Bézier Curves & de Casteljau's
Parametric curve equations, control polygons, and linear interpolation subdivision.
Bernstein Polynomials
A Bézier curve of order $n$ is defined by $n+1$ control points $P_i$. The curve path $B(t)$ is weighted by Bernstein polynomials: $$B(t) = \sum_{i=0}^{n} \binom{n}{i} (1-t)^{n-i} t^i P_i$$ for a parametric variable $t \in [0, 1]$.
de Casteljau's Algorithm
de Casteljau's algorithm evaluates Bézier curves geometrically. For any $t$, linearly interpolate between adjacent control points to find intermediate points. Repeat recursively until a single point remains, which lies exactly on the curve path.
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