# More Bezier splines Math

Bernsein polynomials have next useful properties
i=0,n Bin(t) = 1,     (partition of unity)
Bin(t) ≥ 0,     0 ≤ t ≤ 1

from this it follows
Bin(t) ≤ 1,     0 ≤ t ≤ 1
and
B0n(0) = 1,     Bin(0) = 0,
Bnn(1) = 1,     Bin(1) = 0.

Derivatives
d/dt Bin(t) = n ( Bi-1n-1(t) - Bin-1(t) ).

## Affine Invariance

From the de Casteljau algorithm it follows that, any linear transformation (such as rotation or scaling) or translation of control points defines a new curve that is just the transformation or translation of the original curve (i.e. Bezier curve is affinely invariant with respect to its control points).

## Convex hull

As linear interpolated points are contained in the convex hull of control points, then the Bezier curve is contained in the convex hull of its control points too.

## Linear Precision

If all the control points form a straight line, the curve also forms a line. This follows from the convex hull property; as the convex hull becomes a line, so does the curve.
Moreover, you can test by hand, that for cubic Bernstein polynomials
0 B03(t) + 1/3 B13(t) + 2/3 B23(t) + B33(t) = t.
Therefore for control points with coordinates
P0,x = 0,  P1,x = 1/3,  P2,x = 2/3,  P3,x = 1
we get identical mapping (I used this as "1/3 rule")
Px(t) = t.

## Differentiation of the Bezier curve

Derivative of a curve gives the tangent vector at a point. From
d/dt Bin(t) = n ( Bi-1n-1(t) - Bin-1(t) )
it follows that the derivatives at the endpoints of the Bezier curve are
P'(0) = n (P1 - P0 ),     P'(1) = n (Pn - Pn-1 ).
Therefore the Bezier curve is tangent to the first and last segments of the control polygon, at the first and last control points. In fact, these derivatives are n times the first and last legs of the control polygon.
The second derivatives are
P"(0) = n(n-1)(P2 - 2P1 + P0 ),     P"(1) = n(n-1)(Pn - 2Pn-1 + Pn-2 ).

## Deriving deCasteljau algorithm

We use Bernstein polynomials recurrence relations to get the first step of deCasteljau iterations
P(t) = P0n = ∑i=0,n Bin Pi = ∑i=0,n Pi [ (1 - t)Bin-1 + tBi-1n-1] =
(1 - t)∑i=0,n-1 Pi Bin-1 + t∑i=0,n-1 Pi+1 Bin-1 = (1 - t)P0n-1 + tP1n-1
,
where
Pmk = ∑i=0,k Bik Pi+m .
In a similar way we get the general deCasteljau equation for Pmk.
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updated 7 August 2001