Let’s Bounce! or Physics of Collisions Part 1 Games Fundamentals © by Jarek Francik Kingston...

Let’s Bounce!or

Physics of Collisions Part 1

Games Fundamentals© by Jarek Francik

Kingston University, London

2007 - 2013

What’s on the Agenda?

• Collisions: Elastic vs. Inelastic• Essential Physics:

Momentum, Impulse, Kinematic Energy, Conservation Rules

• Perfectly Inelastic Collision• Elastic Collisions

– Collision with a Stationary Object– Collision between Bills

• Coefficient of Restitution• Implementation Remarks

Elastic vs. Inelastic Collisions

• Physicists classify collisions as:– perfectly inelastic:

– perfectly elastic:

Elastic vs. Inelastic Collisions

• Real collisions are neither perfectly elastic nor inelastic: they are somewhere in-between

• Marbles, bills and many other hard objects may be quite well simulated as perfectly elastic collisions

Elastic Collisions

• Equal mass marbles exchange their velocities:

v‘A = vB v’B = vA

Elastic Collisions

• Maths behind the marbles of different mass is a bit more complex

Elastic Collisions

• But hitting a stationary obstacle, again, is very simple:

v’ = -v

Essential Physics

Momentum, Impulse, Energy

• Momentum

m – massv – velocity

• Impulse

• Kinematic Energy

mvp

pJ

2

2mvEK

2nd Newton’s Law

t

pF

pmv

mt

vF

amF

Two equivalent formulas:

F = am F = Dp / t = J / t

J = Ft

this is how it is widely

known today this is what Isaac Newton

originally wrote

Conservation of Momentum

F1 = -F2 (3rd Newton’s law)

Dp1 / t = Dp2 / t

Dp1 = Dp2

Conservation of Energy

In an Isolated System ENERGY is constant

but

may be converted, for example into heat

Only perfectly elastic collisions conserve the Kinematic Energy.In all real collisions part of Kinematic Energy is lost as colliding objects become hot.

Physics of Perfectly Inelastic Collision

Perfectly Inelastic Collision

before the collision:

vmm

mv

vmmvm

pp

21

1

211

'

')(

'

vmmvmp 121 0

')(' 21 vmmp applying momentum conservation:

after the collision:

Physics of Perfectly Elastic Collision

Perfectly Elastic Collision

Perfectly Elastic Collision

?'

?'

''

'

2

1

22112211

v

v

vmvmvmvm

pp

applying momentum conservation:

after the collision:

before the collision:2211 vmvmp

2211 ''' vmvmp

Elastic collision• Two unknown velocities• Momentum equation is not enough, we

need something more!

• Definition:Elastic colision is a colision in whichthe kinematic energy is conserved

constmv

Ek 2

2

Elastic collision

21

111222

21

222111

22112211

222

211

222

211

2)('

2)('

''2

'

2

'

22

mm

vmmmvv

mm

vmmmvv

vmvmvmvm

vmvmvmvm

Elastic collision of equal masses

source: http://en.wikipedia.org/wiki/Elastic_collision

11

21

1122

2222

1

21

21

111222

21

222111

2

22)('

2

22)('

2)('

2)('

vm

mv

mm

vmmmvv

vm

mv

mm

vmmmvv

mmmif

mm

vmmmvv

mm

vmmmvv

Elastic Collisions

• collision with a stationary object:

marbles bouncing from the walls

• bills bouncing from each other :billiard (or snooker) game

Collision with a Stationary Object

x’v

y’vv’

f

Collision with a Stationary Object

xv

yv v

f

f

f

vv

vv

yy

xx

'

'

v

v’

Collision with a Stationary Object

x’v

y’vv’

f

xv

yv v

f

f

f

NOTICE:CHANGE OF SYMBOLS

NOTICE:CHANGE OF SYMBOLS

vv

vv

yy

xx

'

'

Collision with a Stationary Object

v’

v’||

v’

f

vv||

v

f

ff

xv

x’v

yv

y’v

||'||

'

vv

vv

),( vv yxv

),( nn yxn

n

'? v

||','? vv

||,? vv1n

c

1c

n: normal vector

c: collision vector

),(),( nncc xyyxc

n

c

Collision with a Stationary Object

v’

v’||

v’

f

vv||

v

f

ff

xv

x’v

yv

y’v

||'||

'

vv

vv

),( vv yxv

),( nn yxn

'? v

||','? vv

||,? vv1n

1c

n: normal vector

c: collision vector

),(),( nncc xyyxc

Collision with a Stationary Object

)()('

)()('

)(

)(

nvycvyy

nvxcvxx

yyxxcv

yyxxnv

xyyx

ncv

ncv

cvcv

nvnv

ncnc

v’

v’||

v’

f

v

v

f

ff

xv

x’v

yv

y’v

nv||

c

||'||

'

vv

vv

),( vv yxv

),( nn yxn

)( nvnv

)(|| cvcv

)(' nvnvv

)(' |||| cvcvv

)()(''' || nvncvcvvv

),(),( nncc xyyxc

Collision with a Stationary Object

)()('

)()('

)(

)(

nvycvyy

nvxcvxx

yyxxcv

yyxxnv

xyyx

ncv

ncv

cvcv

nvnv

ncnc

||'||

'

vv

vv

),( vv yxv

),( nn yxn

)( nvnv

)(|| cvcv

)(' nvnvv

)(' |||| cvcvv

)()(''' || nvncvcvvv

),(),( nncc xyyxc

Collision with a Stationary Object

)()('

)()('

)(

)(

nvycvyy

nvxcvxx

yyxxcv

yyxxnv

xyyx

ncv

ncv

cvcv

nvnv

ncnc

||

'||

'

vv

vv

),( vv yxv

),( nn yxn

)( nvnv

)(|| cvcv

)(' nvnvv

)(' |||| cvcvv

)()(''' || nvncvcvvv

),(),( nncc xyyxc

double xv = ball.GetXVelocity();double yv = ball.GetYVelocity();double xn =

sin(DEG2RAD(obj.GetRotation()));double yn =

cos(DEG2RAD(obj.GetRotation()));double xc = -yn;double yc = xn;double vdotn = xv * xn + yv * yn;double vdotc = xv * xc + yv * yc;ball.SetVelocity(xc * vdotc - xn *

vdotn, yc * vdotc - yn * vdotn);

Physics of the Billiard/Snooker Game

Billiard Gamet

v

changenovvvv

exchangevvvv

||2||2||1||1

1221

''

!''

c nv

v||

),( 1212 YYXXt

R

tn

2 ),( nn xyc

)( nvnv )(|| cvcv

)()('''

)()('''

21||21||222

12||12||111

cvcnvnvvvvv

cvcnvnvvvvv

)()('

)()('

212

121

cvcnvnv

cvcnvnv

Billiard Game

// distance vector (t) and radiusdouble xt = ball2.GetPrecX() -

ball1.GetPrecX();double yt = ball2.GetPrecY() -

ball1.GetPrecY();double R = ball1.GetWidth();

// normal and collision vectorsdouble xn = xt / 2 / R;double yn = yt / 2 / R;double xc = -yn;double yc = xn;

// velocity vectorsdouble xv1 = ball1.GetXVelocity();double yv1 = -ball1.GetYVelocity();double xv2 = ball2.GetXVelocity();double yv2 = -ball2.GetYVelocity();

// dot productsdouble v1dotn = xv1 * xn + yv1 * yn;double v1dotc = xv1 * xc + yv1 * yc;double v2dotn = xv2 * xn + yv2 * yn;double v2dotc = xv2 * xc + yv2 * yc;

ball1.SetVelocity(xn * v2dotn + xc * v1dotc, -(yn * v2dotn + yc *

v1dotc));ball2.SetVelocity(xn * v1dotn + xc * v2dotc, -(yn * v1dotn + yc *

v2dotc));

