Bezier Curve (in the works)

Description

The so-called “Bezier Curve” is a method of representing a smooth curve in computer graphics. Note that it is a method of curve representation, and not a curve in the usual sense in math contexts.

In computer graphics, there is a need to represent a smooth curve, in particular, finding a representation of a curve of a few given points entered/drawn by the user. One way is to represent a curve by a set of points that lies on the curve. This method obviously has some flaws. For example, it requires a lot collection of points to represent a curve smoothly, and it will be again jagged after magnification.

The Bezier Curve representation is a method to represent a curve between 2 given points, by a polynomial parametric formula, with the additional idea of using a few “control points” that specifies the tangency of the given points.

The Bezier Curve method is named after the engineer Pierre Bézier.

cubic bezier curve

Cubic Bezier Curve

Example:

Given 4 points P0, P1, P2, P3, find a parametric formula in polynomial such that it passes P0 with tangent vector[P0,P1], and passes P2 with tangent vector[P3,P2].

Answer:

written in complex number notation:

z[t]=(1-t)^3*P0 + 3*t(1-t)^2*P1 + 3*t^2(1-t)*P2 + t^3*P3

We verify that

z[0]==P0, z'[0]==3*(P1-P0)
z[1]=P2, z'[1]==3*(P2-P3)

History

Differential Equations, Mechanics, and Computation

Formulas

Properties

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Bezier curve.

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2006-05