Curbe Bezier - Revision history

Csdexter: /* Interpolarea a patru puncte */ Converted to GeShiCodeTag

2007-02-05T13:42:30Z

‎Interpolarea a patru puncte: Converted to GeShiCodeTag

Csdexter: /* Interpolarea a trei puncte */ Converted to GeShiCodeTag

2007-02-05T13:41:02Z

‎Interpolarea a trei puncte: Converted to GeShiCodeTag

Csdexter: /* Desenarea unei curbe Bézier */ Converted to GeShiCodeTag

2007-02-05T13:39:14Z

‎Desenarea unei curbe Bézier: Converted to GeShiCodeTag

Cbidea at 13:06, 4 February 2007

2007-02-04T13:06:41Z

Cbidea: Început desenare

2007-02-04T12:57:39Z

Început desenare

Cbidea: /* Interpolarea a patru puncte */

2006-12-04T13:19:13Z

‎Interpolarea a patru puncte

Cbidea: /* Interpolarea a patru puncte */

2006-12-04T13:08:03Z

‎Interpolarea a patru puncte

Cbidea: /* Interpolarea a patru puncte */

2006-12-04T13:07:11Z

‎Interpolarea a patru puncte

Cbidea: /* Interpolarea a patru puncte */

2006-12-04T11:04:04Z

‎Interpolarea a patru puncte

Cbidea: /* Interpolarea a patru puncte */

2006-12-04T11:03:46Z

‎Interpolarea a patru puncte

@@ Line 338: / Line 338: @@
 În continuare codul sursă Java care implementează interpolarea a patru puncte folosind o curbă Bézier de ordinul trei.
-<code><java/>import java.awt.geom.Point2D;
+<code java>import java.awt.geom.Point2D;
 public class BezierUtils {

@@ Line 250: / Line 250: @@
 În continuare codul sursă Java care implementează interpolarea a trei puncte folosind o curbă Bézier de ordinul doi.
-<code><java/>import java.awt.geom.Point2D;
+<code java>import java.awt.geom.Point2D;
 /**

@@ Line 115: / Line 115: @@
 '''QuadraticBezier.java'''
-<code><java/>import java.awt.Graphics;
+<code java>import java.awt.Graphics;
 import java.awt.Graphics2D;
 import java.awt.RenderingHints;
@@ Line 183: / Line 183: @@
 '''MainFormDesign.java'''
-<code><java />import javax.swing.JFrame;
+<code java>import javax.swing.JFrame;
@@ Line 206: / Line 206: @@
 '''MainForm.java'''
-<code><java />import javax.swing.SwingUtilities;
+<code java>import javax.swing.SwingUtilities;

@@ Line 225: / Line 225: @@
+	}
 }</code>
 ==Probleme de interpolare folosind curbe Bézier==

← Older revision		Revision as of 12:57, 4 February 2007
Line 95:		Line 95:

