B

from this it follows

and

B

Derivatives

**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 B _{0}^{3}(t) +
1/3 B_{1}^{3}(t) + 2/3 B_{2}^{3}(t) +
B_{3}^{3}(t) = t*.

Therefore for control points with coordinates

we get identical mapping (I used this as

**Differentiation of the Bezier curve**

Derivative of a curve gives the tangent vector at a point. From

^{d}/_{dt} B_{i}^{n}(t) =
n ( B_{i-1}^{n-1}(t) - B_{i}^{n-1}(t) )

it follows that the derivatives at the endpoints of the Bezier curve are

**P**'(0) = n (**P**_{1} - **P**_{0} ),
**P**'(1) = n (**P**_{n} - **P**_{n-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)(**P**_{2} -
2**P**_{1} + **P**_{0} ),
**P**"(1) = n(n-1)(**P**_{n} -
2**P**_{n-1} + **P**_{n-2} ).

(1 - t)S

where

In a similar way we get the general deCasteljau equation for

**Spline formulae standartization**

I'm not a spline expert (I'm only too curious :) Unfortunately I've never seen the standard deBoor textbook. I used information from the Net but different sources have different spline formula notation (and different misprints?). I wouldn't like to mislead users and will try to "sinchronize" cited formulae and Java sources (but it proves almost nothing :) Write me if you have any remarks or suggestions.

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