# Differentials

Manopt.differential_bezier_controlMethod
differential_bezier_control(
M::Manifold,
B::AbstractVector{<:BezierSegment},
T::AbstractVector{Float}
X::AbstractVector{<:BezierSegment}
)

evaluate the differential of the composite Bézier curve with respect to its control points B and tangent vectors X in the tangent spaces of the control points. The result is the “change” of the curve at pts, which are elementwise in $[0,N]$, and each depending the corresponding segment(s). Here, $N$ is the length of B.

See de_casteljau for more details on the curve and [BergmannGousenbourger2018].

source
Manopt.differential_bezier_controlMethod
differential_bezier_control(
M::Manifold,
B::AbstractVector{<:BezierSegment},
t::Float64,
X::AbstractVector{<:BezierSegment}
)

evaluate the differential of the composite Bézier curve with respect to its control points B and tangent vectors Ξ in the tangent spaces of the control points. The result is the “change” of the curve at t $\in[0,N]$, which depends only on the corresponding segment. Here, $N$ is the length of B.

See de_casteljau for more details on the curve.

source
Manopt.differential_bezier_controlMethod
differential_bezier_control(
M::Manifold,
b::NTuple{N,P},
T::Array{Float64,1},
X::BezierSegment,
)

evaluate the differential of the Bézier curve with respect to its control points b and tangent vectors X in the tangent spaces of the control points. The result is the “change” of the curve at the points T, elementwise in $\in[0,1]$.

See de_casteljau for more details on the curve.

source
Manopt.differential_bezier_controlMethod
differential_bezier(M::Manifold, b::BezierSegment, t::Float, X::BezierSegment)

evaluate the differential of the Bézier curve with respect to its control points b and tangent vectors X given in the tangent spaces of the control points. The result is the “change” of the curve at t$\in[0,1]$.

See de_casteljau for more details on the curve.

source
Manopt.differential_forward_logsMethod
Y = differential_forward_logs(M, p, X)

compute the differenital of forward_logs $F$ on the PowerManifold manifold M at p and direction X , in the power manifold array, the differential of the function

$F_i(x) = \sum_{j ∈ \mathcal I_i} \log_{p_i} p_j, \quad i ∈ \mathcal G,$

where $\mathcal G$ is the set of indices of the PowerManifold manifold M and $\mathcal I_i$ denotes the forward neighbors of $i$.

Input

• M – a PowerManifold manifold
• p – a point.
• X – a tangent vector.

Ouput

• Y – resulting tangent vector in $T_x\mathcal N$ representing the differentials of the logs, where $\mathcal N$ is thw power manifold with the number of dimensions added to size(x).
source