**Linear blend:**-
- Line segment from an affine combination of points

- Line segment from an affine combination of points
**Quadratic blend:**-
- Quadratic segment from an affine combination of line segments

- Quadratic segment from an affine combination of line segments
**Cubic blend:**-
- Cubic segment from an affine combination of quadratic segments

- The pattern should be evident for higher degrees.

- Cubic segment from an affine combination of quadratic segments
**Geometric view (deCasteljau Algorithm):**-
- Join the points by line segments
- Join the
*t*: (1-*t*) points of those line segments by line segments - Repeat as necessary
- The
*t*: (1-*t*) point on the final line segment is a point on the curve - The final line segment is tangent to the curve at
*t*

**Expanding Terms (Basis Polynomials):**-
- The original points appear as coefficients of
*Bernstein polynomials* - The Bernstein polynomials of degree
*n*form a basis for the space of all degree-*n*polynomials

- The original points appear as coefficients of
**Recursive evaluation schemes:**-
- To obtain curve points:
- Start with given points and form succesive, pairwise, affine combinations
- The generated points are the
*deCasteljau points*

- Start with given points and form succesive, pairwise, affine combinations
- To obtain basis polynomials:
- Start with 1 and form successive, pairwise, affine combinations
where when

*r*<0 or*r*>*s*

- Start with 1 and form successive, pairwise, affine combinations

- To obtain curve points:
**Recursive triangle diagrams (upward):**- Computing deCasteljau points
- Each node gets the affine combination of the
two nodes entering from below
- Leaf nodes have the value of their respective points

- Each node gets the sum of the path products entering
from below

- Each node gets the affine combination of the
two nodes entering from below
**Recursive triangle diagrams (downward):**- Computing Bernstein (basis) polynomials
- Each node gets the affine combination of the two nodes
entering from above
- Root node has value 1
- For other nodes, missing entries above count as zero

- Each node gets the sum of the path products entering from above

- Each node gets the affine combination of the two nodes
entering from above

University of Waterloo

Computer Graphics Lab