.

See also

Other special orthogonal classes: SpecialOrthogonal2Vectors, SpecialOrthogonal3Vectors, SpecialOrthogonal()

Author

Nicolas Guigui and Nina Miolane

Super classes

rgeomstats::PythonClass -> rgeomstats::Manifold -> rgeomstats::MatrixLieGroup -> SpecialOrthogonalMatrices

Public fields

bi_invariant_metric

An object of class BiInvariantMetric specifying the metric to equip the manifold with.

Methods

Public methods

• SpecialOrthogonalMatrices$new()

• SpecialOrthogonalMatrices$belongs()

• SpecialOrthogonalMatrices$intrinsic_to_extrinsic_coords()

• SpecialOrthogonalMatrices$extrinsic_to_intrinsic_coords()

• SpecialOrthogonalMatrices$projection()

• SpecialOrthogonalMatrices$clone()

Inherited methods

• rgeomstats::PythonClass$get_python_class()
• rgeomstats::PythonClass$set_python_class()
• rgeomstats::Manifold$is_tangent()
• rgeomstats::Manifold$random_point()
• rgeomstats::Manifold$random_tangent_vec()
• rgeomstats::Manifold$regularize()
• rgeomstats::Manifold$set_metric()
• rgeomstats::Manifold$to_tangent()
• rgeomstats::MatrixLieGroup$compose()
• rgeomstats::MatrixLieGroup$exp()
• rgeomstats::MatrixLieGroup$get_identity()
• rgeomstats::MatrixLieGroup$inverse()
• rgeomstats::MatrixLieGroup$lie_bracket()
• rgeomstats::MatrixLieGroup$log()
• rgeomstats::MatrixLieGroup$tangent_translation_map()

---

Method new()

The SpecialOrthogonalMatrices class constructor.

Usage

SpecialOrthogonalMatrices$new(n, ..., py_cls = NULL)

Arguments

n

An integer value specifying the number of rows and columns of the matrices.

...

Extra arguments to be passed to parent class constructors. See MatrixLieAlgebra, LevelSet and Manifold classes.

py_cls

A Python object of class SpecialOrthogonalMatrices. Defaults to NULL in which case it is instantiated on the fly using the other input arguments.

Returns

An object of class SpecialOrthogonalMatrices.

---

Method belongs()

Evaluates if a point belongs to the manifold.

Usage

SpecialOrthogonalMatrices$belongs(point, atol = gs$backend$atol)

Arguments

point

A numeric array of shape \([\dots \times \{\mathrm{dim}\}]\) specifying one or more points to be checked.

atol

A numeric value specifying the absolute tolerance for checking. Defaults to gs$backend$atol.

Returns

A boolean value or vector storing whether the input points belong to the manifold.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$belongs(diag(1, 3))
}

---

Method intrinsic_to_extrinsic_coords()

Converts from intrinsic to extrinsic coordinates.

Usage

SpecialOrthogonalMatrices$intrinsic_to_extrinsic_coords(point_intrinsic)

Arguments

point_intrinsic

A numeric array of shape \([\dots \times \{ \mathrm{dim} \}]\) specifying one or more points on the embedded manifold in intrinsic coordinates.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim_embedding} \}]\) storing the same points on the embedded manifold in extrinsic coordinates.

---

Method extrinsic_to_intrinsic_coords()

Converts from extrinsic to intrinsic coordinates.

Usage

SpecialOrthogonalMatrices$extrinsic_to_intrinsic_coords(point_extrinsic)

Arguments

point_extrinsic

A numeric array of shape \([\dots \times \{ \mathrm{dim_embedding} \}]\) specifying one or more points on the embedded manifold in extrinsic coordinates.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim} \}]\) storing the same points on the embedded manifold in intrinsic coordinates.

---

Method projection()

Project a matrix on \(\mathrm{SO}(n)\) by minimizing the Frobenius norm.

Usage

SpecialOrthogonalMatrices$projection(point)

Arguments

point

A numeric array of shape \([\dots \times n \times n]\) specifying one or more matrices.

Returns

A numeric array of the same shape storing the projected matrices.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(3)
  so3$projection(diag(1, 3))
}

---

Method clone()

The objects of this class are cloneable with this method.

Usage

SpecialOrthogonalMatrices$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples

## ------------------------------------------------
## Method `SpecialOrthogonalMatrices$belongs`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$belongs(diag(1, 3))
}

## ------------------------------------------------
## Method `SpecialOrthogonalMatrices$projection`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(3)
  so3$projection(diag(1, 3))
}