Lecture 21: Physics Simulation

# Creación de Videojuegos

### Physics Simulation

---

# Why Physics?

* Physics is used for a wide variety of purposes in video games 
  
  * On a basic level, games often mirror real(-ish) environments, and players expect "real" physics 
  
  * Even when the "physics" of a game are less involved, many of the same concepts are used 
  
  * For example: Collision detection
  
  * But today we will be talking about masses and forces
  
---

# Physics simulation

* Real-world physics happens in an (arguably) continuous environment 
  
  * Computers are not very good with that 
  
  * Our physics simulations are just approximations to the actual physics 
  
  * Important: Numerically stable, and fast
  
  * Sometimes you can "cheat"
  
---

# Newton's laws of motion

* First law: Objects in motion stay in motion, objects at rest stay at rest, unless a force acts on them 
  
  * Second law: The sum of forces on an object is equal to its mass times its acceleration 
  
  * Third law: Every force has an equal and opposite force 
  
  * Bonus: The law of universal gravitation: Every two bodies attract each other with a force equal to the product of their masses divided by the square of their distance
  
While these laws break at extreme scales (very small objects or very fast movement), in most games we will not encounter these edge-cases.

---

# Newton's laws of motion

$$
\sum \vec{F} = 0 \Leftrightarrow \frac{\operatorname{d}{v}}{\operatorname{d}{t}} = 0
$$

$$ 
\vec{F} = m \cdot \vec{a}
$$

$$
\vec{F_A} = -\vec{F_B}
$$

$$
\vec{F_G} = G \cdot \frac{m_1 m_2}{r^2}
$$

---

# How are these laws useful?

* We have a few objects in our scene 
  
  * None of them should move, unless a force acts on them 
  
  * If a force acts on an object, it is accelerated depending on its mass 
  
  * If two objects bump into each other, the sum of the force vectors on impact is zero
  
  * All objects theoretically pull each other object towards them

---

# Point Mass

* Let's start simple: We have an object that only has a mass, and no size 
  
  * That means it can't rotate
  
  * Also, collisions are easy to determine
  
  * Our point masses will have a mass, position (vector) and velocity (vector)
  
---

# Physics equations

$$
\vec{s} = \int \vec{v} \cdot \operatorname{d}t
$$

$$
\vec{v} = \int \vec{a} \cdot \operatorname{d}t
$$

$$ 
\vec{a} = \frac{\vec{F}}{m}
$$

---

# Discretized Physics

We update our point mass positions in discrete time steps (e.g. every 20ms)

$$
\vec{s} = \sum \vec{v} \cdot \Delta t
$$

$$
\vec{v} = \sum \vec{a} \cdot \Delta t
$$

$$ 
\vec{a} = \frac{\vec{F}}{m}
$$

---

# Discretized Physics

```C#
acceleration = force/mass;

velocity = velocity + acceleration * Time.deltaTime;

transform.position = transform.position + 
                     velocity * Time.deltaTime;

```

---

# Discretized Physics

---

# Rigid Body Physics

* Usually we want our objects to have a size, though
  
  * A rigid body has a volume 
  
  * We can (mostly) use the same math as for a point mass with the same mass, located at the center of mass of the rigid body 
  
  * The center of mass is the geometric center if the object has uniform density (for games that is usually good enough)
  
  * Exceptions: Rotations and collisions

---

# Angular Velocity

* Similar to a forward (movement) velocity, a rigid body can have an angular (rotational) velocity 
  
  * The *force* in this case is called *torque*
  
  * The *mass* is called *moment of inertia*, and describes how hard it is to get an object to rotate
  
$$
\tau = I \alpha
$$

---

# Applying Torque

---

# Torque in 3 Dimensions

* In 3 dimensions the moment of inertia is a 3x3 matrix 
  
  * This matrix describes how hard it is to rotate the object around each axis 
  
  * For example

- A tightrope walker is holding a long pole 
      - The pole is easy to rotate around its core 
      - It helps with balancing, because it is harder to rotate around its length

---

# Collisions

* If two point masses collide, their velocities are converted into potential energy 
  
  * This potential energy is then released as a repulsion force 
  
  * The angle of their movement relative to the normal at the impact location determines their new movement 
  
<img src="/CI-2700/assets/img/elastic.gif" width="60%"/>

---

# Collision energy loss

* In real collisions, the objects do not keep all energy as kinetic energy 
  
  * Some energy is released as heat and sound
  
  * The *coefficient of restitution* determines how much of the incoming velocities is preserved 
  
  * A value of 0 would mean the maximum amount of energy is lost, the two objects would "stick together"
  
  * A value of 1 means a perfectly elastic collision, in which all kinetic energy is preserved

---

# Friction

* When we talk about friction we usually mean "dry friction": Two solid objects rubbing against each other 
 
 * Friction is a force that opposes movement 
 
 * There are no (practical) precise mathematical formulas for frictions
 
 * Instead we use approximations
 
 * Friction is directly dependent on the mass of the objects (or rather, their normal force)
 
 * It is independent of the sliding velocity 
 
---

# The Coulomb Model of Friction

$$
F_f \le \mu \cdot F_n
$$

Friction is less than or equal to the friction coefficient times the normal force.

The friction coefficient defines how much friction two materials have with each other.

The normal force defines how much surface pressure the objects have with each other.

Why "less than or equal to?"

Friction can not move an object backwards.

---

# Static and Kinetic Friction

* It is actually usually harder to get an object to slide across a surface than to keep moving 
  
  * *Static* friction describes how hard it is to get moving
  
  * *Kinetic* friction describes how much a moving object affected
  
  * Both use the same equation, but different constants
  
  * For example: Aluminum and steel have a static friction coefficient of 0.61, and a kinetic friction coefficient of 0.47
  
---

# Drag

* Another common type of friction is air resistance, or drag 
  
  * Unlike dry friction, drag is dependent on the velocity of the object 
  
  * Drag increases quadratically with the speed 
  
  * Additionally, the drag increases with a larger cross section area of the object

---

# Gravity

In the beginning we related the acceleration of an object to its mass and the forces acting on it.

$$ 
\vec{a} = \frac{\vec{F}}{m}
$$

One such force is gravity.

---

# Gravity

* Remember from earlier: The law of universal gravitation: Every two bodies attract each other with a force equal to the product of their masses divided by the square of their distance
  
  * Of course, most of the time these attractions don't matter 
  
  * Most gravitational forces are insignificant next to the one from earth 
  
  * So: Instead of modeling gravity physically correct, just add a force that goes "down"
  
  * Unless your game needs something else
  
---

# Other forces

* Understanding the basics behind physics allows us to develop games that violate them in targeted ways
 
  * Physics can make for very exciting game mechanics 
  
  * In addition to the forces we discussed, it's possible to include others, such as explosions, or magnetism 
  
  * Sometimes just playing with gravity can be fun
  
---

# Celestial dynamics

(0rbitalis)

---

# Playing with gravity

(Prey (2006))

---

# Other forces

(Guildwars 2)
  
---

# Numerical Stability

* One problem with our simplified model of integration is numerical stability 
  
  * If our timesteps are not measured accurately, or there are floating point rounding errors, we may add more to the velocity than expected 
  
  * This problem compounds as the simulation continues
  
  * Ideally, you will have a low, constant physics update rate 
  
  * Problems happen if your simulation takes longer than the time between updates

---

# Physics puzzle

---

# Story time

## From the Quake 3 Source Code

---

# Inverse Square Root

* For Steering Behaviors we often use normalized vectors 
  
  * Turns out normalizing vectors is something that is also used often in graphics (normals, for example)
  
  * How do you normalize a vector? You divide by its length 
  
  * How do you calculate the length? It's the square root of the sum of the squares of its components
  
$$
v_n = \frac{v}{\sqrt{v_x^2 + v_y^2 + v_z^2}}
$$
   
--

As it turns out, 1/sqrt(x) can be calculated "quickly".

---

# Fast Inverse Square Root

(Unedited from the Quake 3 source code)

```C
float Q_rsqrt( float number )
{
  long i;
  float x2, y;
  const float threehalfs = 1.5F;
  x2 = number * 0.5F;
  y  = number;
  i  = * ( long * ) &y; // evil floating point bit level hacking
  i  = 0x5f3759df - ( i >> 1 );  // what the fuck? 
  y  = * ( float * ) &i;
  y  = y * ( threehalfs - ( x2 * y * y ) );  
      // 1st iteration
//	y  = y * ( threehalfs - ( x2 * y * y ) );   
      // 2nd iteration, this can be removed
  return y;
}
```

---

# Fast Inverse Square Root

* Using an integer as a float is approximately the same as calculating the logarithm
  
  * Conversely, casting back to float basically performs an exponentiation
  
  * The line `y  = y * ( threehalfs - ( x2 * y * y ) );` is an iteration of Newton's algorithm on f(y) = 1/y^2 - 2*x2
  
  * And the weird line? It basically divides the number by 2, and performs some scaling and cleanup
  
$$
\log(\frac{1}{\sqrt{x}}) = -\frac{1}{2} \log(\frac{1}{x})
$$
  
---

# References
  
   * [Video Game Physics](https://www.toptal.com/game/video-game-physics-part-i-an-introduction-to-rigid-body-dynamics)
   
   * [List of Moments of Inertia](https://en.wikipedia.org/wiki/List_of_moments_of_inertia)
   
   * [Learn Game Physics](https://www.gamedesigning.org/learn/game-physics/)
   
   * [Elastic Collision in 3D](http://exploratoria.github.io/exhibits/mechanics/elastic-collisions-in-3d/)
   
   * [Stable Rigid-body Physics](https://www.gamasutra.com/features/gdcarchive/2001/rhodes_paper.pdf)
   
   * [Super Planet Crash: Physics Based Game](http://www.stefanom.org/spc/)
   
   * [Understanding Quake's Fast Inverse Square Root](https://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/)