csce 552 spring 2009 math, physics and collision detection by jijun tang
TRANSCRIPT
CSCE 552 Spring 2009
Math, Physics and Collision Detection
By Jijun Tang
Game Entities
What are game entities? Basically anything in a game world that can be interacted
with More precisely, a self-contained piece of logical interactive
content Only things we will interact with should become game entities
Organization Simple list Multiple databases Logical tree Spatial database
Identification and Communication
Identification Strings Pointers Unique IDs or handles
Communication Simplest method is function calls Many games use a full messaging system Need to be careful about passing and allocating
messages
Applied Trigonometry
Trigonometric functions Defined using right triangle
x
yh
Visualize Vectors
The length represents the magnitude The arrowhead indicates the direction Multiplying a vector by a scalar
changes the arrow’s length
V
2V
–V
Vectors Add and Subtraction
Two vectors V and W are added by placing the beginning of W at the end of V
Subtraction reverses the second vector
V
WV + W
V
W
V
V – W–W
Matrix Representation
The entry of a matrix M in the i-th row and j-th column is denoted Mij
For example,
Invertible
An n n matrix M is invertible if there exists another matrix G such that
The inverse of M is denoted M-1
1 0 0
0 1 0
0 0 1
n
MG GM I
Inverse of 4x4
1 1 1 111 12 13
1 1 1 1 1 121 22 23
1
1 1 1 131 32 33
1 0 0 0 1
x
y
z
R R R
R R R
R R R
R T
R R T R TM
R T
0
Transformations
Calculations are often carried out in many different coordinate systems
We must be able to transform information from one coordinate system to another easily
Matrix multiplication allows us to do this
Illustration
R S T
Suppose that the coordinate axes in one coordinate system correspond to the directions R, S, and T in another
Then we transform a vector V to the RST system as follows
2D Example
θ
2D Rotations
Rotation Around Arbitrary VectorRotation about an arbitrary vector Counterclockwise rotation about an arbitrary vector (lx,ly,lz) normalised so that
by an angle α is given by a matrix
where
Transformation matrix
We transform back to the original system by inverting the matrix:
Often, the matrix’s inverse is equal to its transpose—such a matrix is called orthogonal
Translation
A 3 3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed
We must add a translation component D to move the origin:
Homogeneous coordinates
Four-dimensional space Combines 3 3 matrix and translation
into one 4 4 matrix
Direction Vector and Point
V is now a four-dimensional vector The w-coordinate of V determines whether
V is a point or a direction vector If w = 0, then V is a direction vector and the
fourth column of the transformation matrix has no effect
If w 0, then V is a point and the fourth column of the matrix translates the origin
Normally, w = 1 for points
Transformations
The three-dimensional counterpart of a four-dimensional homogeneous vector V is given by
Scaling a homogeneous vector thus has no effect on its actual 3D value
Transformations
Transformation matrices are often the result of combining several simple transformations Translations Scales Rotations
Transformations are combined by multiplying their matrices together
Transformation Steps
Translation
Translation matrix
Translates the origin by the vector T
translate
1 0 0
0 1 0
0 0 1
0 0 0 1
x
y
z
T
T
T
M
Scale
Scale matrix
Scales coordinate axes by a, b, and c If a = b = c, the scale is uniform
scale
0 0 0
0 0 0
0 0 0
0 0 0 1
a
b
c
M
Rotation (Z)
Rotation matrix
Rotates points about the z-axis through the angle
-rotate
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
z
M
Rotations (X, Y)
Similar matrices for rotations about x, y
-rotate
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
x
M
-rotate
cos 0 sin 0
0 1 0 0
sin 0 cos 0
0 0 0 1
y
M
Transforming Normal Vectors
Normal vectors are transformed differently than do ordinary points and directions
A normal vector represents the direction pointing out of a surface
A normal vector is perpendicular to the tangent plane
If a matrix M transforms points from one coordinate system to another, then normal vectors must be transformed by (M-1)T
Orderings
Orderings
Orderings of different type is important A rotation followed by a translation is different from a translation followed by a rotation
Orderings of the same type does not matter
Geometry -- Lines
A line in 3D space is represented by
S is a point on the line, and V is the direction along which the line runs
Any point P on the line corresponds to a value of the parameter t
Two lines are parallel if their direction vectors are parallel
t t P S V
Plane
A plane in 3D space can be defined by a normal direction N and a point P
Other points in the plane satisfy
PQ
N
Plane Equation
A plane equation is commonly written
A, B, and C are the components of the normal direction N, and D is given by
for any point P in the plane
Properties of Plane
A plane is often represented by the 4D vector (A, B, C, D)
If a 4D homogeneous point P lies in the plane, then (A, B, C, D) P = 0
If a point does not lie in the plane, then the dot product tells us which side of the plane the point lies on
Distance from Point to a Line
Distance d from a point P to a lineS + t V
P
VS
d
Distance from Point to a Line
Use Pythagorean theorem:
Taking square root,
If V is unit length, then V 2 = 1
Line and Plane Intersection
Let P(t) = S + t V be the line Let L = (N, D) be the plane We want to find t such that L P(t) = 0
Careful, S has w-coordinate of 1, and V has w-coordinate of 0
x x y y z z w
x x y y z z
L S L S L S Lt
L V L V L V
L S
L V
Formular
If L V = 0, the line is parallel to the plane and no intersection occurs
Otherwise, the point of intersection is
t
L S
P S VL V
Real-time Physics in Game at Runtime:
Enables the emergent behavior that provides player a richer game experience
Potential to provide full cost savings to developer/publisher
Difficult May require significant upgrade of game engine May require significant update of asset creation pipelines May require special training for modelers, animators, and
level designers Licensing an existing engine may significantly
increase third party middleware costs
Engines
Commercial Game Dynamics SDK (Havok.com) Renderware Physics (renderware.com) NovodeX SDK (novodex.com)
Free Open Dynamic Engine (ODE) (ode.org) Tokamak Game Physics SDK (tokamakphysics.
