mfGeomUnitSphere: Geometry of the Unit Sphere

In Almond-S/manifoldboost: Boosting Regression Models for Manifold Valued Data

Geometry of the Unit Sphere

Description

The geometry of the n-dimensional unit sphere, treating the complex sphere is, in case, as a real manifold.

Super classes

manifoldboost::mfGeometry -> manifoldboost::mfGeomEuclidean -> mfGeomUnitSphere

Methods

Public methods

• mfGeomUnitSphere$register()

• mfGeomUnitSphere$register_v()

• mfGeomUnitSphere$log()

• mfGeomUnitSphere$exp()

• mfGeomUnitSphere$transport()

• mfGeomUnitSphere$get_normal()

• mfGeomUnitSphere$clone()

Inherited methods

• manifoldboost::mfGeometry$distance()
• manifoldboost::mfGeometry$unstructure_weights()
• manifoldboost::mfGeomEuclidean$align()
• manifoldboost::mfGeomEuclidean$innerprod()
• manifoldboost::mfGeomEuclidean$structure()
• manifoldboost::mfGeomEuclidean$unstructure()
• manifoldboost::mfGeomEuclidean$validate()

---

Method register()

y vectors are scaled to unit norm

Usage

mfGeomUnitSphere$register(y_)

Arguments

y_

a numeric/complex vector on the sphere

---

Method register_v()

vectors v are orthogonalized with respect to y0_ guaranteeing proper tangent vectors.

Usage

mfGeomUnitSphere$register_v(v_, y0_ = private$.pole_)

Arguments

v_

a numeric/complex tangent vector

y0_

a numeric/complex vector on the sphere

---

Method log()

Usage

mfGeomUnitSphere$log(
  y_,
  y0_ = private$.pole_,
  method = c("simple", "alternative")
)

Arguments

y_

a numeric/complex vector on the sphere

y0_

a numeric/complex vector on the sphere

method

character string, for choosing one of two slightly different methods to implement log-method ("simple" or "alternative").

---

Method exp()

moving v_ along an arc on the sphere.

Usage

mfGeomUnitSphere$exp(v_, y0_)

Arguments

v_

a numeric/complex tangent vector

y0_

a numeric/complex vector on the sphere

---

Method transport()

Usage

mfGeomUnitSphere$transport(
  v0_,
  y0_,
  y1_,
  method = c("general", "horizontal", "simple")
)

Arguments

v0_

a numeric/complex tangent vector

y0_

a numeric/complex vector on the sphere

y1_

a numeric/complex vector on the sphere

method

expression used for parallel transport: "general" corresponds to the one of Cornea et al. 2017, "horizontal" to the one of Dryden & Mardia 2012 p. 76, and "simple" to a simplified version of "general" where v0_ is horizontal to y0_.

---

Method get_normal()

normal vector just corresponds to the pole itself

Usage

mfGeomUnitSphere$get_normal(y0_ = private$.pole_)

Arguments

y0_

a numeric/complex vector on the sphere

---

Method clone()

The objects of this class are cloneable with this method.

Usage

mfGeomUnitSphere$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Almond-S/manifoldboost documentation built on June 23, 2022, 11:06 a.m.