Bezier Talk

Intro

• French automotive
• now everywhere - font, CAD
• Cubic curve
• intuitive
• start and end are explicit
• stays inside polygon
• many ways to define

Vector Math Refresher

• Prefer parametric
• Vector vs Point

As Blossom

• define blossom
• order doesn't matter
• curve is all args equal
• Not efficient - N²
• blossoms will come back

As Sum

• Formula
• binomial refresher
• exponent delta
• can render to polynomial

Derivative

• Degree one less
• Difference of points
• Is a Bezier curve
• Smooth linking of curves

Rasterizing

Rather than computing each point, use divide and conquer. Given the derivative, we can check if the line is a good approximation.

Blossom Subsection

• Can create arbitrary subsection

Line Intersection

• Reason to prefer C4, can be done linearly as polynomial
• C5 is not guaranteed solvable as polynomial
• Can use subsections for divide and conquer

Higher dimensions

• All formulas work in 3D - and higher
• Doesn't have to be spatial, great smoothing for speed, color

Transformation

• 2D review
• 3D 4x4 camera transform: P/w
• Transformation of control points is the same as transformation of the curve

Shortcomings

• Cannot exactly describe a circle, but can get very close quickly
• Cannot exactly describe an "offset" (stroking)

As Building Block

• Piecewise
• Rational
• NURBS