It refers to "time" (perhaps should be denoted with t), and ranges between 0 <= s <= 1. It effectively represents the same variable used as input to a LERP function (and this is exactly the case when n=1).
Will there be different p1 and p2 for the same curve as long as they fit the feature mentioned above?
motoole2
@Doris No---different p1 and p2 will not produce the same curve. The angle of the vector from p0 to p1 (or p3 to p2) affects the tangent of the cubic Bezier curve at p0 (or p3). The magnitude of the vector connecting points p0 and p1 will also affect the curve; for example, a large magnitude will stretch out the curve along the same direction.
Does "s" have an actual meaning here?
It refers to "time" (perhaps should be denoted with
t), and ranges between0 <= s <= 1. It effectively represents the same variable used as input to a LERP function (and this is exactly the case whenn=1).Oh, and check out these neat interactive animations of Bezier curves for n = 1, 2, 3, and 4.
Will there be different p1 and p2 for the same curve as long as they fit the feature mentioned above?
@Doris No---different
p1andp2will not produce the same curve. The angle of the vector fromp0top1(orp3top2) affects the tangent of the cubic Bezier curve atp0(orp3). The magnitude of the vector connecting pointsp0andp1will also affect the curve; for example, a large magnitude will stretch out the curve along the same direction.