Sarnoff, Jeffrey (2022). AngleBetweenVectors (Version 0.3.1) [Source Code].
Open access at https://github.com/JeffreySarnoff/AngleBetweenVectors.jl
https://doi.org/10.5281/zenodo.6745689
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.
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
- points as Tuples
- points as NamedTuples
- points as Vectors
- points as SVectors (StaticArrays)
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}This implementation is more robustly accurate than the usual method.
You can work with points in 2D, 3D, .. 1000D .. ?.
-
The shorter of two angles is given as an unoriented magnitude (0 <= radians < π).
-
Vectors are given by their Cartesian coordinates in 2D, 3D or .. N-dimensions.
-
This follows a note by Professor Kahan Computing Cross-Products 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."