An accurate and stable calculation of the angle separating two vectors.
Author JeffreySarnoff
25 Stars
Updated Last
1 Year Ago
Started In
June 2018


When computing the arc separating two cartesian vectors, this is robustly stable; others are not.

Copyright © 2018-2022 by Jeffrey Sarnoff.    This work is released under The MIT License.

Build Status Coverage Status DOI

preferred citiation
Sarnoff, Jeffrey (2022). AngleBetweenVectors (Version 0.3.1) [Source Code].
Open access at

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

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}

#  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 .. ?.


  • 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."

Required Packages

No packages found.