1.3.4 Definition of Bézier curve and its properties

(1.40) |

The coefficients, , are the

*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.

- The first and last
control points are the endpoints of the curve. In other words,
and
.
*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.

*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:

(1.43)