### Partial Scaling – How to do Half a Multiplication

Programmers have a habit of over-generalizing things, and so it happened that I found myself writing more generalized versions of rotation, translation and scaling, the three most common operations that you’d want to do to 3D objects. These were more generalized in that they had a parameter for how much you want to do a certain operation. Like “translate by (10, 0, 0), but only apply the operation 20%”. This is easy to do for translation: Just multiply by the percentage and only translate by (2, 0, 0). Rotations are also easy in many representations: If the angles are explicit, like in Euler angles, you can just multiply those by the percentage; if you’re using quaternions, you can slerp.

But scaling is more complicated. Internally scaling is just multiplication, but how do you do half a multiplication? What does it mean to say “multiply by 4, but only apply the operation 50%”? My first approach was to multiply by the power, so you’d get “multiply by $4^{0.5}=2$” (or $4^{0.1}$ if you only want to apply 10% of the operation) and that seems to work when you’re close to 1, but the further away you are from 1, the more wrong it gets. The answer ends up being to take the median of multiplication, division and exponentiation, but let me further explain the problem first: