2522437 fisika lecture 5 basic physics

8/3/2019 2522437 Fisika Lecture 5 Basic Physics

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John E. Laird and Sugih Jamin

September 17, 2007

Based on The Physics of the Game, Chapter 13 of Teach

Yourself Game Programming in 21 Days,

pp. 681-715


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Some games dont need any physics

ames ase on t e rea wor s ou oo rea st c,

meaning realistic action and reaction

sliding through a turn in a racecar, sports games, flight simulation, etc.

Running and jumping off the edge of a cliff

Two types of physics: Elastic, rigid-body physics, F = ma, e.g., pong

- , , ,whip, chain, hair, volcanoes, liquid, boomerang


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Approximate real-world physics

We dont want just the equations

Assume fixed discrete simulation constant time step

Must account for actual time passed for variable simulation

Assumptions: 2D physics, usually easy to generalize to 3D (addz)

g o es no e orma on Will just worry about center of mass

Not accurate for all physical effects

Constant time step


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Modeling the movement of objects with velocity

Assume distance unit is in pixels

Position at time tfor an ob ect movin at velocitv, from starting positionx0:

x(t) = x0 + vx t 0 y

Incremental computation per frame, assumingconstant time ste and no acceleration: vx and vy constants, pre-compute

x += vx, y += vy

vyv: velocity

vx

(x0, y0)


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Acceleration (a): change in velocity per unit time

Acceleration

e oc y

Approximate


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= = x x y y

Variable acceleration: use table looku based on other factors:

acceleration = acceleration_value(gear, speed, pedal_pressure)

Cheat a bit: acceleration = acceleration_value(gear, speed) * pedal_pressure*x

ay = sin (v) * acceleration

-


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Gravity is a force between two objects: Force F = G (m1m2)/ D

G = 6.67 x 10-11Nm2kg-2

mi: the mass of the two objects

= s ance e ween e wo o ec s

So both objects have same force applied to them

F=ma --> a=F/m

On earth, assume mass of earth is so large it doesnt

move, andD is constant

ssume un orm acce era on Position of falling object at time t:

x(t) = x0 y(t) = y0 + 1/2 * 9.8 m/s

2 * t2

Incrementally,y += gravity (normalized to frame rate)


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Gravity

Inf uences ot o ies

Can have two bodies orbit each other

Onl si nificant for lar e mass ob ects

ConsiderN-body problem

What ha ens after ou a l a force to an ob ect?

What happens when you shoot a missile from a moving

object?

What types of controls do you expect to have on a

s ace shi ?

What about a flying game?


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Objects represented by their center of mass, not

u e p ys e e s

Center of mass (xc, yc) for a polygon with n vertices:

xc = ximi/mi, i = 0 .. n

yc = yimi/mi, i = 0 .. n

For sprites, put center of mass where pixels are densest


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Conversion of kinetic energy into heat

friction

m = mass, g = 9.8 m/s2

,= frictional coefficient = amount of force to maintain a constant speed

mass(m)Fpush Ffriction

Factual = Fpush - Ffrictionm*g

are u a r c on oesn cause your o ec o move ac war Consider inclined plane

Usually two frictional forces Static friction when at rest (ve ocity = 0). No movement un ess overcome.

Kinetic friction when moving (k

< s)


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Non-linear acceleration

es ng r c on > ro ng r c on

Rolling friction < sliding friction

What controls do ou ex ect to have for a racin ame? Turning requires forward motion!

a a ou o er ypes o rac ng games Boat? Hovercraft?


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Forces

vi = vi cos()

W: wind

W: wind resistance

viy = vi sin()

Reaches apex at t = vi sin()/g,

g: gravityr

vi = initial velocity

hits ground atx = vix * viy g

With wind:

m: mass of projectile

x = vix

y += viy

: angle of inclination

vix += Wrx

v += W + normalizedy y


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Start with lots of point objects (1-4 pixels)

Initialize with random velocities based on velocity ofob ect ex lodin

Apply gravity

Transform color intensity as a function of time

Destroy objects upon collision or after fixed time

Can a vapor tra erent co or, et me, w n


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Modeling liquid (Shrek,

in ing emo

Movement of clothing

Movement of hair

(Monster Inc.) Fire/Explosion effects

Reverse Kinematics


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Havok, AGEIA PhysX, Tokamak, etc.

trengt s Do all of the physics for you as a package

ea nesses Can be slow when there are many objects (use PPU?)

