Noncommutative = md` ## Generalization for noncommutative values This method can also be useful for interpolating on a circular arc in ${$`n`}-dimensional space, with values which might be Euclidean points or (multi-)vectors: for example points on a 2-sphere embedded in 3-dimensional space or unit quaternions (which lie on a 3-sphere). But in cases where the values may not be commutative, we need to be a bit more careful than we were above. Remember, for commutative values we can write (letting ${$`\tilde{t} = 1 - t`}): ${$$` z(t) = \dfrac{(b - m)a\tilde{t} + (m - a)bt} {(b - m)\tilde{t} + (m - a)t} `} For noncommutative values we instead need to put the denominator ${$`q(t) = (b - m)\tilde{t} + (m - a)t`} into the middle of each term separately: ${$$` x(t) = (b - m)q(t)^{-1} a \tilde{t} + (m - a)q(t)^{-1} b t `} We still have, as desired: ${$$` \begin{aligned} x(0) &= (b - m)(b - m)^{-1}a = a \\[.3em] x(1) &= (m - a)(m - a)^{-1}b = b \\[.3em] x\big(\tfrac{1}{2}\big) &= (b - m)(b - a)^{-1}a + (m - a)(b - a)^{-1}b \\[.1em] &= m(b - a)^{-1}(b - a) = m \end{aligned}`} The last of these uses the identity ${$`b(b - a)^{-1}a = \left(a^{-1} - b^{-1}\right)^{-1} = a(b - a)^{-1}b`}. The above, as written, takes a bit more computation than necessary, as it requires more multivector multiplication than before, but is still relatively quite a cheap operation. If we prefer we can write this asymmetrically as: ${$$` \begin{aligned} x(t) &= a + (m - a)q(t)^{-1} (b - a) t \\[.1em] &= b - (b - m)q(t)^{-1} (b - a) \tilde{t} \end{aligned}`} For practical computation, if we assume ${$`a`}, ${$`m`}, and ${$`b`} are Euclidean vectors, and multiplication is the [geometric product](https://en.wikipedia.org/wiki/Geometric_algebra), we can first define ${$`u = m - a`}, ${$`v = b - m`}, ${$`h = b - a`}, and then express our interpolating function as: ${$$` x(t) = a + \dfrac{t}{\operatorname{lerp}(v, u, t)^2}\operatorname{lerp}(uvh, u^2h, t) `} Or if the values are more general invertible multivectors, we can use the multivector reverse ${$`w^\dagger`}, which generalizes the complex conjugate, ${$`w^{-1} = w^\dagger / (w w^\dagger) = w^\dagger / |w|^2`}: ${$$` \begin{aligned} x(t) &= a + \dfrac{t}{|\operatorname{lerp}(v, u, t)|^2}\operatorname{lerp}(uv^\dagger h, |u|^2h, t) \\[1em] &= b - \dfrac{1 - t}{|\operatorname{lerp}(v, u, t)|^2}\operatorname{lerp}(|v|^2h, vu^\dagger h, t) \tag{5} \end{aligned}`} The initial setup cost is 3 vector subtractions, 2 multivector–multivector multiplications, 1 dot product, and 1 scalar–vector multiplication. The per-point runtime cost is 2 vector lerps, 1 dot product, 1 scalar division, 1 scalar–vector multiplication, and 1 vector addition. `