Bézier Curve
Given a set of 
control points, the corresponding Bézier curve (or Bernstein-Bézier Curve) is given by
where
is a Bernstein Polynomial and 
. A ``rational'' Bézier curve is defined by
where
is the order, 
are the Bernstein Polynomials, 
are controlpoints, and the weight 
of is the last ordinate of the homogeneous point 
. These curvesare closed under perspective transformations, and can represent Conic Sections exactly.The Bézier curve always passes through the first and last control points and lies within the Convex Hull of thecontrol points. The curve is tangent to 
and 
at the endpoints. The``variation diminishing property'' of these curves is that no line can have more intersections with a Bézier curve thanwith the curve obtained by joining consecutive points with straight line segments. A desirable property of these curves isthat the curve can be translated and rotated by performing these operations on the control points.
Undesirable properties of Bézier curves are their numerical instability for large numbers of control points, and thefact that moving a single control point changes the global shape of the curve. The former is sometimes avoided bysmoothly patching together low-order Bézier curves. A generalization of the Bézier curve is the B-Spline.
See also B-Spline, NURBS Curve- Black Spleenwort Fern
- Blancmange Function
- Blaschke Conjecture
- Blaschke's Theorem
- Blecksmith-Brillhart-Gerst Theorem
- Blichfeldt's Lemma
- Blichfeldt's Theorem
- BLM/Ho Polynomial
- Bloch Constant
- Bloch-Landau Constant
- Block
- Block Design
- Block Growth
- Block Matrix
- Block (Set)
- Blow-Up
- Blue-Empty Coloring
- Blue-Empty Graph
- Board
- Boatman's Knot
- Bochner Identity
- Bochner's Theorem
- Bode's Rule
- Bogdanov Map
- Bogomolov-Miyaoka-Yau Inequality