Page 120 - DCAP504_Computer Graphics

Unit 8: 3-D in Computer Graphics

B is Bernstein polynomial and

, this is a binomial coefficient.

For a 3-D Bezier surface,

Here, each is a Bezier curve.

Source: http://en.wikipedia.org/wiki/B%C3%A9zier_curve

The B here stands for basis. Not to be confused with beta or Bezier.

B-spline Surface

The B–spline surface is obtained by the Cartesian product resulting from the extension of B-spline
curve. B-spline surface is sectioned by dividing the polygon grid lines in one or both parametric
direction. Its flexibility is improved by raising the order of the basis function resulting in defining the
polygon/grid lines. It allows the generation of curves for any degree of continuity. They are the
preferred way to describe smooth curves.
Properties of B-spline

1. The highest order in each parametric direction depends on the number of defining polygon
vertices in that direction.
2. The surface being continued in each parametric direction is k-2, l-2.
3. The surface does not change based on affine transformation.
4. The polygon net vertex influences the parametric direction in the range ±k/2, ±1/2.

5. The B spline surface reduces to a Bezier surface if the number of polygon net vertices is equal to
the order of basis in that direction and if there exists no interior knot values.
8.1.3 Curves

A continuous map from a one-dimensional space to an n dimensional space is known as a curve in
mathematical terms. It is made up of a number of continuous points. The main property of a curve is
that every point in a curve has a neighbor. The infinite and closed curves do not have neighbors. (End
points).
Curves can be described in three different ways.
Implicit –these representations define the set of points on a curve by providing them with a procedure
to check if a point is positioned on the curve or not. It is defined by an implicit function of the form,

F(x, y) =0
It is a scalar function.

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