Ver10.1. TSBK03 Technics for Advanced Gomputer Games: Physics Ht2010. Sergiy Valyukh, IFM
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1 Ver10.1 TSBK03 Technics for Adanced Gomputer Games: Physics Ht010 Sergiy Valyukh, IFM
2 Outline i. Introduction (physics and its role in game industry) 1. Kinematics, Newton s mechanics. Work, Energy and Power 3. Projectiles 4. Collisions 5. Sports Simulations 6. Cars and Motorcycles TSBK03: Physics, Ht010
3 i. Introduction (physics and its role in game industry) i.1about the course i. Physics as a natural science i.3 Modelling of the physical word in a game TSBK03: Physics, Ht010 3
4 i.1 about the course physics for game programers Purposes To inroduce into basics of physics, in order to model the real world in computer games Sources Webpage of the course (Ingemar R) Ragnemalm, PFNP-SHCWMTS [IR] G. Palmer, Physics for Game Programmers, Apress, 005 Witkin, Baraff, Kass, lectures from Pixar SIGGRAPH 001 Course notes [Pix], Ian Millingston, Game Physics Engine Deelopment, Elseier, 007 C. Hecker, Behind the Screen, D. Eberly, Game physics, Elseier, 004 D. Bourg, Physics for Game Deelopers, O Reilly, 00 TSBK03: Physics, Ht010 4
5 i. Physics as a natural science Classical Physics Mechanics Molecular physics Electromagnetism TSBK03: Physics, Ht010 5
6 i.3 Modelling of the physical word 90% of games applied physical simulations use: 3D objects and 3D scenes Moement Rigid objects Rotation Friction Air and water resistance Graity Collisions and explosions Springy things Waes Criterion for a physical model in a game: If it is looks right on the screen, that s good enough! TSBK03: Physics, Ht010 6
7 i.3 Modelling of the physical word Physics Will Keep Your Games from Looking Fake Adding Physics-Based Realism Is Easier Than You Might Think Adding Physics Won t Affect Game Performance Knowing Some Physics Will Make You a Better Game Programmer TSBK03: Physics, Ht010 7
8 Basic Concepts Systems of Units TSBK03: Physics, Ht010 8
9 Basic Concepts Coordinate Systems and Frames of Reference TSBK03: Physics, Ht010 9
10 Basic Concepts Coordinate Systems and Frames of Reference TSBK03: Physics, Ht010 10
11 TSBK03: Physics, Ht Basic Concepts Scalars and Vectors zk yj xi R k V j V V i V z y x k z z j y y i x x R R ) ( ) ( ) ( Sum of ectors: Magnitude of a ector: z y x R Vector scalar product: z z y y x x R R Vector cross product: k y x y x j x z x z i z y z y R R ) ( ) ( ) (
12 Basic Concepts Matrices Deriaties Differential Equations TSBK03: Physics, Ht010 1
13 TSBK03: Physics, Ht Basic Kinematics X Y X Y t ( o ) t r ) r (t ) ( ) ( ) ( t r dt t d t a acceleration s m ) ( ) ( ) ( t r dt t dr t Instantaneous elocity s m o o aerage t t t r t r ) ( ) ( Aerage elocity s m ) ( ) ( ) ( o t r t r t r Displacement m 1 t t dr s Distance m t t t t o o t t o o o o dt t a t t t r dt t t r t r ) ( ) ( ) ( ) ( ) ( ) (
14 Basic Newtonian Mechanics Newton s three laws of motion Some special types of forces graitational, friction, centripetal, and spring The concept of a force ector Force balances and force diagrams Work Energy Power Rotational Motion Many-Particle Interactions TSBK03: Physics, Ht010 14
15 Basic Newtonian Mechanics Newton s First Law of Motion: Inertia Eery body preseres in its state of rest, or of uniform in a right line, unless it is compelled to change that state by forces impressed thereon. TSBK03: Physics, Ht010 15
16 Basic Newtonian Mechanics Newton s Second Law of Motion: Force, Mass, and Acceleration F ma The alteration of motion is eer proportional to the motie force impressed TSBK03: Physics, Ht010 16
17 Basic Newtonian Mechanics Newton s Third Law of Motion: Equal and Opposite Forces To eery action there is always opposed an equal reaction TSBK03: Physics, Ht010 17
18 Basic Newtonian Mechanics Types of Forces Centripetal Force F m r TSBK03: Physics, Ht010 18
19 Basic Newtonian Mechanics Types of Forces Graitational Force F G M M R 1 R M g G Nm kg 4 11 Nm kg N g kg 6 6,37510 m kg TSBK03: Physics, Ht010 19
20 Basic Newtonian Mechanics Types of Forces Graitational Force Equations of motion Quantity Differential Equation Solution Acceleration None a g Velocity d dt g 0 gt Location d r dt dr dt g 0 gt r r o t 0 1 gt TSBK03: Physics, Ht010 0
21 Basic Newtonian Mechanics Types of Forces Graitational Force Equations of motion for projections X Z g Y Quantity Differential Equation Solution Acceleration None a z g, a x 0, a y 0 Velocity d dt z az g z z0 gt, x, x0 y y0 Location d z dt dz dt a z g z z0 gt 1 z zo z0t gt x xo x0t, y y t o y0, TSBK03: Physics, Ht010 1
22 Basic Newtonian Mechanics Types of Forces Friction F F F N TSBK03: Physics, Ht010
23 Basic Newtonian Mechanics Friction Friction Coefficients for Some Common Surface Interactions TSBK03: Physics, Ht010 3
24 Basic Newtonian Mechanics Springs Hooke s Law F kx Equation of motion: m x kx x x 0 k m Solution: x( t) Asin( t o) TSBK03: Physics, Ht010 4
25 Basic Newtonian Mechanics Force balance TSBK03: Physics, Ht010 5
![[, Nm, J]](/docs-images/91/107519254/images/26-1.jpg "TSBK03:")
26 Work W F r W r r 1 Fdr kgm s [, Nm, J] TSBK03: Physics, Ht010 6
27 Energy Kinetic Energy E k m p m Kinetic energy is the energy of an object due to its motion TSBK03: Physics, Ht010 7
28 Energy Potential Energy E p kx E p mgh Potential energy is the energy of an object due to its location TSBK03: Physics, Ht010 8
29 Energy Other Forms of Energy Thermal energy Chemical energy Nuclear energy Electromagnetic energy TSBK03: Physics, Ht010 9
![Power N W t J [, watt] s Power is defined as the amount](/docs-images/91/107519254/images/30-0.jpg "of work performed per unit time.")
30 Power N W t J [, watt] s Power is defined as the amount of work performed per unit time. TSBK03: Physics, Ht010 30
31 Rotational Motion Definitions Angular position: (t) Angular elocity: ( t) ( t) 1 s Frequency: Tangential elocity: f 1 s Angular acceleration: ( t) ( t) ( t) s ( t) ( t) r ( t) d Tangential acceleration: a( t) ( t) r ( t) dt ( t) r ( t) ( t) ( t) r ( t) m s m s TSBK03: Physics, Ht010 31
32 Rotational Motion Torque Torque: M r F Nm TSBK03: Physics, Ht010 3
33 TSBK03: Physics, Ht Rotational Motion Moment of inertia I [kgm ] ] [kgm V dm r I ] [kgm i m i r i I Kinetic energy I T zz yz xz yz yy xy xz xy xx I I I I I I I I I J Inertia tensor J L M J J L ˆ J M ˆ N k k k k xx z y m I 1 ) ( N k k k k yy z x m I 1 ) ( N k k k k zz y x m I 1 ) ( N k k k k yx xy y x m I I 1 N k k k k zx xz z x m I I 1 N k k k k zy yz z y m I I 1
34 Many-Particle Interactions For an N particles system, in equilibrium F 3 3 P pi i i m i i L i r i p i i r i m i f 3 f 31 f 3 0 f 13 f 1 resultant force 0 F 1 f 1 1 r i F i 0 i i F i resultant moment (torque) 0 TSBK03: Physics, Ht010 34
35 Conseration laws i i m E i i const const P i i const i L i 0 TSBK03: Physics, Ht010 35
36 Projectiles Topics The graity-only model Aerodynamic drag Laminar and turbulent flow Wind effects Spin effects Details on specific types of projectiles including bullets, cannonballs, and arrows TSBK03: Physics, Ht010 36
37 Projectiles The graity-only model Equation of motion: d d r g or dt dt g or Solution: gt 0 1 r r o 0 t gt y y0 gt, x x0 y x y x o o 1 y0t gt x0t,, TSBK03: Physics, Ht010 37
38 Projectiles The graity-only model Golf Game Jaa_Code\Chapter05_Projectile\GolfGame.jaa (from TSBK03: Physics, Ht010 38
39 Projectiles The graity-only model Summary o The only force on the projectile is due to graity, which acts in the ertical. o The motion in the three coordinate directions is independent. o The projectile trajectory is independent of mass and projectile geometry. o The elocity in the x- and y-directions is constant oer the entire trajectory and is equal to the initial elocities in the x- and y-direction. o The shape of the projectile trajectory is a parabola. TSBK03: Physics, Ht010 39
40 Projectiles Aerodynamic drag Basic Concepts F Drag FD, pressure FD, Friction Total drag Pressure drag Friction drag (or skin drag) TSBK03: Physics, Ht010 40
41 Projectiles Aerodynamic drag Drag Coefficient Drag force Drag coefficient Effectie area Velocity Density of the fluid TSBK03: Physics, Ht010 41
42 Projectiles Aerodynamic drag Drag Coefficient o Laminar and Turbulent Flow CD C D (Re) Re L The drag coefficient of a sphere as a function of Reynolds number TSBK03: Physics, Ht010 4
43 Projectiles Aerodynamic drag Drag Coefficient o Laminar and Turbulent Flow CD C D (Re) Laminar and Turbulent Drag Coefficients Re L TSBK03: Physics, Ht010 43
44 Projectiles Aerodynamic drag Drag Coefficient o Altitude Effects on Density Values of Air Density As a Function of Altitude TSBK03: Physics, Ht010 44
45 Projectiles Aerodynamic drag Equations of motion F mg FD r g F D m r r TSBK03: Physics, Ht010 45
46 Projectiles Aerodynamic drag Golf Game Version Jaa_Code\Chapter05_Projectile\GolfGame.jaa (from TSBK03: Physics, Ht010 46
47 Projectiles Aerodynamic drag Summary o Drag force acts in the opposite direction to the elocity. The magnitude of the drag force is proportional to the square of the elocity. o The three components of motion are coupled when drag is taken into account. o The drag force is a function of the projectile geometry. o The acceleration due to drag is inersely proportional to the mass of the projectile. o The drag on an object is proportional to the density of the fluid in which it is traeling. TSBK03: Physics, Ht010 47
48 Projectiles Wind Effect Equations of motion F mg FD r g F D m r r Apparent elocity is the ector sum of the projectile elocity and wind elocity. TSBK03: Physics, Ht010 48
49 Projectiles Wind Effect Golf Game Version 3 The effects of headwind or tailwind on a golf ball trajectory Jaa_Code\Chapter05_Projectile\GolfGame3.jaa (from TSBK03: Physics, Ht010 49
50 Projectiles Wind Effect Summary o The presence of wind changes the apparent elocity seen by the projectile in flight. A headwind will increase the apparent elocity. A tailwind will decrease it. o The wind elocity affects the drag force in all three coordinate directions een if the wind elocities themseles are only in the x- and z-planes. TSBK03: Physics, Ht010 50
51 Projectiles Projectiles Spin Effect A spinning object generates lift Bernoulli s equation: pressure 1 p gh const the fluid density elocity altitude p 1 (h=const) const TSBK03: Physics, Ht010 51
52 Projectiles Spin Effect Magnus force F M C L A The Magnus force lift coefficient Equation of motion For a sphere: For a cylinder: F mg F D C L C L r r C L r g F D m r r r CL m r r TSBK03: Physics, Ht010 5
53 Projectiles Spin Effect Golf Game Version 4 The effect of spin on golf ball flight Jaa_Code\Chapter05_Projectile\GolfGame4.jaa (from TSBK03: Physics, Ht010 53
54 Projectiles Spin Effect Golf Game Version 4 A tilt in the spin axis causes the ball to cure Jaa_Code\Chapter05_Projectile\GolfGame4.jaa (from TSBK03: Physics, Ht010 54
55 Projectiles Spin Effect Summary: o An object gien backspin will generate a lifting force. An object gien topspin will generate a force that will push the object downwards. o The acceleration that results from Magnus force is inersely proportional to mass. A heaier object will experience less acceleration than a similar, lighter object. o The magnitude of Magnus force depends on the geometry. All other things being equal, larger objects will generate a larger Magnus force than will smaller objects. TSBK03: Physics, Ht010 55
56 Projectiles Details on Specific Types of Projectiles Bullets Shadowgraph of.308 Winchester FMJ bullet traeling at approximately 850 m/s (from Bullets usually hae a yaw angle during flight TSBK03: Physics, Ht010 56
57 Projectiles Details on Specific Types of Projectiles Bullets TSBK03: Physics, Ht010 57
