I’ve been thinking a bit about interpolation recently. In animation programming, we often use interpolation as a way to ensure continuity when transitioning from one animation to another. The most basic type of interpolation is linear interpolation, given by $$ a +_s b = sa + (1-s)b\,, $$ for vectors \(a, b\) and weight \(s\). Another common type is spherical interpolation, where we interpolate between two rotations (represented as unit quaternions) along a shared great circle in the \(3\)-sphere.
This got me thinking – can we axiomatize the common properties of different interpolation operators? Let \(+_t\) be a family of binary operators on some set \(A\), indexed by \([0, 1]\). What properties might we ask such a family of operators to satisfy in order to qualify as a form of interpolation? We can ask for a kind of commutativity:
We might also want a kind of idempotence: $$ a +_t a = a\,, $$ and special behaviour for the endpoints \(1\) and \(0\): $$ a +_0 b = a\,. $$