I do not have experience of Mathematics past a-level, so please excuse the incorrect terminology.
I am trying to better understand the fundamentals of how a cubic bezier curve works. I am going by this (parametric?) formula... (I formatted, and labelled to make my point clear)
t = time along path r = 1-t a = start co-ordinate b = first handle c = second handle d = end co-ordinate a * r * r * r + b * r * r * t * 3 + c * r * t * t * 3 + d * t * t * t = value of x or y
As far as I understand it, the tension passes from t, to r as time passes, so that early points/handles become less influential over time, and late ones increase.
I understand that - but what I don't understand is why the value 3 is used. As far as I can see, it would work equally well with any other arbitrary value.
I expect that using another value would break the method of calculating points by tracing lines between the points at set intervals, like at
t = 0.25 for example, being able to make a line between ab and bc intersecting at their respective 25% marks, and finding the point at
t = 0.25 of the new line.
So, what I want to understand, is the relationship between the number 3, and the ability to calculate the points along the path in said manner.