The atomic constituents of paths are segments, which are single straight lines or cubic Bezier curves. Segments are translationally invariant, that is, they have no particular "location" and are unaffected by translations. They are, however, affected by other transformations such as rotations and scales.

straight v constructs a translationally invariant linear segment with direction and length given by the vector v.

bezier3 v1 v2 v3 constructs a translationally invariant cubic Bezier curve where the offsets from the first endpoint to the first and second control point and endpoint are respectively given by v1, v2, and v3.

atParam yields a parametrized view of segments as continuous functions [0,1] -> v, which give the offset from the start of the segment for each value of the parameter between 0 and 1. It is designed to be used infix, like seg `atParam` 0.5.

Compute the offset from the start of a segment to the end. Note that in the case of a Bezier segment this is not the same as the length of the curve itself; for that, see arcLength.

splitAtParam splits a segment s into two new segments (l,r) at the parameter t where l corresponds to the portion of s for parameter values from 0 to t and r for s from t to 1. The following should hold for splitting:

 paramSplit s t u
   | u < t     = atParam s u == atParam l (u / t)
   | otherwise = atParam s u == atParam s t ^+^ atParam l ((u - t) / (1.0 - t))
   where (l,r) = splitAtParam s t

That is to say, the parameterization scales linearly with splitting.

arcLength s m approximates the arc length of the segment curve s with accuracy of at least plus or minus m. For a Cubic segment this is computed by subdividing until the arc length of the path through the control points is within m of distance from start to end.