)()('

)()('

212

121

cvcnvnv

cvcnvnv

),( 1212 YYXXt

R

tn

2 ),( nn xyc

)( nvnv )(|| cvcv

Billiard GameLife Demo

Coefficient of Restitution

General Case

• Newton’s Law of Restitution

e – coefficient of restitution

Notice that we use relative velocityand relative normal velocity:

nevnv '

nvvnv

vvv

)( 21

21

evv '

Implementation Remarks

1.Ball Sinking

2. Bouncing & Sinking: a simple solution for balls and boxes

Ball Sinking in the Floor

In the real world:

before hit! after

Ball Sinking in the Floor

In the simulated world:

before hit! after

Ball Sinking in the Floor

or even a worse scenario:

before hit! after

WRONGBounce condition may be wrongly detected!

SOLUTION: only bounce when moving DOWN

Ball Sinking in the Floor

General solution:

only apply the bounce if v • n > 0

v

n

Ball Sinking in Another Ball

Ball Sinking in Another Ball

• In the real world:

before hit! after

Ball Sinking in Another Ball

• In the simulated world:

before hit! after

Ball Sinking in Another Ball

• In the nasty case:

#1 #2 #3

In frame #3 the balls still overlap. This may lead to repeated application of the ‘bounce’ effect. The balls may end up ‘glued’!

Ball Sinking in Another Ball

• In the nasty case:

no collision yet

frame #1

Ball Sinking in Another Ball

• In the nasty case:

collision detected

frame #2

Ball Sinking in Another Ball

• In the nasty case:

collision detected

bouncing applied

frame #2

Ball Sinking in Another Ball

• In the nasty case:

collision detected (again!)

frame #3

Ball Sinking in Another Ball

• In the nasty case:

collision detected (again!)

bouncing applied

frame #3

WRONG

Ball Sinking in Another Ball

• Solution:

Take:- t: position vector (of the other ball)- v: velocity vector

if (v ∙ t < 0) don’t bounce!

frame #3t

v

Ball Sinking in Another Ball

• Solution:

frame #3

Take:- t: position vector (of the other ball)- v: velocity vector

if (v ∙ t < 0) don’t bounce!

// distance vector (t) and radiusdouble xt = ball2.GetPrecX() -

ball1.GetPrecX();double yt = ball2.GetPrecY() -

ball1.GetPrecY();double R = ball1.GetWidth();

// normal and collision vectorsdouble xn = xt / 2 / R;double yn = yt / 2 / R;double xc = -yn;double yc = xn;

// velocity vectorsdouble xv1 = ball1.GetXVelocity();double yv1 = -ball1.GetYVelocity();double xv2 = ball2.GetXVelocity();double yv2 = -ball2.GetYVelocity();

// dot productsdouble v1dotn = xv1 * xn + yv1 * yn;double v1dotc = xv1 * xc + yv1 * yc;double v2dotn = xv2 * xn + yv2 * yn;double v2dotc = xv2 * xc + yv2 * yc;

ball1.SetVelocity(xn * v2dotn + xc * v1dotc, -(yn * v2dotn + yc *

v1dotc));ball2.SetVelocity(xn * v1dotn + xc * v2dotc, -(yn * v1dotn + yc *

v2dotc));

t

v

Ball Sinking in Another Ball

• Solution:

frame #3

Take:- t: position vector (of the other ball)- v: velocity vector

if (v ∙ t < 0) don’t bounce!