	==Desenarea unei curbe Bézier==		==Desenarea unei curbe Bézier==
		+	Desenarea unei curbe Bézier nu este o sarcină trivială şi asta nu din cauza calculelor, pe care le-am lămurit până la acest moment ci din cauza afişării curbei astfel încât să aibe un aspect foarte lin.
		+
		+	Algoritmul cel mai simplu de desenare este următorul (pentru curba cubica).
		+
		+	Date de intrare: P1, P2, P3, P4 (P2, P3 - puncte de control)
		+	B03, B13, B23, B33 sunt functii care intorc valorile polinoamelor Bernstein de ordin 3 si indice k (k = 0..3).
		+
		+	<pre>
		+	1. PP <- P1 ;initializam punctul anterior (Previous Point) cu P1
		+	2. IS <- .01 ;setam pasul de incrementare
		+	3. pentru t = IS; t < 1; t <- t + IS executa
		+	4. CP.x = P1.x * B03(t) + P2.x * B13(t) + P3.x * B23(t) + P4.x * B33(t);
		+	5. CP.y = P1.y * B03(t) + P2.y * B13(t) + P3.y * B23(t) + P4.y * B33(t);
		+	6. drawLine(PP.x, PP.y, CP.x, CP.y);
		+	7. PP = CP;
		+	8. end pentru
		+	</pre>
		+	În continuare codul scris în Java pentru pseudocodul anterior.
		+
		+	'''QuadraticBezier.java'''
		+	<code><java/>import java.awt.Graphics;
		+	import java.awt.Graphics2D;
		+	import java.awt.RenderingHints;
		+	import java.awt.geom.Point2D;
		+
		+	import javax.swing.JPanel;
		+
		+	public class QuadraticBezier extends JPanel {
		+	private static final long serialVersionUID = -6991920282065229392L;
		+
		+	public void paint(Graphics g) {
		+	Graphics2D g2 = (Graphics2D)g;
		+	super.paint(g2);
		+
		+	drawCubicBezier(g2, new Point2D.Float(100,100),
		+	new Point2D.Float(130,150),
		+	new Point2D.Float(150,200),
		+	new Point2D.Float(200,130));
		+	}
		+
		+	/**
		+	* Aceasta functie deseneaza o curba Bezier cubica.
		+	* @param g2 Graphics2D
		+	* @param p1 BezPoint 1
		+	* @param p2 Control Point 1
		+	* @param p3 Control Point 2
		+	* @param p4 BezPoint 2
		+	*/
		+	public void drawCubicBezier(Graphics2D g2,
		+	Point2D.Float p1,
		+	Point2D.Float p2,
		+	Point2D.Float p3,
		+	Point2D.Float p4) {
		+	float is = 0.01f; //pasul de incrementare
		+	Point2D.Float prevPoint = p1; //Punctul calculat anterior
		+	Point2D.Float curPoint = new Point2D.Float(0,0); //Punctul curent
		+
		+	for (float t = is; t <= 1; t += is) {
		+	curPoint.x = B03(t)p1.x + B13(t)p2.x + B23(t)p3.x + B33(t)p4.x;
		+	curPoint.y = B03(t)p1.y + B13(t)p2.y + B23(t)p3.y + B33(t)p4.y;
		+	System.out.println(prevPoint.distance(curPoint));
		+	if (prevPoint.distance(curPoint) > 5) {
		+	g2.drawLine((int)prevPoint.x, (int)prevPoint.y, (int)curPoint.x, (int)curPoint.y);
		+	prevPoint.x = curPoint.x;
		+	prevPoint.y = curPoint.y;
		+	}
		+	}
		+	}
		+
		+
		+	private float B03(float t) {
		+	return (float) Math.pow(1-t, 3);
		+	}
		+
		+	private float B13(float t) {
		+	return (float) (3tMath.pow(1-t, 2));
		+	}
		+
		+	private float B23(float t) {
		+	return (3tt*(1-t));
		+	}
		+
		+	private float B33(float t) {
		+	return (float) Math.pow(t,3);
		+	}
		+	}</code>
		+
		+	'''MainFormDesign.java'''
		+	<code><java />import javax.swing.JFrame;
		+
		+
		+	public abstract class MainFormDesign extends JFrame {
		+	private QuadraticBezier quadraticBezier = null;
		+
		+	public MainFormDesign() {
		+	setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
		+	setTitle("Test Bezier");
		+	setSize(400, 400);
		+
		+	setContentPane(getQuadraticBezier());
		+	}
		+
		+	private QuadraticBezier getQuadraticBezier() {
		+	if (null == quadraticBezier) {
		+	quadraticBezier = new QuadraticBezier();
		+	}
		+	return quadraticBezier;
		+	}
		+	}</code>
		+
		+	'''MainForm.java'''
		+	<code><java />import javax.swing.SwingUtilities;
		+
		+
		+	public class MainForm extends MainFormDesign {
		+	private static final long serialVersionUID = 1L;
		+
		+	public MainForm() {
		+	super();
		+	}
		+
		+	public static void main(String[] args) {
		+	SwingUtilities.invokeLater(new Runnable() {
		+	public void run() {
		+	MainForm mf = new MainForm();
		+	mf.setVisible(true);
		+	}
		+	});
		+	}
		+	}</code>

	==Probleme de interpolare folosind curbe Bézier==		==Probleme de interpolare folosind curbe Bézier==

