Math / Physics 101

GAM 376

Robin Burke

Fall 2006

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Lab

Begin math / physics 101

Vectors Coordinate spaces Kinematics

Vector

A multi-dimensional quantityrepresented by a tuple of numbers

We will deal mostly with 2-D vectors<2, 5>

3-D vectors are more mathematically complexbut not fundamentally differentfor AI purposes

Vectors

can represent positions<1,1> is a position 1 unit north and 1

unit east of the origin can also represent velocities

<3,-4> is a speed of 5 units in a northwesterly direction

other quantitiesorientation, acceleration, angle, etc.

Vector operations

Magnitude v = <x1, y1> | v | = magnitude(v) = sqrt(x12+y12) the length of the vector

Adding <x1, y1> + <x2, y2> = <x1+x2, y1+y2>

Scalar multiplication <x1, y1> * z = <x1*z, y1*z> changes only the length

Normalization v / |v| makes a vector of length 1 does not change the orientation of the vector

More vector operations

dot product <x1, y1> ● <x2, y2> =

• x1 * x2 + y1 *y2 scalar quantity related to the angle between vectors cos (angle) = u ● v / | u | | v |

Note if u and v are normalized (length = 1) then u ● v is the cosine of the angle between

them

Example we have two agents: a, b

each has a vector position and a normalized vector orientation

• the direction they are facing turn to face an enemy

agent: Pa, Oa

enemy: Pb, Ob vector from agent to enemy

Pba = Pb-Pa normalize it

compute dot product norm(Pba) ● Oa

compute inverse cosine this is the angle to turn

Pa

Pb

Pb-Pa

cos-1(Oa • norm(Pb-Pa))

Coordinate transformations

We will always want to move back and forth between coordinate systems objects each have their own local coordinate systems related to each other in "world space"

Example NPC

• a character is at a particular position, facing a particular direction

• he defines a coordinate system Door

• the door has its own coordinate system• in that system, the handle is in a particular spot

To make the character reach for the handle• we have to make the two coordinate systems interact

Transformations

We know where the handle is in door space where the agent's hand is in agent space what is the vector in agent space

Set of transformations transform handle location L0 to world space

Lw transform handle location to agent space La

Each transformation is a translation (changing the origin) and a rotation changing the orientation

Affine transformations

Usually coordinate transformations are done through matrix multiplications we multiply a vector by a matrix and get

another vector out The curious thing is

that to do this we have to use vectors and matrices one size larger than our current number of dimensions

you will see <x, y, 1> as the representation of 2-D vector and <x,y,z,1> for 3-D

Example

rotation around the origin by angle θ matrix

multiply by current vector [1, 0] becomes ?

answer x = cos(θ) – sin (θ) + 0 y = sin(θ) + cos(θ) + 0

100

0)cos()sin(

0)sin()cos(

x' = x cos(θ) – y sin(θ) y' = x sin(θ) + y cos(θ)

(1, 2)

θ = -π / 4

(2.12, 0.70)

Translation

Achieved by adding values at the ends of the top two rows

This rotates and translates by (3, -4)

100

4)cos()sin(

3)sin()cos(

Efficiency

Mathematical operations are expensive especially trigonometric ones (inverse cosine) also square root multiplication and division, too

Normalization is particularly expensive square root, squaring and division

We want to consider ways to make our calculations faster especially if we are doing them a lot we can do calculations in "squared space" never have to do square roots

Kinematics

the physics of motion we worry about this a lot in computer

games• moving through space, collisions, etc.

our NPCs have to understand this too in order to not look stupid

Basic kinematics

position a vector in space (point position) for a large object we use a convenient point

• classically the center of mass velocity

the change of position over time• expressed as a vector

pt+1 = pt + vΔt if we define the time interval Δt to be 1 we avoid a multiplication

Basic kinematics II

Acceleration change in velocity over time

• also expressed as a vector

vt+1 = vt + a Δt Pair of difference equations

Numerical methods

We want to run a simulation that shows what happens But now we have a problem

there is a lag of 1 time step before acceleration impacts position

not physically accurate Example

deceleration to stop t = 0, d = 0, v = 10, a = 0 t = 1, d = 10, v = 10, a = -5 t = 2, d = 20, v = 5, a = -5 t = 3, d = 25, v = 0, a = 0

Correct distance traveled Δd = v0 Δt + ½ a Δt 2

Δd = 10 d = 20

We can improve our simulation finer time step (very expensive) improved estimation methods

for example instead of using vt, we can use the average of vt and vt+1

Example t=0, d=0, v=10, a =0 t=1, d=10, v=10, a = -5 t=2, d=17.5, v=5, a=-5 t=3, d=20, v=0, a=0

better we slow down in the first time step correct final answer but only works if acceleration is constant take GAM 350 for better ways

What does this have to do with AI? Imagine a NPC baseball player

he has to decide when to swing that judgment has to be based on an

estimate of the ball's trajectory When NPC take physical actions

we have to know what the parameters of those actions should be

very often physics calculations are required not as detailed as the calculations needed to

run the game

Approximation in Prediction

NPC will frequently need to predict the future where will the player be by the time my rocket gets

there? Assumptions

current velocity stays constant we know the speed of the rocket

Correct way vector algebra lots of square roots and trig functions

Typical approximation use the current distance won't change that much if the rocket is fast

Accuracy?

Accuracy may be overrated for NPC enemies

We want an enjoyable playing experience no warnings sniper attack is realistic but no fun a game has to give the player a chance

Built-in inaccuracy many games have enemies deliberately miss

on the first couple of shots others build random inaccuracy into all

calculations

Force

Netwon's lawF = m*a

In other wordsacceleration is a function of force and

mass Force is also a vector quantity

Example

When a bat hits a ballwe will want to determine the force

imparted to the ball• we could simulate this with the mass of

the bat and its speed• more likely, we would just have a built-in

valuedirection of the force

• we need to know the angle of the bat when it strikes the ball

More complex motions

To handle rotationwe also have to worry about torque

and angular momentum But for the purposes of this class

we will deal only with "particles"game physics is often simplified this

way

Example

rocket-jumping

Wednesday

Finite state machines