58 Collisions Specific topics Linear momentum and impulse Conseration of linear momentum Two-body linear collisions Elastic and inelastic collisions Collision detection Collision response Collisions with Friction TSBK03: Physics, Ht010 58
59 Collisions Linear momentum and impulse The linear momentum Newton s nd law P F m dp dt Conseration of Linear Momentum i P i const TSBK03: Physics, Ht010 59
60 Collisions Elastic Collisions A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. i i i m i m i i Before After TSBK03: Physics, Ht010 60
61 Collisions Inelastic Collisions An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision i i i m Before the collision i m i i After the collision TSBK03: Physics, Ht010 61
62 Collisions Elastic and Inelastic Collisions coefficient of restitution e e 1 Jaa_Code\Chapter06_Collision\SphereCollision.jaa (from TSBK03: Physics, Ht010 6
63 Collisions Collision detection Gilbert-Johnson-Keerthi (GJK) algorithm Minkowski Addition The Minkowski sum of a box and a sphere. TSBK03: Physics, Ht010 63
64 Collisions Collision detection Gilbert-Johnson-Keerthi (GJK) algorithm Minkowski difference (Configuration space obstacle (CSO) ) TSBK03: Physics, Ht010 64
65 Collisions Collision detection Gilbert-Johnson-Keerthi (GJK) algorithm The two shapes for our GJK example. The negated shape -B and the Minkowski sum A (-B). TSBK03: Physics, Ht010 65
66 Collisions Collision detection Gilbert-Johnson-Keerthi (GJK) algorithm The first step of the GJK algorithm, on separate objects (left) and combined (right) TSBK03: Physics, Ht010 66
67 Collisions Collision detection Gilbert-Johnson-Keerthi (GJK) algorithm The first step of the GJK algorithm, on separate objects (left) and combined (right) TSBK03: Physics, Ht010 67
68 TSBK03: Physics, Ht Collisions Collision response r A n r B p A X B X A B (t) X p r A A ) (t X p r B B A A A pa r B B B pb r pb pa rel jn P imp A A A m jn / B B B m jn / n r ji jn r I A A A A A A A 1 1 ˆ ˆ n r ji jn r I B B B B B B B 1 1 ˆ ˆ rel rel n r r n B B B A A A pb pa rel ) ( ) ( n r r n B B B A A A pb pa rel ) ( ) ( n r n r I m n r n r I m n j B B B B A A A A rel ) ˆ / ˆ / ( 1 1 n r n r I r n r I j m j m j B B B A A A B A rel ) ˆ ( ˆ / / 1) ( 1 1 n r n r I r n r I m m j B B B A A A B A rel ) ˆ ( ˆ 1/ 1/ 1) ( 1 1 (1) () (3) (4) (5) (6) (7) (9) (8) (10)
69 Collisions Collisions with Friction When two objects collide obliquely, they will slide against each other for a brief period of time. Fˆ n P b P a The frictional impulse ( ) n n 1 Fˆ I ˆ 11 r 1 Fˆ I ˆ r The frictional impulse acts in the direction normal to the line of action and causes rotations of the objects TSBK03: Physics, Ht010 69
70 Collisions Summary The change in elocity that results from a collision can be characterized by a linear or angular impulse. The post-collision elocities of two objects after a collision can be determined from the principle of conseration of momentum and the coefficient of restitution for the collision. For frictionless collisions, only the elocity in the direction of the line of action of a collision is affected by the collision. The other elocity components normal to the line of action are unchanged. For collisions that inole friction, the resulting frictional impulse reduces the magnitude of the elocity in the direction normal to the line of action and causes the objects to spin. TSBK03: Physics, Ht010 70
71 Sports Simulations Topics to be considered Golf Soccer Basketball Other games TSBK03: Physics, Ht010 71
72 Sports Simulations Golf Golf Ball Specifications A model for simulation includes the mass of the club head, the mass of the ball, the elocity of the club head at impact, and the angle of the impact. cp C lub nn cn Club Club nn Schematic of a club head golf ball collision Ball (1 e) mc mc lub m lub Ball C lub n n TSBK03: Physics, Ht010 7
