# Circle

`Manifolds.Circle`

โ Type`Circle{๐ฝ} <: AbstractManifold{๐ฝ}`

The circle $๐^1$ is a manifold here represented by real-valued points in $[-ฯ,ฯ)$ or complex-valued points $z โ โ$ of absolute value $\lvert z\rvert = 1$.

**Constructor**

`Circle(๐ฝ=โ)`

Generate the `โ`

-valued Circle represented by angles, which alternatively can be set to use the `AbstractNumbers`

`๐ฝ=โ`

to obtain the circle represented by `โ`

-valued circle of unit numbers.

`Base.exp`

โ Method`exp(M::Circle, p, X)`

Compute the exponential map on the `Circle`

.

\[\exp_p X = (p+X)_{2ฯ},\]

where $(โ )_{2ฯ}$ is the (symmetric) remainder with respect to division by $2ฯ$, i.e. in $[-ฯ,ฯ)$.

For the complex-valued case, the same formula as for the `Sphere`

$๐^1$ is applied to values in the complex plane.

`Base.log`

โ Method`log(M::Circle, p, q)`

Compute the logarithmic map on the `Circle`

`M`

.

\[\log_p q = (q-p)_{2ฯ},\]

where $(โ )_{2ฯ}$ is the (symmetric) remainder with respect to division by $2ฯ$, i.e. in $[-ฯ,ฯ)$.

For the complex-valued case, the same formula as for the `Sphere`

$๐^1$ is applied to values in the complex plane.

`Base.rand`

โ Method`Random.rand(M::Circle{โ}; vector_at = nothing, ฯ::Real=1.0)`

If `vector_at`

is `nothing`

, return a random point on the `Circle`

$\mathbb S^1$ by picking a random element from $[-\pi,\pi)$ uniformly.

If `vector_at`

is not `nothing`

, return a random tangent vector from the tangent space of the point `vector_at`

on the `Circle`

by using a normal distribution with mean 0 and standard deviation `ฯ`

.

`Manifolds.complex_dot`

โ Method`complex_dot(a, b)`

Compute the inner product of two (complex) numbers with in the complex plane.

`Manifolds.manifold_volume`

โ Method`manifold_volume(M::Circle)`

Return the volume of the `Circle`

`M`

, i.e. $2ฯ$.

`Manifolds.sym_rem`

โ Method`sym_rem(x,[T=ฯ])`

Compute symmetric remainder of `x`

with respect to the interall 2*`T`

, i.e. `(x+T)%2T`

, where the default for `T`

is $ฯ$

`Manifolds.volume_density`

โ Method`volume_density(::Circle, p, X)`

Return volume density of `Circle`

, i.e. 1.

`ManifoldsBase.check_point`

โ Method`check_point(M::Circle, p)`

Check whether `p`

is a point on the `Circle`

`M`

. For the real-valued case, `p`

is an angle and hence it checks that $p โ [-ฯ,ฯ)$. for the complex-valued case, it is a unit number, $p โ โ$ with $\lvert p \rvert = 1$.

`ManifoldsBase.check_vector`

โ Method`check_vector(M::Circle, p, X; kwargs...)`

Check whether `X`

is a tangent vector in the tangent space of `p`

on the `Circle`

`M`

. For the real-valued case represented by angles, all `X`

are valid, since the tangent space is the whole real line. For the complex-valued case `X`

has to lie on the line parallel to the tangent line at `p`

in the complex plane, i.e. their inner product has to be zero.

`ManifoldsBase.distance`

โ Method`distance(M::Circle, p, q)`

Compute the distance on the `Circle`

`M`

, which is the absolute value of the symmetric remainder of `p`

and `q`

for the real-valued case and the angle between both complex numbers in the Gaussian plane for the complex-valued case.

`ManifoldsBase.embed`

โ Method`embed(M::Circle, p, X)`

Embed a tangent vector `X`

at `p`

on `Circle`

`M`

in the ambient space. It returns `X`

.

