Angle between two vectors via the dot product

Computes the angle in \([0, \pi]\) between two numeric vectors using the dot-product/acos formula, with the dot product clamped to \([-1, 1]\) for numerical stability near colinear and anti-colinear configurations.

Usage

angle_between(v1, v2)

Arguments

v1

Numeric vector. Unit-normalized inputs are recommended for speed; otherwise the caller is responsible for prescaling.

v2

Numeric vector of the same length as v1.

Value

A single numeric value in \([0, \pi]\): the angle between v1 and v2 in radians.

Details

For unit vectors the formula is $$\theta = \arccos(v_1 \cdot v_2);$$ for general vectors, divide the dot product by the product of the norms before applying acos. The implementation assumes pre-normalised input and only clamps the dot product to the unit interval.

References

Strang, G. (2016). Introduction to Linear Algebra, 5th edition. Wellesley-Cambridge Press. ISBN 978-0980232776.

See also

lhuilier_angle for the spherical excess of a triangle whose sides are returned by this function; spherical_triangle_area for the Van Oosterom-Strackee form; cross_product_3d for the associated 3D vector product.

Examples

angle_between(c(1, 0, 0), c(0, 1, 0))                # pi / 2
#> [1] 1.570796
angle_between(c(1, 1, 1) / sqrt(3), c(2, 2, 2) / sqrt(12))  # ~ 0
#> [1] 0
angle_between(c(1, 0, 0), c(-1, 0, 0))               # pi
#> [1] 3.141593