The proof of the touching point equation in the environment modeling tutorial of Pixar in a Box is very neat, but it took me a second round to understand. Here I’m writing a note hoping making the proof intuitively understandable.

We need to prove that the touching point P of the parabolas arc and line QR is controlled by t on line QR.

The proof starts with writing P as controlled by a random value s on line QR, where s controls line Q’R’. Then, reforming the equation of P controlled by s on line QR results in another equation, in which P is controlled by t on line Q’R’.

The result of the reformation indicates that P lies on line Q’R’ too. Hence, P is the intersection point of QR and Q’R’.

From the observation of section 4, P becomes the touching point of QR when s is equal to t.

There’s no proof of the observation though. But it’s a very neat one that I want to explain. Now that for any point P in QR, P is either the intersection point with line Q’R’, or the touching point. If s doesn’t equal to t, we can always define the intersection point. Otherwise, P has to be the touching point, because we cannot define this one single intersection point.

Proof completed! DAH!