This algebraic formula for Bézier curves appears to imply that given the same control points, the resulting Bézier curve is always the same. Is there any method of "tightening" or "loosening" a Bézier curve (via, say, modifying the formula) without shifting control points? I suspect that moving control points closer together tightens the curve and that the opposite loosens the curve.
moridin22
I think the point of having control points is that given the points, the curve is completely determined; therefore, instead of storing the curve using some complicated equation structure or by keeping track of all the pixels it goes through, you can just store the control points and reconstruct the curve based on those alone.
DavidVakshlyak
Fun, irrelevant fact: you can interpret the bernstein polynomial B_i^n as the probability of getting i heads in n coin flips times the number of ways to get i heads in n flips. where t is the probability of heads
Jordanwyli
@Hsifnus, I also suspect that if you were to compute your bezier curve for degree n-1 instead of n, you would be able to to "tighten" or "loosen", depending on your definition, the curves that you plot. Perhaps there are some benefits to that, such as not overfitting our curves to the sample points, or producing a curve that appears to be smoother or less change.
jeshlee121
For a cubic bezier curve with control points p0, p1, p2, and p3, you can apply de Casteljau's algorithm using matrix multiplication, where s = 1-t:
[ s t ] [ s t 0 ] [ s t 0 0 ] [ p0 ]
[ 0 s t ] [ 0 s t 0 ] [ p1 ]
[ 0 0 s t ] [ p2 ]
[ p3 ]
This algebraic formula for Bézier curves appears to imply that given the same control points, the resulting Bézier curve is always the same. Is there any method of "tightening" or "loosening" a Bézier curve (via, say, modifying the formula) without shifting control points? I suspect that moving control points closer together tightens the curve and that the opposite loosens the curve.
I think the point of having control points is that given the points, the curve is completely determined; therefore, instead of storing the curve using some complicated equation structure or by keeping track of all the pixels it goes through, you can just store the control points and reconstruct the curve based on those alone.
Fun, irrelevant fact: you can interpret the bernstein polynomial B_i^n as the probability of getting i heads in n coin flips times the number of ways to get i heads in n flips. where t is the probability of heads
@Hsifnus, I also suspect that if you were to compute your bezier curve for degree n-1 instead of n, you would be able to to "tighten" or "loosen", depending on your definition, the curves that you plot. Perhaps there are some benefits to that, such as not overfitting our curves to the sample points, or producing a curve that appears to be smoother or less change.
For a cubic bezier curve with control points p0, p1, p2, and p3, you can apply de Casteljau's algorithm using matrix multiplication, where s = 1-t: