Polygonal Curves

Intuitive methods to draw curves : Polygonal Curves

Sungbae Kim
(All equations are made using m i m e T e X)

Introduction

When designers try to draw cuves, they want to draw and control them as they want. In this lecture, I show how programmers can implement tools to generate curves by intuitive methods, which are uniform cubic b-spline, 4-point curve and Jarek's curve. In addition, I provide demo programs to enhance your understading.

Uniform Cubic B-spline applet
4-point curve applet
Jarek's curve applet

Uniform Cubic B-spline

Polyloop

A Polyloop is defined by a cyclic sequence of vertices and represents like below :
$\left\{v_i\middle|0\leq i \leq n-1\right\}$

Refinement View applet

Refinement is to insert a new vertex in each edge to a existing polyloop.

Fig. 2 By repeating refinement, we can get sufficient samples. View applet

Smoothing view applet

Smoothing is to move the old vertices halfway towards the average of their neighbors.

Fig. 3 By smoothing, vertices move towards the average of their neighbors.

Smoothing algorithm is like this:

  For each i
    vector $L_{i}$
    $L_{i}=\frac{v_i v_{i-1} + v_i v_{i+1}}4$
  End For

  For each i
     $v_i = v_i + L_i$
  End For

By repeating smoothing, a polyloop collapses.

Fig. 4 Repeating smoothing. View applet

Uniform Cubic B-spline View applet

By repeating refinement and smoothing, we can get a uniform cubic B-spline from a polygon.

Fig. 5 Repeating refinement & smoothing. View applet

Uniform Cubic B-spline is $C^2$ continous and non-interpolating, and converses quickly.

4-point curve

Bulging

Bulging is to move the new vertices by one-quarter away from the average of their new second-degree neighbors.

Fig. 6 How to bulge

4-point Curve View applet

By repeating refinement and bulging instead of smoothing,, we can get a 4-point curve from a polyloop.

Fig. 7 Repeating refinement & bulging. View applet

4-point Curve is $C^1$ continuous and interpolating.

Jarek's curve

Jarek's tweak

Jarek's tweak is to move the old vertices by half of the smoothing and the new vertices by half of the bulging.

Fig. 8 How to Jarek's tweak

Jarek's curve View applet

By repeating refinement and Jarek's tweak, we can get a Jarek's curve.

Fig. 9 Repeating refinement & Jarek's tweak. View applet

Jarek's curve is a close approximatin to the original polygon than either of these two.

Reference

Jarek Rossignac, Education-Driven Research in CAD, CAD, vol. 37, 2004
Jarek Rossignac, Polyloop Curves for Computer Graphics lecture note