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why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) \geq 0 $) and also due to the Partition of Unity/sum up to one ($ \sum_{k=0}^n B_{k,n}(x) = 1, for\ all\ x \in [0,1] $) implies Bezier curve is guaranteed to lie within the convex hull of its control points $ CH=(\{p_{0},p_{1},p_{2},...,p_{n}\})$

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Here are the facts:

  • A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form $ \sum_{k=0}^n \lambda_k p_k $. where $\lambda_k \ge 0$ and $\sum_{k=0}^n \lambda_k = 1$.

  • The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.

  • A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.

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