Bezier Talk: Difference between revisions
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* French automotive | * French automotive | ||
* now everywhere - font, CAD | * now everywhere - font, CAD | ||
* | * Cubic curve | ||
* intuitive | * intuitive | ||
* start and end are explicit | |||
* stays inside polygon | * stays inside polygon | ||
* many ways to define | * many ways to define | ||
| Line 23: | Line 24: | ||
* exponent delta | * exponent delta | ||
* can render to polynomial | * 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 == | == Blossom Subsection == | ||
| Line 31: | Line 41: | ||
* C5 is not guaranteed solvable as polynomial | * C5 is not guaranteed solvable as polynomial | ||
* Can use subsections for divide and conquer | * Can use subsections for divide and conquer | ||
== Higher dimensions == | == Higher dimensions == | ||
| Line 45: | Line 50: | ||
* 3D 4x4 camera transform: P/w | * 3D 4x4 camera transform: P/w | ||
* Transformation of control points is the same as transformation of the curve | * 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 | |||
Latest revision as of 19:19, 18 October 2019
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