Background.

    The Bézier Curve is the original computer generated "French Curve" and it's discovery is attributed to the French engineer, Pierre Bézier.  Let us first review ideas concerning the cubic polynomial of degree 3 passes through the 4  points for    and is shown in Figure 1.  

             Figure 1.

Example 1.  Find the cubic polynomial or that passes through the four points
  and satisfies
          
Solution 1.

Background for Hermite Interpolating Polynomial.

    The cubic Hermite polynomial  p(x)  has the interpolative properties          and    both the function values and their derivatives are known at the endpoints of the interval  .  The Hermite polynomial is referred to as a "clamped cubic," where "clamped" refers to the slope at the endpoints being fixed.  This situation is illustrated in Figure 2.  
    

             Figure 2.

Example 2.  Find the cubic Hermite polynomial or "clamped cubic" that satisfies  
          
Solution 2.

The Bézier Curve

    The Bézier curve was originally used by  Pierre Bézier  for CAD/CAM operations at Renault motor car company.  Bézier curves are the basis of the entire Adobe PostScript drawing model which is used in the software products: Adobe Illustrator, Macromedia Freehand and Fontographer.  

Construction of the Bézier Curve

    A cubic Bézier curve is defined by four points. Two of the points are endpoints to the curve, is the starting point and  is the destination point. The points are control points or "handles."  A parametric formula for the curve is  
    
     where are cubic equations defined in the interval   as shown in Figure 3.

             Figure 3.

Theorem (Bézier Curve).  The parametric equations for   for the Bézier curve   are given by   

    ,  
and
        for  .  

Proof  The Bézier Curve  The Bézier Curve  

Example 3.  Find the Bézier curve which has the starting at the point and destination point which has the control points and , respectively.  Use the parametric equations to form the  Bézier curve.
Solution 3.

More background

    If a control point is chosen farther away from a endpoint, (but in the same direction), then the Bézier curve will more closely approximate the tangent line near that endpoint.  

Example 4.  Find the Bézier curve which starts at and ends at which has the control points and , respectively.  
Solution 4.

Example 5.  Compare the Bézier curve which starts at and ends at given in examples 3 and 4.  
Solution 5.

Bernstein polynomials.

Definition (Bernstein Polynomials).  The Bernstein polynomials of degree are  
    
    , for ,  where .  
Exploration.

Construction of the Bézier Curve using Bernstein polynomials.

    The coordinate functions   and   for the  Bézier curve we are using can written as a linear combinations of the Bernstein polynomials
:     

        ,  
    and
        .  

Verify that the two functions    and are the same.  

Similarly, the two functions    and   are the same.  
To verify we have the correct end conditions, we can evaluate    and its derivative at and ,  and see if it has the required properties.  

Recall that this is what we wanted in the first construction.  

Justification for the factor 3.

    Since we want the two constructions of the Bézier curve to be the same, it is common practice to use the term in the development of the parametric equations.  Similarly the term   is justified.  

    The construction of a Bézier curve using Bernstein polynomials is more appealing mathematically because the coefficients in the linear combination are just the coordinates of the given four points.  

Example 6.  Find the Bézier curve which starts at and ends at which has the control points and , respectively.
Use Bernstein polynomials.  
Solution 6.

Computer Programs  The Bézier Curve  The Bézier Curve  

Mathematica Subroutine (Bézier Curve).  Construct the Bézier given endpoints and  and control points .  
For illustration purposes the variables pts, ctr and lin are used to help form dots and control lines for the graph we will draw.    
This is for pedagogical purposes, and usually they would not be necessary.  

Example 7.  Find composite Bézier curve for the data given below.

Solution 7.

Example 8.  Let use Mathematica's Bézier curve fit procedure.
The following example is found in the  "Help"  menu under  Add-ons;  Standard Packages;  NumericalMath;  SplineFit.  
Find the "spline fit" through  (0,0), (1,2), (-1,3), (0,1), (3,0).  

Solution 8.

Example 9.  What is the nature of the  "spline fit"  constructed by Mathematica ?

Solution 9.

Research Experience for Undergraduates

The Bézier Curve  The Bézier Curve  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook The Bézier Curve

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(c) John H. Mathews 2004