Ma have trouble with small vs. bi ob ect interactions

Have trouble with boundary cases

Source: AGEIA


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Steps of analysis for different types of collisions

,

Two circles/spheres Rigid bodies

e orma e

Model the simplest - dont build a general engine


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Detect that a collision has occurred

Determine the time of the collision So can back up to point of collision

Determine where the objects were at time of collision

Determine the collision angle off the collision normal

Determine the velocity vectors after collision


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Simplest case

-

Assume circle hitting an immovable barrier

Detect that a collision occurred

Reformulate as a point about to hit a bigger wall

If vertical and horizontal walls, simple test of x, y

r


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What if more complex background: pinball?

For comp ex surfaces, pre-compute an fi an array wit

collision points (and surface normals)


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Determine the time of collision (tc):

x1,y1

xh

c = i h- 1 2- 1

ti = initial timet= time increment collision normal

Determine where the objects are when they touch = - - * -

2, 2

c 1 1 2 c i

Determine the collision angle against collision normal Collision normal is the surface normal of the wall in this case

Compute angle of line using (x1-xh) and (y1-yc)


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Determine the velocity vectors after collision

=

angle equal and opposite off the surface normal If surface is co-linear with thex- ory-axes:

er ca - c ange s gn o x ve oc y

Horizontal - change sign ofy velocity

Corner - change sign of both

Use t - tc to calculate new position from collision point

Determine chan es in rotation None!

Is t s wort t Depen s on spee o s mu at on,


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- Another important special case

Good step for your games

Many techniques developed here can be

Assume elastic collisions: Conservation of momentum

Conservation of kinetic energy

Non-elastic collision converts kinetic

energy into heat and/or mechanicaldeformations


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If the distance between two circles is less than the sum of

Trick: avoid square root in computing distance! Instead of checking (r1+r2) > D, whereD = sqrt((x1-x2)

2 + (y1-y2)2)

ec r1 r2 x1-x2 y1-y2

x1, y1

x2, y2

r21

Unfortunately, this is still O(N2) comparisons,Nnumber of

objects


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With non-circles, gets more complex and more

expens ve or eac pa r-w se compar son

Use bounding circles/spheres and check for overlap Pretty cheap

Not great for thin objects


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General approach:

O servations: co isions are rare

Most of the time, objects are not colliding

Use various filters to remove as many objects as possible

from the comparison set


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Avoid most of the calculations by using a grid:

ze o ce = ameter o ggest o ect

Test objects in cells adjacent to objects center Can be com uted usin mods of ob ects coordinates:

bin sort

Linear in number of objects


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Alternative if many different sizes

Ce size can e ar itrary

E.g., twice size of average object

Must determine all the cells the object touches

Works for non-circles also


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- Determine the time of the collision

Interpolate based on old and newpositions of objects

Determine where objects are when

they touch

Determine the collision normal

through the colliding intersection collision normal

collision surface


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- Determine the velocity: assume elastic, no

r ct on, ea on co s on

Conservation of Momentum (mass * velocity): m1v1 + m2v2 = m1v1 + m2v2

Conservation of Ener Kinetic Ener : m1v1

2 + m2v22 = m1v1

2 + m2v22

Final Velocities v1 = (2m2v2 + v1(m1-m2))/(m1+m2)

v2 = (2m1v1 + v2(m1-m2))/(m1+m2) at equa mass, m1 = m2 What if m2 is infinite mass?


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ir l - ir l lli i n R nFor non-head on collision, but still no friction:

Velocity change: Maintain conservation of momentum

collision surface


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Round-off error in floating point arithmetic can throw off

computat on Careful with divides

spec a y w o ec s o very eren masses


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For simple collisions, dont do the math

Two i entica a s swap ve ocities

For collisions between dissimilar objects rea e a co s on ma r x

paddle

ball

side

brick

bottom

2522437 fisika lecture 5 basic physics

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