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.