@@ Line 239: / Line 239: @@
 		result[1] = new Point2D.Float();
 		result[2] = new Point2D.Float();
-		result[1].x = ((float) (-(f3(t4) * x5.getX() - f3(t5) * x4.getX() + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.getX() + (f1(t4) * f3(t5) - f3(t4)
+		result[1].x = ((float) (-(f3(t4) * x5.x - f3(t5) * x4.x + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.x + (f1(t4) * f3(t5) - f3(t4)
 				* f1(t5))
-				* x0.getX()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
+				* x0.x) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
-		result[2].x = ((float) ((f2(t4) * x5.getX() - f2(t5) * x4.getX() + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.getX() + (f1(t4) * f2(t5) - f2(t4)
+		result[2].x = ((float) ((f2(t4) * x5.x - f2(t5) * x4.x + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.x + (f1(t4) * f2(t5) - f2(t4)
 				* f1(t5))
-				* x0.getX()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
+				* x0.x) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
-		result[1].y = ((float) (-(f3(t4) * x5.getY() - f3(t5) * x4.getY() + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.getY() + (f1(t4) * f3(t5) - f3(t4)
+		result[1].y = ((float) (-(f3(t4) * x5.y - f3(t5) * x4.y + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.y + (f1(t4) * f3(t5) - f3(t4)
 				* f1(t5))
-				* x0.getY()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
+				* x0.y) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
-		result[2].y = ((float) ((f2(t4) * x5.getY() - f2(t5) * x4.getY() + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.getY() + (f1(t4) * f2(t5) - f2(t4)
+		result[2].y = ((float) ((f2(t4) * x5.y - f2(t5) * x4.y + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.y + (f1(t4) * f2(t5) - f2(t4)
 				* f1(t5))
-				* x0.getY()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
+				* x0.y) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
 		return result;
+	}
 	/**
@@ Line 262: / Line 264: @@
 		return Math.pow((1 - t), 3);
+	}
 	/**
@@ Line 271: / Line 275: @@
 		return 3 * t * Math.pow(1 - t, 2);
+	}
 	/**
@@ Line 280: / Line 286: @@
 		return 3 * Math.pow(t, 2) * (1 - t);
+	}
 	/**

← Older revision		Revision as of 13:08, 4 December 2006
Line 203:		Line 203:

	<math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>		<math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>
		+
		+	În continuare codul sursă Java care implementează interpolarea a patru puncte folosind o curbă Bézier de ordinul trei.

	<code><java/>import java.awt.geom.Point2D;		<code><java/>import java.awt.geom.Point2D;

← Older revision		Revision as of 13:07, 4 December 2006
Line 203:		Line 203:

	<math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>		<math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>
		+
		+	<code><java/>import java.awt.geom.Point2D;
		+
		+	public class BezierUtils {
		+
		+
		+	/**
		+	* Ecuatiile sunt rezolvate in Maxima 0.7.0a folosind urmatoarele
		+	* eqn: x4 = x0f1(t4) + x1f2(t4) + x2f3(t4) + x3f4(t4);
		+	* eqn1: x5 = x0f1(t5) + x1f2(t5) + x2f3(t5) + x3f4(t5);
		+	* algsys([eqn, eqn1], [x1, x2]);
		+	*
		+	* Functiile f1, f2, f3 si f4 sunt polinoamele Bernstein de ordinul 3.
		+	*
		+	* @param x0
		+	* @param x4
		+	* @param x5
		+	* @param x3
		+	* @return
		+	*/
		+	public static Point2D.Float[] getBezierInterpPoints(Point2D.Float x0, Point2D.Float x4, Point2D.Float x5, Point2D.Float x3) {
		+	Point2D.Float[] result = new Point2D.Float[4];
		+	result[0] = new Point2D.Float();
		+	result[0].x = x0.x;
		+	result[0].y = x0.y;
		+	result[3] = new Point2D.Float();
		+	result[3].x = x3.x;
		+	result[3].y = x3.y;
		+
		+	float t4 = (float) (x0.distance(x4) / (x0.distance(x4) + x4.distance(x5) + x5.distance(x3)));
		+	float t5 = (float) ((x0.distance(x4) + x4.distance(x5)) / (x0.distance(x4) + x4.distance(x5) + x5.distance(x3)));
		+
		+	result[1] = new Point2D.Float();
		+	result[2] = new Point2D.Float();
		+	result[1].x = ((float) (-(f3(t4) * x5.getX() - f3(t5) * x4.getX() + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.getX() + (f1(t4) * f3(t5) - f3(t4)
		+	* f1(t5))
		+	* x0.getX()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
		+	result[2].x = ((float) ((f2(t4) * x5.getX() - f2(t5) * x4.getX() + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.getX() + (f1(t4) * f2(t5) - f2(t4)
		+	* f1(t5))
		+	* x0.getX()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
		+	result[1].y = ((float) (-(f3(t4) * x5.getY() - f3(t5) * x4.getY() + (f4(t4) * f3(t5) - f3(t4) * f4(t5)) * x3.getY() + (f1(t4) * f3(t5) - f3(t4)
		+	* f1(t5))
		+	* x0.getY()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
		+	result[2].y = ((float) ((f2(t4) * x5.getY() - f2(t5) * x4.getY() + (f4(t4) * f2(t5) - f2(t4) * f4(t5)) * x3.getY() + (f1(t4) * f2(t5) - f2(t4)
		+	* f1(t5))
		+	* x0.getY()) / (f2(t4) * f3(t5) - f3(t4) * f2(t5))));
		+	return result;
		+	}
		+
		+	/**
		+	* Polinomul Bernstein de indice 0 si de ordin 3
		+	*
		+	* @param t
		+	*/
		+	private static double f1(double t) {
		+	return Math.pow((1 - t), 3);
		+	}
		+
		+	/**
		+	* Polinomul Bernstein de indice 1 si de ordin 3
		+	*
		+	* @param t
		+	*/
		+	private static double f2(double t) {
		+	return 3 * t * Math.pow(1 - t, 2);
		+	}
		+
		+	/**
		+	* Polinomul Bernstein de indice 2 si de ordin 3
		+	*
		+	* @param t
		+	*/
		+	private static double f3(double t) {
		+	return 3 * Math.pow(t, 2) * (1 - t);
		+	}
		+
		+	/**
		+	* Polinomul Bernstein de indice 3 si de ordin 3
		+	*
		+	* @param t
		+	*/
		+	private static double f4(double t) {
		+	return Math.pow(t, 3);
		+	}
		+
		+	public static void main(String[] args) {
		+	Point2D.Float P1 = new Point2D.Float(0,0);
		+	Point2D.Float Pc1 = new Point2D.Float(50,50);
		+	Point2D.Float Pc2 = new Point2D.Float(100,60);
		+	Point2D.Float P4 = new Point2D.Float(100,0);
		+
		+	Point2D.Float[] bezCoords = getBezierInterpPoints(P1, Pc1, Pc2, P4);
		+	System.out.println("Curba Bezier cubica este data de:");
		+	System.out.println(bezCoords[0]);
		+	System.out.println(bezCoords[1]);
		+	System.out.println(bezCoords[2]);
		+	System.out.println(bezCoords[3]);
		+	}
		+	}</code>
	[[Category:Pending]]		[[Category:Pending]]