com) Newton Game Dynamics SDK (newtondynamics.
com)
Particle Physics
What is a Particle? A sphere of finite radius with a perfectly smooth,
frictionless surface Experiences no rotational motion (or assume the
sphere has no size) Particle Kinematics
Defines the basic properties of particle motion Position, Velocity, Acceleration
Location of Particle in World Space SI Units: meters (m)
Changes over time when object moves
Particle Position
zyx ppp ,,p
Particle Velocity and Acceleration
Velocity (SI units: m/s) First time derivative of position:
Acceleration (SI units: m/s2) First time derivative of velocity Second time derivative of position
)()()(
lim)(0
tdt
d
t
tttt
tp
ppV
)()()(2
2
tdt
dt
dt
dt pVa
Newton’s 2nd Law of Motion
Paraphrased – “An object’s change in velocity is proportional to an applied force”
The Classic Equation:
m = mass (SI units: kilograms, kg) F(t) = force (SI units: Newtons)
tmt aF
What is Physics Simulation?
The Cycle of Motion: Force, F(t), causes acceleration Acceleration, a(t), causes a change in velocity Velocity, V(t) causes a change in position
Physics Simulation:Solving variations of the above equations over time to emulate the cycle of motion
Concrete Example: Target Practice
F = w eig ht = m gTarget
Projectile LaunchPosition, pinit
Choose Vinit to Hit a Stationary Target ptarget is the stationary target location We would like to choose the initial velocity, Vinit, required to hi
t the target at some future time, thit. Here is our equation of motion at time thit:
Target Practice
22
1inithitinithitinitinittarget tttt gVpp
Equation Problem
Solution in general is a bit tedious to derive…
Infinite number of solutions! Hint: Specify the magnitude of Vinit, sol
ve for its direction
969.31
Example 1
Vinit = 25 m/sValue of Radicand of tan equation:Launch angle : 19.4 deg or 70.6 deg
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
0.00 20.00 40.00 60.00
Horizontal Position (m)
Ver
tica
l P
osi
tio
n (
m) Projectile Launch
Position
Target Position
Trajectory 1 - HighAngle, Slow Arrival
Trajectory 2 - LowAngle, Fast Arrival
Finite Difference Methods-I
The Explicit Euler Integrator:
Properties of object are stored in a state vector, S Use the above integrator equation to incrementally update S over
time as game progresses Must keep track of prior value of S in order to compute the new For Explicit Euler, one choice of state and state derivative for particle:
)( 2
derivative statestateprior statenew
tOtdt
dtttt
SSS
pVS ,m VFS ,dtd
Finite Difference Methods-II
The Verlet Integrator:
Must store state at two prior time steps, S(t) and S(t-t) Uses second derivative of state instead of the first Valid for constant time step only (as shown above) For Verlet, choice of state and state derivative for a particle:
pS aFS mdtd /22
derivative state
2
22
2 stateprior 1 stateprior state new
)(2
t
dt
dtttttt SSSS
Errors
Exact
Euler
Generalized Rigid Bodies
Key Differences from Particles Not necessarily spherical in shape Position, p, represents object’s center-of-mass location Surface may not be perfectly smooth and friction forces
may be present Experience rotational motion in addition to translational
(position only) motion
Center of Mass
worldX
worldZ
objectX
objectZ
Additional forces
Linear Spring Viscous Damping Aerodynamic Drag Friction …
Linear Springs
dllkF restspring )(
Viscous Damping
ddVVcF epepdamping ))(( 12
Aerodynamic Drag
S: projected front area
CD: drag coefficient
Friction
Collision Detection and Resolution
What is Collision Detection
A fundamental problem in computer games, computer animation, physically-based modeling, geometric modeling, and robotics.
