info/BezierInfo-2
1
0
Fork 0
mirror of https://github.com/Pomax/BezierInfo-2.git synced 2025-08-31 03:59:58 +02:00
Code Issues Releases Wiki Activity

Edits to canonical section (#191)

Browse Source
This commit is contained in:
Simon Cozens
2019-06-11 01:39:48 +01:00
committed by Pomax
parent eec7f90731
commit 9114a00913
1 changed files with 2 additions and 2 deletions
Download Patch File Download Diff File Expand all files Collapse all files

4
components/sections/canonical/content.en-GB.md
View File

@@ -1,8 +1,8 @@
# Canonical form (for cubic curves)
While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature controlled by more than one control points, it exhibits all kinds of features like loops, cusps, odd colinear features, and up to two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?
While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?
As it so happens, the answer is yes and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper ["A Geometric Characterization of Parametric Cubic curves"](http://graphics.pixar.com/people/derose/publications/CubicClassification/paper.pdf). It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell which features a curve will have. So how does it work?
As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper ["A Geometric Characterization of Parametric Cubic curves"](http://graphics.pixar.com/people/derose/publications/CubicClassification/paper.pdf). It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?
The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown: