The Bezier Curves are used in computer-aided design and are named after a mathematician working in the automotive industry.

A cubic Bezier curve is determined by four control points , , , and , and is defined by the parametric equations

where is in [ ].

Notice that when we have and when we have
, so the curve starts at and ends at .
Here we graph the Bezier curve with control points , , , and .

(Here is a procedure to graph Bezier curves )

> display( G, t0, t1, t2, t3, pp );

Then on the same screen, graph the line segments , , and . Notice that the middle control points and do not lie on the curve; the curve starts at heads toward and without reaching them, and ends at .

(Here is a procedure to graph Bezier curves with line segments connecting control points.)

> x0 := 4:    y0 := 1:
> x1 := 28:  y1 := 48:
> x2 := 50:  y2 := 42:
> x3 := 40:  y3 := 5:
> Bezier_L( x0, y0, x1, y1, x2, y2, x3, y3 );

From the graph above, it appears that the tangent at passes through and the tangent at passes through .

To produce a Bezier curve with a loop, we can change the second control point .

> display( seq( Bezier_L( x0, y0, 28+5*n, 48-4.3*n, x2, y2, x3, y3 ), n=0..10 ), insequence=true);

 

Some laser printer use Bezier curves to represent letters and other symbols. A reasonable representation of the letter C can be done as follows:

> x0 := 35.5:  y0 := 22:
> x1 := 28:     y1 := 30:
> x2 := 25:     y2 := 10:
> x3 := 36:     y3 := 17:
> Bezier( x0, y0, x1, y1, x2, y2, x3, y3 );

More complicated shapes can be represented by piecing together two or more Bezier curves. Suppose the first Bezier curve has control points , , , and the second one has control points , , , . If we want these two pieces to join together smoothly, then the tangents at should match and so the points , and all have to lie on this common tangent line. We can use this principle to find control points for a pair of Bezier curves that represent the letter S.

> x0 := 30:     y0 := 13:
> x1 := 26.5:  y1 := 16:
> x2 := 24:     y2 := 10:
> x3 := 29:     y3 := 9.5:
> b1 := Bezier( x0, y0, x1, y1, x2, y2, x3, y3 ):
> x4 := 34:     y4 := 9:
> x5 := 29.5:  y5 := 0.5:
> x6 := 26:     y6 := 7:
> b2 := Bezier( x3, y3, x4, y4, x5, y5, x6, y6 ):
> display( b1, b2 );