Including algorithms: To check for collision, i.e. intersection, of
two given objects To calculate trajectories, impact times and
impact points in a physical simulation.
Collision Detection
Complicated for two reasons Geometry is typically very complex, potentially
requiring expensive testing Naïve solution is O(n2) time complexity, since
every object can potentially collide with every other object
Two basic techniques Overlap testing: Detects whether a collision has
already occurred Intersection testing: Predicts whether a collision
will occur in the future
Overlap Testing (a posteriori)
Overlap testing: Detects whether a collision has already occurred, sometime is referred as a posteriori
Facts Most common technique used in games Exhibits more error than intersection testing
Concept For every (small) simulation step, test every pair
of objects to see if they overlap Easy for simple volumes like spheres, harder for
polygonal models
Overlap Testing Results
Useful results of detected collision Pairs of objects will have collision Time of collision to take place Collision normal vector
Collision time calculated by moving object back in time until right before collision Bisection is an effective technique
Bisect Testing: collision detected
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Bisect Testing: Iteration I
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Bisect Testing: Iteration II
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Bisect Testing: Iteration III
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Bisect Testing: Iteration IV
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Bisect Testing: Iteration V
B B
t1
t0.375
t0.25
B
t0
I te r a tio n 1F o r w ar d 1 /2
I te r a tio n 2Bac k w ar d 1 /4
I te r a tio n 3F o r w ar d 1 /8
I te r a tio n 4F o r w ar d 1 /1 6
I te r a tio n 5Bac k w ar d 1 /3 2
I n itia l O v er lapT es t
t0.5t0.4375 t0.40625
BB B
A
A
A
AA A
Time right before the collision
Overlap Testing: Limitations
Fails with objects that move too fast Thin glass vs. bulltes Unlikely to catch time slice during overlap
t0t -1 t1 t2
b u lle t
w in d o w
Solution for This Limitation
Speed of the fastest object multiplies the time step should be smaller than the smallest objects in the scene
Possible solutions Design constraint on speed of objects: ha
rd to apply without affecting the play Reduce simulation step size: too expensi
ve
Intersection Testing (a priori)
Predict future collisions When predicted:
Move simulation to time of collision Resolve collision Simulate remaining time step
Intersection Testing:Swept Geometry
Extrude geometry in direction of movement Swept sphere turns into a “capsule” shape
t0
t1
Intersection Testing:Sphere-Sphere Collision
Q 1
Q 2
P 1
P 2
P
Q
t= 0
t= 0
t= 1
t= 1
t
,
2
2222
B
rrΑBt
QP
BAΒΑ .QQPPB
QPA
1212
11
d
Special Cases
No collision:
B2 = 0: both objects are stationary, or they are traveling at parallel
When will collision occur?
02222 QP rrΑBBA
02222 QP rrΑBBA
Intersection Testing:When to Collide
Smallest distance ever separating two spheres:
If
there is a collision
2
222
BAd
BA
22QP rrd
Intersection Testing:Limitations
Issue with networked games Future predictions rely on exact state of
world at present time Due to packet latency, current state not
always coherent Assumes constant velocity and zero
acceleration over simulation step Has implications for physics model and
choice of integrator
Dealing with Complexity
Two issues1. Complex geometry must be simplified
2. Reduce number of object pair tests
Simplified Geometry
Approximate complex objects with simpler geometry, like this ellipsoid or bounding boxes
Minkowski Sum
By taking the Minkowski Sum of two complex volumes and creating a new volume, overlap can be found by testing if a single point is within the new volume
Minkowski Sum
Y}B and :{ XABAYX
X Y =YX X Y =
Using Minkowski Sum
t0
t1
t0
t1
Bounding Volumes
Bounding volume is a simple geometric shape Completely encapsulates object If no collision with bounding volume, no
more testing is required Common bounding volumes
Sphere Box
Box Bounding Volumes
Ax is - Alig n ed Bo u n d in g Bo x O r ien ted Bo u n d in g Bo x
More Examples
Using Bounding Box in Game
Complex objects can have multiple bounding boxes Human object can have one big bounding box
for the whole body Human object can have one bounding box per
limb, head, etc Bounding box can be hierarchical:
Test the big first if possible collision, test the smaller ones
Reduce Number of Detections
O(n) Time Complexity can be achieved.
One solution is to partition space
Achieving O(n) Time Complexity
Another solution is the plane sweep algorithm
Requires (re-)sorting in x (y) coordinate
C
B
R
A
x
y
A 0 A 1 R 0 B0 R 1 C 0 C 1B1
B0
B1A 1
A 0
R 1
R 0
C 1
C 0