Affine Invariance
Invariance under affine parameter transformations
Convex hull property
Endpoint Interpolation
Symmetry
Invariance under barycentric combinations
Linear Precision
Pseudolocal 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 4control 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((1t)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((1t)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, bn1,..., 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.
Pseudolocal control
Mathematical expression: L
Geometric meaning: T
Common usage: 1.
