将多项式曲线转换为贝塞尔曲线控制点

问题描述

在给定曲线以幂形式的情况下，如何计算控制点?假设我有p(t)=(x(t)，y(t))和4个控制点.

x(t) = 2t
y(t) = (t^3)+3(t^2)

解决方案

您始终可以从幂基础转换为伯恩斯坦基础.这始终是可行的，并将为您提供精确的结果.请参阅此链接的第3.3节( http://cagd.cs.byu .edu/〜557/text/ch3.pdf ).

由于上述链接不再可用，因此我在下面列出了公式:

式中，M是贝斯坦数的基数，如果i ＜0，则0 ＜＝ k ＜＝ M，b_i，k ＝ 0. k.

以公共立方Berstein基础(其中M = 3)为例，我们将

How do I compute the control points given a curve in the form of power form? Say I have p(t)=(x(t),y(t)) and 4 control points.

x(t) = 2t
y(t) = (t^3)+3(t^2)

解决方案

You can always convert from power basis to Bernstein basis. This is always doable and will give you the precise result. Refer to section 3.3 of this link (http://cagd.cs.byu.edu/~557/text/ch3.pdf) for details.

EDIT:Since the above link is no longer available, I am listing the formula below:

where M is the degree of the Berstein basis, 0 <= k <= M and b_i,k=0 if i < k.

Using the common cubic Berstein basis (where M=3) as an example, we will have

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