This post is a little long so I will post it in two parts, the fist one is about NURBS in 2D (Curves) and the next one will be about NURBS in 3D (surfaces)…

Ever since I have been working in Flash I wanted to write one Class that would allow me to work with NURBS curves and surfaces the way I used to work them in 3dsMax, but getting to understand how they work is something somehow difficult because there is some math that needs to be applied in order to make it right.

NURBS is the acronym for Non Uniform Ration B-Splines, Nurbs curves and surfaces represent a precise mathematical representation of a free form curve or surface. All the theory needed to understand them is well explained here, but i´ll make a short introduction of the Nurbs concepts for a better undestanding of the example below.

**B-Splines:**

B-Splines are a generalization of the Bezier curves, these kind of curves use a set of interpolation functions called Basis Funcions , hence the “B” on “B-Splines”, the kind of interpolation of these function has the advantage of allowing local control on the curve with the control points unlike the bezier curves. This means that meanwhile the all the bezier curve is affected by all the control points, the control points of the B-Spline curve only affect one portion of the whole curve. The construction of the basis function can be read here.

**Rational B-Splines:**

Rational B-Splines are the generalization of the B-Splines, the rational factor enables to change the control point influence in the curve using weight factors on each control point, changing the weight of one control point will make the curve get closer to or farther from the control point edited. Using rational b-splines conic sections can be represented exactly.

**Non Uniform Rational B-Spline:**

The non uniform part of the acronym is not another generalization, it is the way rational b-splines start and end on the desired end points. The basis functions needed to interpolate the spline require a set of knots that define the way each basis function interacts with the global interpolation, the creation of the knots can be done in a uniform distribution, using the same distance among them, or using a non uniform distribution that force the interpolation start in the first control point and finish in the last control point. You can see the way the knots affect the resultant curve here.

The example from below has two curves with seven control points, the red one is a Nurbs curve and the blue one is a Bezier curve, the light blue lines represent the convex hull of both curves. You can move the control points to edit both curves and see the differences between them, you can also change the weights of the control points and the degree of the Nurbs curve to see how the degree change the general interpolation.

One of the most interesting things with Nurbs curves and Bezier curves is that for a given set of control points “n” (all with the same weight), the Nurbs curve of degree n-1 is equal as the Bezier curve created with the same control points. The lower the degree of the Nurbs curve the closer the curve will be to the convex hull. Finally if a degree of 1 is defined the curve will be exactly as the convex hull.

**The AS3 Class:**

The actionScript class calculates nurbs points on a curve and nurbs points on a surface. It has three static functions: Nurbs.nurbs(), Nurbs.nurbsCurve() and Nurbs.surface(), the parameters needed for each function depends on the theory for the Nurbs curves and surfaces. The whole class is not here, only the first function that allows to find the points on a Nurbs curve given a defined parameter.

The function Nurbs.nurbs() requieres four parameters:

p… parameter (value between 0 and 1).

controlPoints… vector of vectors 3D that define all the control points in the curve.

degree… degree of the curve interpolation.

order… the order alters the way the knots vector are created (uniform or non-uniform), usually it should not be used.

[as]

package math {

import flash.geom.Vector3D;

/**

* @author miaumiau.cat

*/

public class Nurbs {

//Function that calculates a point on a Nurbs curve…

public static function nurbs(p : Number, controlPoints : Vector.<Vector3D>, degreeParam : uint = 3, orderParam : int = -1) : Vector3D {

var cp : uint = controlPoints.length;

var degree : uint = degreeParam < 1 ? 1 : degreeParam > cp ? cp : degreeParam;

var order : uint = orderParam == -1 ? degree + 1 : orderParam < 0 ? degree + 1 : orderParam > degree + 1 ? degree + 1 : orderParam;

var totalKnots : uint = cp + degree + 1;

var i : uint;

var j : uint;

var baseVector : Vector.<Number> = new Vector.<Number>();

var knotsVector : Vector.<Number> = new Vector.<Number>();

var recurtion : uint;

var output : Vector3D = new Vector3D();

var f : Number;

var g : Number;

var rangoFinal : Number;

var t : Number;

i = 0;

while(i < order) {

knotsVector[i] = 0;

i += 1;

}

i = order;

while(i < cp) {

knotsVector.push(i – order + 1);

i += 1;

}

i = cp;

while(i < totalKnots) {

knotsVector.push(cp – order + 1);

i += 1;

}

var data : Number = 1;

rangoFinal = knotsVector[cp + 1];

t = p * (rangoFinal);

//Generate the Basis Functions…

i = 0;

recurtion = totalKnots – 1;

while(i < recurtion) {

baseVector[i] = (knotsVector[i] <= t && t <= knotsVector[i + 1] && knotsVector[i] < knotsVector[i + 1]) ? 1 : 0;

i += 1;

}

recurtion -= 1;

j = 1;

while(j <= degree) {

i = 0;

while(i < recurtion) {

f = (knotsVector[i + j] – knotsVector[i]);

g = (knotsVector[i + j + 1] – knotsVector[i + 1]);

f = f != 0 ? (t – knotsVector[i]) / f : 0;

g = g != 0 ? (knotsVector[i + j + 1] – t) / g : 0;

baseVector[i] = f * baseVector[i] + g * baseVector[i + 1];

i += 1;

}

if(p == data) {

var salida : String = “”;

for(var u : uint = 0;u < recurtion; u++) {

salida += baseVector[u] + “,”;

}

}

j += 1;

recurtion -= 1;

}

//Calculate the Rational points…

i = 0;

var divider : Number = 0;

while(i < cp) {

output.x += baseVector[i] * controlPoints[i].x * controlPoints[i].w;

output.y += baseVector[i] * controlPoints[i].y * controlPoints[i].w;

output.z += baseVector[i] * controlPoints[i].z * controlPoints[i].w;

divider += baseVector[i] * controlPoints[i].w;

i += 1;

}

output.x /= divider;

output.y /= divider;

output.z /= divider;

return output;

}

}

}

[/as]