// distance vector (t) and radiusdouble xt = ball2.GetPrecX() -

ball1.GetPrecX();double yt = ball2.GetPrecY() -

ball1.GetPrecY();double R = ball1.GetWidth();

// normal and collision vectorsdouble xn = xt / 2 / R;double yn = yt / 2 / R;double xc = -yn;double yc = xn;

// velocity vectorsdouble xv1 = ball1.GetXVelocity();double yv1 = -ball1.GetYVelocity();double xv2 = ball2.GetXVelocity();double yv2 = -ball2.GetYVelocity();

// check velocity dot positiondouble vx = xv1 - xv2;double vy = yv1 - yv2;if (vx * xt + vy * yt < 0)

return;

// dot productsdouble v1dotn = xv1 * xn + yv1 * yn;double v1dotc = xv1 * xc + yv1 * yc;double v2dotn = xv2 * xn + yv2 * yn;double v2dotc = xv2 * xc + yv2 * yc;

ball1.SetVelocity(xn * v2dotn + xc * v1dotc, -(yn * v2dotn + yc *

v1dotc));ball2.SetVelocity(xn * v1dotn + xc * v2dotc, -(yn * v1dotn + yc *

v2dotc));

t

v

• Bouncing from boxes and walls is quite common in games – simple and advanced

a scene from the Marble,

a workshop task for today...

Bouncing & Sinking: a simple solution for balls & boxes

• GFC offers an effective hit test function for rectangular sprites against circular shapes, for which position and radius are known:

rect.HitTest(pos, radius);

Bouncing & Sinking: a simple solution for balls & boxes

• Problem:when the ball is overlapping the box it is too late to determine which side it entered (why?)

Bouncing & Sinking: a simple solution for balls & boxes

• Solution:track the previous frame position.

Bouncing & Sinking: a simple solution for balls & boxes

• Example:the ball is hitting the left hand edge of the box if:– there is a hit in frame #n– it was to the left of the box in frame #n-1

Bouncing & Sinking: a simple solution for balls & boxes

• Solution Draft:1. Store in a safe place the position of the ball before the

update. This should be the position after the previous frame, before the current one – (x0, y0)

2. Just after update, check the HitTest. If there is no collision there is no bounce.

3. Now, when we know there is a hit:• Assume there was no hit with (x0, y0)*

• Check on which side of the box the ball was (in the prev frame)• Apply the appropriate bounce• Amend the position of the ball so that to prevent sinking

* this assumption will be satisfied if you prevent sinking!

Bouncing & Sinking: a simple solution for balls & boxes

• Here is the solution:1. Store in a safe place the position of the ball before

the update. This should be the position after the previous frame, before the current one – (x0, y0)

2. Just after update, check the HitTest. If there is no collision there is no bounce.

3. Now, when we know there is a hit:• Assume there was no hit with (x0, y0)*.

• Check on which side of the box the ball was.• Apply the appropriate bounce.• Amend the position of the ball so that to prevent sinking.

* this assumption will be satisfied if you prevent sinking!

let’s have a look at thispoint in greater detail...

Bouncing & Sinking: a simple solution for balls & boxes

Decompose ball’s velocity into vx and vy

• if vx > 0 the ball was travelling rightwardsand we only check the left side of the box

• if vx < 0 the ball was travelling leftwardsand we only check the right side of the box

• if vy > 0 the ball was travelling upwardsand we only check the bottom side of the box

• if vy < 0 the ball was travelling downwardsand we only check the upper side of the box

Bouncing & Sinking: a simple solution for balls & boxes

Decompose ball’s velocity into vx and vy

• if vx > 0 the ball was travelling rightwardsand we only check the left side of the box

• if vx < 0 the ball was travelling rightwardsand we only check the left side of the box

• if vy > 0 the ball was travelling upwardsand we only check the bottom side of the box

• if vy < 0 the ball was travelling downwardsand we only check the upper side of the box

Bouncing & Sinking: a simple solution for balls & boxes

Decompose ball’s velocity into vx and vy

• if vx > 0 the ball was travelling rightwardsand we only check the left side of the box

if (vx > 0 && x0 + r <= pWall->GetLeft()){

vx *= -damp; // bounce with dampingx1 = pWall->GetLeft() - r; // amend the position

// (to prevent sinking)

}// Repeat for remaining three sides...// and, finally, apply all the changestheBall.SetVelocity(vx, vy);theBall.SetPosition(x1, y1);

Bouncing & Sinking: a simple solution for balls & boxes