Affine Invariance

Invariance under affine parameter transformations

Convex hull property

Endpoint Interpolation

Symmetry

Invariance under barycentric combinations

Linear Precision

Pseudo-local control

 


Affine Invariance --

Mathematical expression: Recall a Bezier curve with control points {b0, b1,..., bn} can be obtained by the de Castlejau Algorithm. Let F be an affine map, then Since the algorithm is composed of a sequence of linear interpolations and linear interpolations are invariant under affine map, thus is invariant under an affine map F.

Geometric meaning: This means the following two procedures yield the same result: (1) first, from starting control points {b0, b1,..., bn} compute the curve and then apply an affine map to it; (2) first apply an affine map to the control points {b0, b1,..., bn} to obtain new control points {F(b0),...,F(bn)} and then find the curve with these new control points.

Common usage: Let us discuss a practical aspect of affine invariance. Suppose we plot a cubic curve (t) by evaluating a 100 points and then plotting the resulting point array. Suppose now that we would like to plot the curve after a rotation has been applied to it. We can take the 100 computed points, apply the rotation to each of them, and plot. Or, we can apply the rotation to the 4-control points, then evaluate a 100 times and plot. The first method needs 100 applications of the rotation, while the second needs only 4!

Invariance under affine parameter transformations --

Mathematical expression: Let be a Bezier curve defined over an arbitrary interval [a,b]. Then b(u) = b((1-t)a+tb).

Geometric meaning: This means that if we transform the interval [a,b] to [0,1], the Bezier curve stays invariant. Since the transition from the interval [a,b] to [0,1] is an affine map, we say that Bezier curves are invariant under affine parameter transformations.

Common usage: Because of this proposition, we often think of a Bezier curve as being defined over the interval [0,1]. It is usually convenient and saves calculations if we transform a Bezier curve b(u) defined over [a,b] to a Bezier curve b'(t)= b((1-t)a+tb) defined over [0,1].

Convex Hull Property Mathematical expression:

Let be a Bezier curve. Then for any point on the curve, we can write as a convex combination of the control points {b0, b1,...,bn} satisfy (*junk*)

Geometric meaning: This means for t [0,1], lies in the convex hull of the control points.

Common usage:
1. Because of this proposition, we can gaurantee that a planar control polygon always generates a planar curve.
2. We can use this property to check whether two Bezier curves intersect each other. (This is known as interference checking ). For a real life example, please click here.

Endpoint Interpolation Mathematical expression:

Let b(t) be a Bezier curve defined over [0,1] with control points {b0, b1,..., bn} then one can easily verify that (*junk*)

Geometric meaning: The Bezier curve b(t) always passes through the first and last control points b0 and bn.

Common usage: In a design situation, the endpoints of a curve are certainly two very important points. It is therefore essential to have direct control over them, which is assured by endpoint interpolation.

Symmetry Mathematical expression:

Let two Bezier curves be generated by ordered Bezier (control) points labelled by {b0,b1,...,bn} and {bn, bn-1,..., b0} respectively, then it is easy to check the following identity holds: (*junk*)

Geometric meaning: This means that the curves corresponding to the two different orderings of control points look the same; they differ only in the direction in which they are traversed.

Common usage: Given a Bezier curve b(t). To design a curve with reversed orientation, we only need to first, reverse its control points, then generate the curve.

Invariance under barycentric combinations Mathematical expression:

Let (*junk*) and (*junk*) be two Bezier curves. We can take a barycentric combination of the two curves which then satisfies the following equation: (*junk*)

Geometric meaning: This means that we can construct the weighted average of the two Bezier curves either by taking the weighted average of the correspoding points on the curves, or by taking the weighted average of corresponding control vertices and then computing the curve. Of course, the latter one saves time in general.

Linear Precision Mathematical expression:

Suppose the control polygon vertices are uniformly distributed on a straight line joining two points p and q: Then one can check that the curve generated by these control points satisfies (*junk*)

Geometric meaning: This means if we start with an evenly distributed control points on a line decided by two points p and q, then the generated curve is the straight line between p and q, i.e. the initial straight line is reproduced. This property is called linear precision.

Pseudo-local control

Mathematical expression: L

Geometric meaning: T

Common usage: 1.