AngleBetweenVectors.jl
When computing the arc separating two cartesian vectors, this is robustly stable; others are not.
Copyright © 20182020 by Jeffrey Sarnoff. This work is released under The MIT License.
AngleBetweenVectors exports angle
.
angle(point1, point2)
determines the angle of their separation. The smaller of the two solutions is used. π
obtains If the points are opposed, [(1,0), (1,0)]; so 0 <= angle(p1, p2) <= pi
.
This function expects two points from a 2D, 3D .. ManyD space, in Cartesian coordinates. Tuples and Vectors are handled immediately (prefer Tuples for speed). To use another point representations, just define a Tuple
constructor for it. NamedTuples and SVectors have this already.
Most software uses acos(dot(p1, p2) / sqrt(norm(p1) norm(p2))
instead. While they coincide often; it is exceedingly easy to find cases where angle
is more accurate and then, usually they differ by a few ulps. Not always.
provides
angle( point₁, point₂ )
 points are given as Cartesian coordinates
 points may be of any finite dimension >= 2
 points may be any type with a Tuple constructor defined
point representations that just work
 points as Tuples
 points as NamedTuples
 points as Vectors
 points as SVectors (StaticArrays)
working with other point representations
Just define a Tuple
constructor for the representation. That's all.
# working with this?
struct Point3D{T}
x::T
y::T
z::T
end
# define this:
Base.Tuple(a::Point3D{T}) where {T} = (a.x, a.y, a.z)
# this just works:
angle(point1::Point3D{T}, point2::Point3D{T}) where {T}
why use it
This implementation is more robustly accurate than the usual method.
You can work with points in 2D, 3D, .. 1000D .. ?.
notes

The shorter of two angle solutions is returned as an unoriented magnitude (0 <= radians < π).

Vectors are given by their Cartesian coordinates in 2D, 3D or .. Ndimensions.

This follows a note by Professor Kahan in Computing CrossProducts and Rotations (pg 15): "More uniformly accurate .. valid for Euclidean spaces of any dimension, it never errs by more than a modest multiple of ε."