`ManifoldsBase.embed`

โ Method`embed(M::Circle, p)`

Embed a point `p`

on `Circle`

`M`

in the ambient space. It returns `p`

.

`ManifoldsBase.get_coordinates`

โ Method`get_coordinates(M::Circle{โ}, p, X, B::DefaultOrthonormalBasis)`

Return tangent vector coordinates in the Lie algebra of the `Circle`

.

`ManifoldsBase.get_vector_orthonormal`

โ Method`get_vector(M::Circle{โ}, p, X, B::DefaultOrthonormalBasis)`

Return tangent vector from the coordinates in the Lie algebra of the `Circle`

.

`ManifoldsBase.injectivity_radius`

โ Method`injectivity_radius(M::Circle[, p])`

Return the injectivity radius on the `Circle`

`M`

, i.e. $ฯ$.

`ManifoldsBase.inner`

โ Method`inner(M::Circle, p, X, Y)`

Compute the inner product of the two tangent vectors `X,Y`

from the tangent plane at `p`

on the `Circle`

`M`

using the restriction of the metric from the embedding, i.e.

\[g_p(X,Y) = X*Y\]

for the real case and

\[g_p(X,Y) = Y^\mathrm{T}X\]

for the complex case interpreting complex numbers in the Gaussian plane.

`ManifoldsBase.is_flat`

โ Method`is_flat(::Circle)`

Return true. `Circle`

is a flat manifold.

`ManifoldsBase.manifold_dimension`

โ Method`manifold_dimension(M::Circle)`

Return the dimension of the `Circle`

`M`

, i.e. $\dim(๐^1) = 1$.

`ManifoldsBase.parallel_transport_to`

โ Method` parallel_transport_to(M::Circle, p, X, q)`

Compute the parallel transport of `X`

from the tangent space at `p`

to the tangent space at `q`

on the `Circle`

`M`

. For the real-valued case this results in the identity. For the complex-valud case, the formula is the same as for the `Sphere`

`(1)`

in the complex plane.

\[\mathcal P_{qโp} X = X - \frac{โจ\log_p q,Xโฉ_p}{d^2_{โ}(p,q)} \bigl(\log_p q + \log_q p \bigr),\]

where `log`

denotes the logarithmic map on `M`

.

`ManifoldsBase.project`

โ Method`project(M::Circle, p, X)`

Project a value `X`

onto the tangent space of the point `p`

on the `Circle`

`M`

.

For the real-valued case this is just the identity. For the complex valued case `X`

is projected onto the line in the complex plane that is parallel to the tangent to `p`

on the unit circle and contains `0`

.

`ManifoldsBase.project`

โ Method`project(M::Circle, p)`

Project a point `p`

onto the `Circle`

`M`

. For the real-valued case this is the remainder with respect to modulus $2ฯ$. For the complex-valued case the result is the projection of `p`

onto the unit circle in the complex plane.

`Statistics.mean`

โ Method`mean(M::Circle{โ}, x::AbstractVector[, w::AbstractWeights])`

Compute the Riemannian `mean`

of `x`

of points on the `Circle`

$๐^1$, reprsented by complex numbers, i.e. embedded in the complex plane. Comuting the sum

\[s = \sum_{i=1}^n x_i\]

the mean is the angle of the complex number $s$, so represented in the complex plane as $\frac{s}{\lvert s \rvert}$, whenever $s \neq 0$.

If the sum $s=0$, the mean is not unique. For example for opposite points or equally spaced angles.

`Statistics.mean`

โ Method`mean(M::Circle{โ}, x::AbstractVector[, w::AbstractWeights])`

Compute the Riemannian `mean`

of `x`

of points on the `Circle`

$๐^1$, reprsented by real numbers, i.e. the angular mean

\[\operatorname{atan}\Bigl( \sum_{i=1}^n w_i\sin(x_i), \sum_{i=1}^n w_i\sin(x_i) \Bigr).\]