TD 2 -Bézier curves

Files to download today

• geometry.zip: some classes for manipulating geometric objects (such as points, vectors and transformations in 2D).
• Curve.java: abstract class defining main methods for computing the plot of the curve
  
• DrawCurve.java: main drawing class, allowing to handle mouse events (adding, moving and deleting points).

Java documentation

• Jama documentation;
• Processing documentation;
• TD2 documentation.

class drawCurve

public void setup() {
	size(400,400); // size of the window
	// choose the scheme and curves to draw
	this.scheme=new Bezier(this, this.points);    
}

public void draw() {
	background(220);
	    
	if(points.isEmpty()) return; // no control points
	scheme.plotCurve(1./1000.);
}

2.1 Implementing the de Casteljau algorithm

• write a class Bezier which extends class  Curve.
  
• implement method Point_2 evaluate(double t) which computes the point b(t).
• implement method plotCurve(double dt) which draw the curve in the frame, by evaluating the curve with a given step.
  
• 
  

1. implement method Point_2 recursiveDeCasteljau(int r, int i, double t) which computes the i-th Bézier point (at setp r), for value t, with the recursive definition.
2. implement method Point_2 iterativeDeCasteljau(double t) which computes the Bézier curve with the iterative scheme (in linear space).

You can follow the code below:

public class Bezier extends Curve {

	Transformation_2 transformation;

	public Bezier(DrawCurve frame, LinkedList<Point_2> p) {
		super(frame, p);
		transformation=null;
	}

	public Bezier(DrawCurve frame, LinkedList<Point_2> p, Transformation_2 transformation) {
		super(frame, p);
		this.transformation=transformation;
	}
	
	public Bezier(DrawCurve frame, Point_2[] points, Transformation_2 transformation) {
		super(frame, points);
		this.transformation=transformation;
	}

	...
	...
	...

	public Bezier[] subdivide(double t) {
		Point_2[] controlPolygon=this.points;
		int n=this.points.length-1; // degree and number of edges of the control polygon

		Point_2[] b0=new Point_2[n+1]; // first control polygon
		Point_2[] b1=new Point_2[n+1]; // second control polygon
		Bezier[] result=new Bezier[2]; // the pair of Bezier curves to return as result

		//throw new Error("To be completed: TD3");

		// store and return the pair of new control polygons
		result[0]=new Bezier(this.frame, b0, this.transformation);
		result[1]=new Bezier(this.frame, b1, this.transformation);
		return result;
	}

}

2.2 Compute Bézier curves with the efficient (nested) evaluation

1. implement method Point_2 BernsteinBezier(double t) which computes the Bézier curve with nested evaluation of the Bernstein form of a Bézier curve.

Remark:

What do you observe implementing and comparing these methods?

2.3 Render a Bézier curve with the subdivision method

1. implement method public Bezier[] subdivide(double t) which subdivides the Bézier curve into a couple of Bézier curves, with parameter t. Modifiy the plot method of your class, in order to visualize the corresponding control polygons of the two curves (see the picture on the right).
2. implement method public void subdivisionRendering(int n) which renders the Bézier curve (subdivide n times with parameter t=0.5). Remove comments from file DrawCurve.java in order to plot the curve with the subdivision method.

• modify your class Bezier in order to plot the original curve and the curve after affine transformation.

1. firstly, plot the apply the transformation to all points of b(t)
2. apply the transformation only to the control points, and then compute the curve
   

Suggestion:
 add a transformation variable Transformation_2 transformation  to your class  Bezier.

• repeat your experience testing with projective transformations (bonus).

3.1 Implementing rational de Casteljau algorithm

• complete class RationalBezier which extends class  Curve.
  
• implement method Point_2 evaluate(double t) which computes the point b(t).
• implement method plotCurve(double dt) which draw the curve in the frame, by evaluating the curve with a given step.
  

1. implement method Point_2 rationalDeCasteljau(int r, int i, double t) which computes the Bézier point, at value t, with the recursive definition.

3.2 Test projective invariance of rational Bézier curves

• modify your class RationalBezier in order to plot the original curve and the curve after projective transformation.