3 advanced topics in animation presented with JOGL2 tủ tài liệu bách khoa

Advanced topicsin animation kinematic animation: time interpolation of degrees of freedom dynamic animation: time integration of degrees of freedom dynamic animation: all-pairs n-body pr

Advanced topics

in animation

kinematic animation: time interpolation of degrees of freedom dynamic animation: time integration of degrees of freedom dynamic animation: all-pairs n-body problem and gravitation

(for practical introduction to JOGL 2.0)

Advanced Animation and Rendering Techniques – Theory and Practice

Alan Watt, Mark Watt; Addison-Wesley, ACM press

Michael Thomas Flanagan's Java Scientific and Numerical Library

http://www.ee.ucl.ac.uk/~mflanaga/java/index.html

http://www.ee.ucl.ac.uk/~mflanaga/java/CubicSpline.html (for cubic spline interpolation)

http://farside.ph.utexas.edu/teaching/329/lectures/node35.html for Runge-Kutta method

http://www.ast.cam.ac.uk/~sverre/web/pages/nbody.htm

for n-body problem

http://www.scholarpedia.org/article/N-body_simulations_(gravitational)

kinematic animation: time interpolation of degrees of freedom

cubic spline interpolation:

install Flanagan's scientific library:

♦ put flanagan.jar into folder C:\Program Files\Java\jdk1.6.0_23\jre\lib\ext

♦ add to system variable CLASSPATH (already exists, was created for JOGL)

♦ Computer  Properties  Advanced system settings  Environment Variables

♦ edit system variable CLASSPATH (already exists, was created for JOGL)

 add at the end of it:

;C:\Program Files\Java\jdk1.6.0_23\jre\lib\ext\flanagan.jar

import statement:

import flanagan.interpolation.*;

cubic spline with Nk control points ( tk , xk ) :

double[] xk = new double[Nk];

double[] tk = new double[Nk];

CubicSpline csx = new CubicSpline(tk , xk);

interpolation of function x(t) :

double x = csx.interpolate(t);

for every control point k , csx.interpolate(tk[k]) yields value xk[k]

example: interpolated curve with interactive selection and modification of control points

.

import flanagan.interpolation.*;

public class P

{ public static void main(String[] arg)

{ Gui gui = new Gui(); } }

class Gui

{

JFrame f; DrawingPanel p;

JPanel ps; JSlider skx , sky , skz , skt; JLabel lkx , lky , lkz , lkt;

int Nk = 8; int ks = 0; // ks = k of selected key

double[] xk = new double[Nk] , yk = new double[Nk] , zk = new double[Nk] , tk = new double[Nk];

GLU glu; GLUquadric quad;

float[] RED = new float[] { 1.0f , 0.0f , 0.0f , 1.0f } , GREEN = new float[] { 0.0f , 1.0f , 0.0f , 1.0f }

, WHITE = new float[] { 1.0f , 1.0f , 1.0f , 1.0f };

, GLLightingFunc.GL_AMBIENT_AND_DIFFUSE , (k == ks ? RED : GREEN) , 0);

gl.glPushMatrix(); gl.glTranslatef((float)xk[k] , (float)yk[k] , (float)zk[k]);

glu.gluSphere(quad , 2.0f , 10 , 10);

gl.glPopMatrix();

}

//// - INTERPOLATED POINTS

gl.glMaterialfv(GL.GL_FRONT_AND_BACK

, GLLightingFunc.GL_AMBIENT_AND_DIFFUSE , WHITE , 0);

CubicSpline csx = new CubicSpline(tk , xk)

, csy = new CubicSpline(tk , yk)

Gui()

{

f = new JFrame(); f.setFocusable(true); f.setVisible(true);

p = new DrawingPanel(); f.getContentPane().add(p , BorderLayout.CENTER);

f.addMouseListener(new MouseAdapter()

{ public void mouseClicked(MouseEvent e)

{ int bt = e.getButton();

switch (bt)