@@ Line 201: / Line 201: @@
 Rezolvând sistemul se obţine:
-p2=-(b23tc1*pc2-b23tc2*pc1+(b23tc2*b33tc1-b23tc1*b33tc2)*p4+(b03tc1*b23tc2-b03tc2*b23tc1)*p1)/(b13tc1*b23tc2-b13tc2*b23tc1),
 <math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>
 [[Category:Pending]]

← Older revision		Revision as of 11:03, 4 December 2006
Line 200:		Line 200:
	<math>\begin{cases}P_{C1} = B_{0,3}(t_{c1})P_1 + B_{1,3}(t_{c1})P_2 + B_{2,3}(t_{c1})P_3 + B_{3,3}(t_{c1})P_4 \\ P_{C2} = B_{0,3}(t_{c2})P_1 + B_{1,3}(t_{c2})P_2 + B_{2,3}(t_{c2})P_3 + B_{3,3}(t_{c2})P_4\end{cases}</math>		<math>\begin{cases}P_{C1} = B_{0,3}(t_{c1})P_1 + B_{1,3}(t_{c1})P_2 + B_{2,3}(t_{c1})P_3 + B_{3,3}(t_{c1})P_4 \\ P_{C2} = B_{0,3}(t_{c2})P_1 + B_{1,3}(t_{c2})P_2 + B_{2,3}(t_{c2})P_3 + B_{3,3}(t_{c2})P_4\end{cases}</math>

		+	Rezolvând sistemul se obţine:
		+	p2=-(b23tc1pc2-b23tc2pc1+(b23tc2b33tc1-b23tc1b33tc2)p4+(b03tc1b23tc2-b03tc2b23tc1)p1)/(b13tc1b23tc2-b13tc2b23tc1),
		+	p3=(b13tc1pc2-b13tc2pc1+(b13tc2b33tc1-b13tc1b33tc2)p4+(b03tc1b13tc2-b03tc2b13tc1)p1)/(b13tc1b23tc2-b13tc2b23tc1)]]
		+	<math>\begin{cases}P_2 = - \frac{B_{2,3}(t_{c1})P_{C2} - B_{2,3}(t_{c2})P_{C1} +(B_{2,3}(t_{c2})B_{3,3}(t_{c1}) - B_{2,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{2,3}(t_{c2}) - B_{0,3}(t_{c2})B_{2,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})} \\ P_3 = \frac{B_{1,3}(t_{c1})P_{C2} - B_{1,3}(t_{c2})P_{C1} +(B_{1,3}(t_{c2})B_{3,3}(t_{c1}) - B_{1,3}(t_{c1})B_{3,3}(t_{c2}))P_4 + (B_{0,3}(t_{c1})B_{1,3}(t_{c2}) - B_{0,3}(t_{c2})B_{1,3}(t_{c1}))P_1}{B_{1,3}(t_{c1})B_{2,3}(t_{c2}) - B_{1,3}(t_{c2})B_{2,3}(t_{c1})}\end{cases}</math>
	[[Category:Pending]]		[[Category:Pending]]