73 Sports Simulations Golf Friction Effects I m( n n ) r r n Friction between the ball and club face causes the ball to spin n n I 1 mr The friction force does two things: 1) it reduces the relatie elocity between the club and ball, and ) it generates a torque on the ball that causes it to spin. I mr n n 5 n 7r Bx Cx mc m m C B ( 1 e)cos sin 7 mc m m 5 sin cose Bz Cx 7 C B TSBK03: Physics, Ht010 73
74 Sports Simulations Golf Modeling the Golf Ball in Flight F M CL A C L r Force diagram for a golf ball in flight Experimental data C L r Experimental and computed lift coefficients for a standard golf ball TSBK03: Physics, Ht010 74
75 Sports Simulations Golf A blow-up shot results from too much spin on the ball Jaa_Code\Chapter07_Sports\GolfGame.jaa (from TSBK03: Physics, Ht010 75
76 Sports Simulations Soccer Modeling the Soccer Ball in Flight Laminar and Turbulent Drag F D C D A The Reynolds number: L Re The iscosity of air: T T Drag coefficient of a nonspinning soccer ball TSBK03: Physics, Ht010 76
77 Sports Simulations Soccer Modeling the Soccer Ball in Flight Magnus Force F M C L C L r Experimental data Experimental and computed soccer ball lift coefficients TSBK03: Physics, Ht010 77
78 Sports Simulations Soccer Free-Kick Game The Free-Kick Game screen shot Jaa_Code\Chapter07_Sports\FreeKick.jaa (from TSBK03: Physics, Ht010 78
79 Sports Simulations Basketball Equipment Specifications The Radius, Diameter, and Mass of Regulation Men s Basketballs Court Dimensions A schematic of the location of the basket, lane, and free-throw line TSBK03: Physics, Ht010 79
80 Sports Simulations Basketball Equipment Specifications Basket and backboard schematics Basket and Backboard Dimensions TSBK03: Physics, Ht010 80
81 Sports Simulations Basketball Ealuating the Forces on a Basketball in Flight Force and Acceleration Components Acting on a Basketball For a shot to be good, it must trael through the hoop TSBK03: Physics, Ht010 81
82 Sports Simulations Basketball A Free-Throw Game A screen shot of the Free-Throw Game Jaa_Code\Chapter07_Sports\FreeThrow.jaa (from TSBK03: Physics, Ht010 8
83 Sports Simulations Specific of simulation of other games Baseball Football Hockey Tennis TSBK03: Physics, Ht010 83
84 Sports Simulations Summary When a ball (or person for that matter) is in the air, it can be treated as projectile and will be subject to the forces due to graity, aerodynamic drag, wind, and spin. The Magnus force due to spin is ery important for the sports of golf, soccer, and baseball. The magnitude of the force due to spin can be obtained by determining the lift coefficient for the object in question. At times the effects of wind and spin can be ignored, for example, when simulating the flight of a basketball. There are also instances, for example soccer and baseball, when it is probably better for game programming purposes not to try to model the initial collision, but rather to begin the simulation by specifying the post-collision elocity, spin rate, and spin axis of the ball. TSBK03: Physics, Ht010 84
85 Cars and Motorcycles Topics to be considered The basic force diagram of a car Power and Torque TSBK03: Physics, Ht010 85
86 Cars and Motorcycles Basic Force Diagram F g T r w F f F D 0 TSBK03: Physics, Ht010 86
87 Cars and Motorcycles Torque The engine torque is a function of the rate at which the engine is turning oer A typical torque cure TSBK03: Physics, Ht010 87
88 Cars and Motorcycles Power and Torque P e T e e The power cure for the 004 Boxster S TSBK03: Physics, Ht010 88
89 Cars and Motorcycles Gears and Wheel Torque A cross-section of a transmission (Photo courtesy of Daimler-Chrysler). Gears are used to change angular elocity and torque. TSBK03: Physics, Ht010 89
90 Cars and Motorcycles Gears and Wheel Torque Porsche Boxster S Gear Ratios Theoretical Maximum Velocity for Each Gear for the Boxster S TSBK03: Physics, Ht010 90
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