{ case MouseEvent.BUTTON1: ks ; if (ks == -1) ks = Nk - 1; break;

case MouseEvent.BUTTON3: ks++; if (ks == Nk) ks = 0; break; }

skx = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(skx);

lkx = new JLabel(" x of selected key"); ps.add(lkx);

sky = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(sky);

lky = new JLabel(" y of selected key"); ps.add(lky);

skz = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(skz);

lkz = new JLabel(" z of selected key"); ps.add(lkz);

skt = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(skt);

lkt = new JLabel(" t of selected key"); ps.add(lkt);

skx.addChangeListener( { xk[ks] = -L/2+L*skx.getValue()/1000; f.repaint(); } } ); sky.addChangeListener( { yk[ks] = -L/2+L*sky.getValue()/1000; f.repaint(); } } ); skz.addChangeListener( { zk[ks] = -L/2+L*skz.getValue()/1000; f.repaint(); } } ); skt.addChangeListener(

example: interpolated trajectory of a ball: interactive creation of key frames over time followed by spline driven animation

.

import flanagan.interpolation.*;

public class P

{ public static void main(String[] arg)

{ Gui gui = new Gui();

boolean perform_anim = false;

synchronized void write_perform_anim(boolean value) { perform_anim = value; }

synchronized boolean read_perform_anim() { return perform_anim; }

long dt_real = System.currentTimeMillis() - t_start;

if (dt_real < dt) try {Thread.sleep(dt - dt_real);} catch(InterruptedException ee){}

else System.out.println("PC too slow; please increase dt"); }

}

class DrawingPanel extends GLJPanel

{

GLU glu; GLUquadric quad;

float[] RED = new float[] { 1.0f , 0.0f , 0.0f , 1.0f } , GREEN = new float[] { 0.0f , 1.0f , 0.0f , 1.0f } , WHITE = new float[] { 1.0f , 1.0f , 1.0f , 1.0f };

, GLLightingFunc.GL_AMBIENT_AND_DIFFUSE , (k == nk - 1 ? RED : GREEN) , 0);

gl.glPushMatrix(); gl.glTranslatef((float)xk[k] , (float)yk[k] , (float)zk[k]);

CubicSpline csx = new CubicSpline(tk , xk)

, csy = new CubicSpline(tk , yk)

Gui()

{

f = new JFrame(); f.setFocusable(true); f.setVisible(true);

p = new DrawingPanel(); f.getContentPane().add(p , BorderLayout.CENTER);

String s = JOptionPane.showInputDialog(f , "number of ctrl pts over " + T + "s ; at least 3");

Nk = Integer.valueOf(s).intValue(); if (Nk<3) { System.out.println("Nk < 3"); System.exit(0); }

nk = 0;

xk = new double[Nk]; yk = new double[Nk]; zk = new double[Nk]; tk = new double[Nk];

ps = new JPanel(); ps.setLayout(new GridLayout(0 , 2));

f.getContentPane().add(ps , BorderLayout.EAST);

skx = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(skx);

lkx = new JLabel(" x of new key"); ps.add(lkx);

sky = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(sky);

lky = new JLabel(" y of new key"); ps.add(lky);

skz = new JSlider(JSlider.HORIZONTAL , 0 , 1000 , 0); ps.add(skz);

lkz = new JLabel(" z of new key"); ps.add(lkz);

skx.addChangeListener(new ChangeListener()

{ public void stateChanged(ChangeEvent e)

{ if (nk > 0) { xk[nk - 1] = L * skx.getValue() / 1000; f.repaint(); } } } );

sky.addChangeListener(new ChangeListener()

{ public void stateChanged(ChangeEvent e)

{ if (nk > 0) { yk[nk - 1] = L * sky.getValue() / 1000; f.repaint(); } } } );

{ public void actionPerformed(ActionEvent e)

{ if (nk < Nk) JOptionPane.showMessageDialog(f , "you have not yet recorded all the keys!"); else write_perform_anim(true); } } );

f.setSize(new Dimension(1000 + 16 , 600 + 38));

}

}

example : angles of an articulated structure: interactive creation of key frames over time followed by spline driven animation

.

import flanagan.interpolation.*;

public class P

{ public static void main(String[] arg)

{ Gui gui = new Gui();

JFrame f; DrawingPanel p; JPanel ps;

