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Bezier Curves

(Some of the theory behind BCs and diagrams are yet to be added. But the applet is interesting.)

Introduction

The Bezier Curve(BC) was developed in the early 1970s by Pierre Bezier of the French automobile company Renault. He was attempting to use computers to design automobiles.
Another important use of Bezier Curves is in the design of type fonts for reproducing text on a computer printer. Companies that sell type font softwares rely heavily on Bezier Curves to define the shape of each character in each different type face.

Spline Curves

The Spline Curve is perhaps the single most important curve in both the aircraft and ship building industries. A drafting tool called a "Spline" is a strip of plastic or other material that is easily flexed to pass through a series of key design points already established on a drawing. Weights called "ducks" hold the spline in place while the draftsman uses the spline as a guide to draw a smooth curve formed by it through the design points. A spline curve can be drawn through any set of n points that imply a smooth curve.

B-Spline Curves

The mathematical equivalent of a spline, the B-Spline, is a cubic polynomial that has one more degree of continutiy than the BC and therefore is inherently smoother. It would be a better choice if it were as easy to use. Unfortunately, the B-Spline curve does not pass through the beginning and the ending points for a desired curve. It is possible to repeat the first and last control points several times and thus bring the beginning and ending points of the curve nearer to the first and last points, but the solution is not very satifactory.

Theory of Bezier Curves

Bezier started with the principle that any point on a curve segment must be given by a parametric function of the form
where the vectors pi represent the (n+1) vertices of a "characteristic polygon" (or control points). He then set forth certain properties that the fi(u) blending functions must have and then looked for specific functions to meet these requirements.

1. The functions must interpolate the first and the last vertex points.
2. The tangent at p0 must be given by p1-p0 and the tangent at pn by pn - pn-1.
3. The function fi(u) must be symmetric with reference to u and (1-u). This means we can reverse the sequence of vertex points without changing the shape of the curve.

Please enter a value for the number of points to be plotted and hit the "Draw" button.
The number of points has been limited to 10.

Applet for demonstrating Bezier Curves
(This applet fits a curve for randomly generated points. I shall modify it sometime later to accept user defined points.)

Properties

1. The curve in general does not pass through any of the control points except the first and last.
2. The curve is always contained within the convex hull of the control points.
3. If there are only two control points p0 and p1, ie n=1, then the formula reduces to a line segment between the two points.
4. Closed curves can be generated by making the last control point the same as the first control point.

Click here to delve into more theory: Explicit Equations, Implicit Equations, Parametric Equations

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