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Bézier Curves
Urs Oswald osurs@bluewin.ch http://www.ursoswald.ch September 11, 2002
Quadratic Bézier curves
From the above equations, it follows that
,
therefore
Cubic Bézier curves
From (2), we get
From (1), which, after expansion, yields In fig. 2, if moves about the segment , then moves on the quadratic Bézier curve determined by points , while moves on the quadratic Bézier curve determined by points .
Bézier curves of arbitrary order
For distinct points
, the Bézier curve of order (
)
can be recursively defined by
Theorem 1 (Bézier curves of order 1)
For points , the Bézier curve of order 1 is given by the equation
PROOF: By the above definition,
Theorem 2
For any nonnegative integer , the Bézier curve of order is given by the equation
PROOF: By induction on . For , the theorem claims
which is correct by the first part of the definition. For , we have by definition. By induction hypothesis, We get
, as
 