JSlider[] ska = new JSlider[Na]; JButton b_new_key , b_perform_anim;

boolean perform_anim = false;

synchronized void write_perform_anim(boolean value) { perform_anim = value; }

synchronized boolean read_perform_anim() { return perform_anim; }

long dt_real = System.currentTimeMillis() - t_start;

if (dt_real < dt) try {Thread.sleep(dt - dt_real);} catch(InterruptedException ee){}

else System.out.println("PC too slow; please increase dt"); }

}

class DrawingPanel extends GLJPanel

float[] ai = new float[Na];

for (int a = 0 ; a < Na ; a++) ai[a] = (float)(ak[a][nk - 1]);

articulated_arm(gl , glu , quad , ai);

}

//// - ANIMATED ARM

if (read_perform_anim())

{

CubicSpline[] csa = new CubicSpline[Na];

for (int a = 0 ; a < Na ; a++) csa[a] = new CubicSpline(tk , ak[a]);

float[] ai = new float[Na];

for (int a = 0 ; a < Na ; a++) ai[a] = (float)(csa[a].interpolate(time));

articulated_arm(gl , glu , quad , ai);

Gui()

{

f = new JFrame(); f.setFocusable(true); f.setVisible(true);

p = new DrawingPanel(); f.getContentPane().add(p , BorderLayout.CENTER);

String s = JOptionPane.showInputDialog(f , "n of ctrl pts over " + T + "s ; at least 3");

Nk = Integer.valueOf(s).intValue(); if (Nk<3) { System.out.println("Nk<3"); System.exit(0); }

nk = 0;

ak = new double[Na][Nk]; tk = new double[Nk];

ps = new JPanel(); ps.setLayout(new GridLayout(0 , 1));

f.getContentPane().add(ps , BorderLayout.EAST);

for (int a = 0 ; a < Na ; a++)

{ ska[a] = new JSlider(JSlider.HORIZONTAL , 0 , 90 , 0); ps.add(ska[a]);

{ public void actionPerformed(ActionEvent e)

{ if (nk < Nk) JOptionPane.showMessageDialog(f , "you have not yet recorded all the keys!"); else write_perform_anim(true); } } );

f.setSize(new Dimension(800 + 16 , 600 + 38));

}

void articulated_arm(GL2 gl , GLU glu , GLUquadric quad , float[] ai)

{

gl.glMaterialfv(GL.GL_FRONT_AND_BACK , GLLightingFunc.GL_AMBIENT_AND_DIFFUSE , new float[] { 1.0f , 1.0f , 0.0f , 1.0f } , 0);

gl.glRotatef(ai[0] , 0.0f , 0.0f , 1.0f); gl.glRotatef( 90 - ai[1] , 1.0f , 0.0f , 0.0f);

dynamic animation: time integration of degrees of freedom

example: balls under the influence of gravity and bouncing off the floor

.

public class P

{ public static void main(String[] arg)

{ Gui gui = new Gui();

this.x = x; this.y = y; this.z = z;

this.vx = vx; this.vy = vy; this.vz = vz; }

}

class Gui

{

JFrame f; DrawingPanel p;

int N; Ball[] ball;

double V = 30.0 , DELTA = 10.0 , ALPHA = 0.9; //// V in m/s; DELTA in degrees; ALPHA <= 1.0 void animation()

{

double delta_t = 0.05; long dt = (long)(delta_t * 1000);

while (true)

{ long t_start = System.currentTimeMillis();

for (int i = 0 ; i < N ; i++)

long dt_real = System.currentTimeMillis() - t_start;

if (dt_real < dt) try {Thread.sleep(dt - dt_real);} catch(InterruptedException ee){}

else System.out.println("PC too slow; please increase dt"); }

float[] RED = new float[] { 1.0f , 0.0f , 0.0f , 1.0f } , GREEN = new float[] { 0.0f , 1.0f , 0.0f , 1.0f }

, BLUE = new float[] { 0.0f , 0.0f , 1.0f , 1.0f }, WHITE = new float[] { 1.0f , 1.0f , 1.0f , 1.0f };

class DrawingPanel extends GLJPanel

Gui()

{

f = new JFrame(); f.setFocusable(true); f.setVisible(true);

p = new DrawingPanel(); f.getContentPane().add(p , BorderLayout.CENTER);

