Have you ever heard of Bezier curves? They are a type of parametric
curve that is widely used in graphic design, engineering, and other application
fields. A Bezier curve is defined using a set of points in the plane,
called control points, which dictate the shape of the curve. Moving the
control points alters the shape of the curve. This proves to be the perfect
tool for interactive graphic design. Once control points have been defined,
the user can move these points interactively to refine the shape of the
desired curve. The figure below shows several versions of a Bezier curve
with slightly different positions of the control points. You can interactively
manipulate Bezier curves online at
One of the attractions of Bezier curves is their simple mathematical representation. As an illustration
of the main ideas, consider five control points, A, B, C, D, and E, thought of as vectors in the plane.
The corresponding Bezier curve F(t) is defined as a weighted sum of the vectors A, B, C, D, E.
The weights are the terms from the binomial expansion for
[t + (1-t)]4, so the equation for F is:
F(t) = At4 + 4Bt3(1-t)+ 6Ct2 (1-t)2+ 4Dt(1-t)3 + E(1-t)4
This is a curve that is at point E at time 0, at point A at time 1, and generally bends in the
direction of the other points, (although does not generally go through those points). For any value of
t the weights on the points add up to 1, because they are the terms of
[t + (1-t)]4 = 1. As t varies from 0 to 1, the weights also vary, so that E gets maximal weight at the start, A gets all the weight at the end, and in between, each point takes a turn as recipient of the dominant weight for a different subinterval of [0,1].
A recent calculus 2 class was introduced to Bezier curves by one of the students, Alex, who investigated these curves for a term project. According to Alex, all of the outline fonts used by Adobe Acrobat are described by combinations of Bezier curves. Alex gave a couple of on-line references to articles about Bezier curves: http://mathworld.wolfram.com/BezierCurve.html and http://mathworld.wolfram.com/BernsteinPolynomial.html