1.3.4 Definition of Bézier curve and its properties

A Bézier curve is a parametric curve that uses the Bernstein polynomials as a basis. A Bézier curve of degree (order ) is represented by

The coefficients, , are the control points or Bézier points and together with the basis function determine the shape of the curve. Lines drawn between consecutive control points of the curve form the control polygon. A cubic Bézier curve together with its control polygon is shown in Fig. 1.3 (a). Bézier curves have the following properties:

• Geometry invariance property: Partition of unity property of the Bernstein polynomial assures the invariance of the shape of the Bézier curve under translation and rotation of its control points.
• End points geometric property:
  • The first and last control points are the endpoints of the curve. In other words, and .
    
  • The curve is tangent to the control polygon at the endpoints. This can be easily observed by taking the first derivative of a Bézier curve
    
    
    In particular we have and . Equation (1.41) can be simplified by setting :
    
    
    The first derivative of a Bézier curve, which is called hodograph, is another Bézier curve whose degree is lower than the original curve by one and has control points , . Hodographs are useful in the study of intersection (see Sect. 5.6.2) and other interrogation problems such as singularities and inflection points.
• Convex hull property: A domain is convex if for any two points and in the domain, the segment is entirely contained in the domain [335]. It can be shown that the intersection of convex domains is a convex domain. The convex hull of a set of points is the boundary of the smallest convex domain containing . There are several efficient algorithms for computing the convex hull of a set of points [335,66,292].

  Using the above definitions and facts, the convex hull of a Bézier curve is the boundary of the intersection of all the convex sets containing all vertices or the intersection of the half spaces generated by taking three vertices at a time to construct a plane and having all other vertices on one side. The convex hull can also be conceptualized at the shape of a rubber band in 2-D or a sheet in 3-D stretched taut over the polygon vertices [75]. The entire curve is contained within the convex hull of the control points as shown in Fig. 1.3 (b). The convex hull property is useful in intersection problems (see Fig. 1.4), in detection of absence of interference and in providing estimates of the position of the curve through simple and efficiently computable bounds.

• Variation diminishing property:
  • 2-D: The number of intersections of a straight line with a planar Bézier curve is no greater than the number of intersections of the line with the control polygon. A line intersecting the convex hull of a planar Bézier curve may intersect the curve transversally, be tangent to the curve, or not intersect the curve at all. It may not, however, intersect the curve more times than it intersects the control polygon. This property is illustrated in Fig. 1.5.
  • 3-D: The same relation holds true for a plane with a space Bézier curve.
  From this property, we can roughly say that a Bézier curve oscillates less than its control polygon, or in other words, the control polygon's segments exaggerate the oscillation of the curve. This property is important in intersection algorithms and in detecting the fairness of Bézier curves.
• Symmetry property: If we renumber the control points as , or in other words relabel from to and using the symmetry property of the Bernstein polynomial (1.20) the following identity holds:
  
  
  

Figure 1.3: A cubic Bézier curve: (a) with control polygon, (b) with convex hull

Figure 1.4: Comparison of convex hulls of Bézier curves as means of detecting intersection

Figure 1.5: Variation diminishing property of a cubic Bézier curve