just use `z` instead of `0 + z * t` in section 11.

[Pomax]

It's more confusing than useful.

[Haiderahandali]

Here is my logic behind it We have our range of t as follows: 0 <= t<= 1 We want to split it (let's say at t =1/3) such that, As t varies from 0 to 1, our Our range varies from 0 to to 1/3 (the original way I did it is multiplying the range above by value z, so our range becomes: 0 <= t*z <= z, which is exactly what we wanted, but I knew this would not work for the second case, because if wanted to start from z and not 0, we have to shift our entire range by z so it becomes z <= t+z <= 1+z But this is wrong because at t = 1 our range is 1+z, so we have to multiply t by some constant ( say c) such that z + ct=1 So at t = 1, c =1-z So as t varies from 0 to 1 z+t(1-z) varies from z to 1)

Below is a more systematic way of arriving at the same result.

So we have to replace t by the simplest possible equation, the linear equation! a + bt, for some constants a and b. (actually it's affine but everyone calls it linear) We want to start from t=0 our range starts at 0, so we have a + bt = 0 (at t =0) a + 0*b=0 Then a = 0 We want at t = 1, our range becomes 1/3 So we have a + bt = 1/3 Sub a = 0, t =1 Then our equation becomes instead of t, we have 1/3t.

So now, in order to split the original range from 0 to 1 we replace t by t*z and we get our range to vary from 0 to z.

In the exact same manner if we split at value z, and want to transform our range from 0 to 1 into a new range from z to 1 We have: a +bt. At t = 0 we want to start at value z, a + bt = z Sub t = 0 a +b 0= z, then a = z At t = 1, we want our range to be 1: a + bt = 1 Sub a=z, t=1 z = 1b = 1 b = 1-z Then to map our range from (originally from 0 to 1) into z to 1 we replace t by z + (1-z)t

That is why we replace t by t*z in our first range And t by z+t(1-z) In the second range I hope it made sense.