String s = JOptionPane.showInputDialog(f , "number of balls; at least 1");

N = Integer.valueOf(s).intValue(); if (N < 1) { System.out.println("N < 1"); System.exit(0); } ball = new Ball[N];

for (int i = 0 ; i < N ; i++)

{ double v = V * (0.2 * Math.random() + 0.8);

double a = 2 * Math.PI * Math.random();

double b = DELTA * (Math.PI / 180) * Math.random();

ball[i] = new Ball( (i%3==0 ? RED : i%3==1 ? GREEN : BLUE)

dynamic animation: all-pairs n-body problem and gravitation

all-pairs n-body problem:

n bodies fully interacting with each other:

position pi and velocity vi of body i :

time derivative of the system's global state:

i j

i

i i

vz vy vx z y x

v

p pv

pv pv

D

pv pv

dt d

1 n

0

1 n 0

time integration:

global state s(t) is known at time t

Euler method:

inaccurate + numerical instability: errors pile up over time

second-order Runge-Kutta method:

accurate + numerically stable = excellent

fourth-order Runge-Kutta method is often preferred

D

D1=

( ) ( s t D Δt / 2 , t Δt / 2 )

m v

v p

dt

d t

i i

ε

i j

i j

p p

p p G

case study: collision of two galaxies - simulation parameters and initial state:

total mass of the two galaxies = 4×1042 kg

initial diameter of each galaxy H = 1021 m

class encapsulating the simulation parameters:

class Para

{

static double G = 6.67384e-11;

static double M = 4.0e42; // total mass of the two galaxies

static double H = 1.0e21; // diameter of each galaxy

static double EPS = H / 100;

static double DX = H; // distance between the two galaxies over X axis

static double DY = H; // distance between the two galaxies over Y axis

static double V = 350.0 * 1.e3; // velocity between the two galaxies along X axis

static double DT = 1.0e5 * 31536000; // 100,000 years

}

classes describing the state of the n-body system:

PositionVelocity(double x , double y ,double z , double vx , double vy , double vz)

{ this.x = x; this.y = y; this.z = z;

this.vx = vx; this.vy = vy; this.vz = vz; }

initialization of the state:

if (i < n / 2) { gal_x = - Para.DX / 2; gal_y = - Para.DY / 2; gal_z = 0.0;

gal_vx = Para.V / 2; gal_vy = 0.0; gal_vz = 0.0; }

else { gal_x = Para.DX / 2; gal_y = Para.DY / 2; gal_z = 0.0;

gal_vx = - Para.V / 2; gal_vy = 0.0; gal_vz = 0.0; }

m[i] = Para.M / n;

do

{

pv[i] = new PositionVelocity( gal_x + Para.H * ( -0.5 + Math.random())

, gal_y + Para.H * ( -0.5 + Math.random())

, gal_z + Para.H * ( -0.5 + Math.random())

, gal_vx , gal_vy , gal_vz);

h = Math.sqrt( (pv[i].x - gal_x) * (pv[i].x - gal_x)

+ (pv[i].y - gal_y) * (pv[i].y - gal_y)

+ (pv[i].z - gal_z) * (pv[i].z - gal_z));

}

while (h > Para.H / 2);

dpv[i] = new PositionVelocity();

pv_[i] = new PositionVelocity();

}

}

time integration with Runge-Kutta:

pv1[i].x = pv0[i].x + dpv[i].x * k;

pv1[i].y = pv0[i].y + dpv[i].y * k;

pv1[i].z = pv0[i].z + dpv[i].z * k;

pv1[i].vx = pv0[i].vx + dpv[i].vx * k;

pv1[i].vy = pv0[i].vy + dpv[i].vy * k;

pv1[i].vz = pv0[i].vz + dpv[i].vz * k;

}

}

D 1 s(t) + D

1 × ∆t / 2

D 2 s(t) + D 2 × ∆t

time derivatives of positions and velocities:

//// calculate dpv for pv = state.pv or state.pv_

void derivatives_positions_velocities(PositionVelocity[] pv)

{

double pipjx , pipjy , pipjz , d2 , q;

for (int i = 0 ; i < n ; i++)