Lecture 19: Physics

# Creación de Videojuegos

### Physics

---

# Review

Given a plane going through the point (1,2,3) with normal (0.71,0,-0.71), determine where a ray starting at (0,0,3) in direction (1,0,0) intersects that plane.

Recall:
$$
t = \frac{(\vec{p} - \vec{o}) \cdot \vec{n}}{\vec{d} \cdot \vec{n}}
$$

---

# Bounding Volumes

---

# Bounding Volumes

* Computing intersections between all pairs of arbitrary objects is expensive
  
  * Idea: Put a bigger object around each object for which intersections are easy to compute 
  
  * Different bounding volumes:
       - Spheres: Easy to compute intersections, but may be too big
       - Boxes: Harder to compute intersections, but can be fitted pretty well
       - Axis-Aligned Boxes: Easy to compute intersections, better fit than spheres 
       
---

# Axis-Aligned Bounding Boxes

To test intersections: Check if x-projections *and* y-projections (and z-projections, in 3D) overlap.

Sort-And-Sweep Algorithm: Sort start- and end-values of each interval, and iterate over them in sorted order. If there are two or more consecutive starts of intervals, the projections overlap.

---

# Hierarchical Representations

* The idea of using larger bounding volumes to quickly eliminate unnecessary tests can be extended 
  
  * For example, objects in one room will never collide with objects in a completely different room 
  
  * Idea: Put multiple objects into a single bounding volume. If two such bounding volumes don't intersect, no object from one of them intersects with any object in the other.
  
  * Start with two objects, and put them into a combined bounding volume, then combine this larger bounding volume with another, etc.
  
  * We get a tree hierarchy of bounding volumes
  
---

# Bounding Volume Tree

---

# Quadtrees/Octrees

* Another idea is to split the space into fixed size parts, say cut each axis in half 
  
  * This gives us 4 (in 2D), or 8 (in 3D) total parts 
  
  * For any part in which there are objects, we can repeat this process
  
  * When any box only contains one object (and is sufficiently small), we can stop splitting
  
  * We can now test if something may intersect any object by checking the cell in which it falls in the tree
  
---

# Octree

---

# Let there be light

---

# The Phong Reflection Model

* The Phong Reflection Model is an empirical model for how lights are reflected off of surfaces 
  
  * Consists of ambient, diffuse and specular components
  
<img src="/CI-2700/assets/img/phongcomponents.png" width="100%"/>

---

# The Phong Reflection Model

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$$
I = k_a \cdot i_a + \sum_m \left(k_d (L_m \cdot N) i_d + k_s (R_m \cdot V)^\alpha i_s \right)
$$

---

# How do we use this information?

* Flat shading: Calculate one value for the center of each triangle, and color the entire triangle that way 
  
  * Gouraud shading: Calculate one value for each vertex of a triangle and interpolate the color in between
  
  * Phong shading: Interpolate normals between the vertices of a triangle, and calculate the color for each pixel
  
<img src="/CI-2700/assets/img/Phong-shading-sample.jpg" width="60%"/><img src="/CI-2700/assets/img/Gouraud_low_anim.gif" width="30%"/>

---

# Note

* Phong reflection *model*: Equation to determine how light affects the color of a point 
    
  * Phong shading/Gouraud shading/Flat shading: Methods to *use* this equation
  
  * Shader: The *implementation*/*program* that performs shading
  
---

# Alternatives: Non-Photorealistic Shading

* Toon/Cel shaders!
  
<img src="/CI-2700/assets/img/unitytoon.png" width="100%"/>

---

# Interpolation

---

# Where do we use Interpolation

* Whenever we have two (or more) values where we want intermediate values, we use interpolation
  
  * Values can be numbers, positions, rotations, colors, etc.
  
  * For example: A platform that moves between two end points
  
  * The simplest case: Linear interpolation
  
  * However, we can generalize!
  
---

# The problem

We have two values: a and b, and we want a function

$$
f: [0,1] \mapsto [a,b]
$$

such that

$$
f(0) = a
$$
$$
f(1) = b
$$

(And values between 0 and 1 are between a and b)

---

# Linear interpolation

One such function, which we have called Linear Interpolation (Lerp) is:

$$
f(t) = (1-t)\cdot a + t \cdot b
$$

But this results in constant speed over time, meaning we start and stop abruptly.

Unless ...

What if we start and stop slowly?

---

# Composing functions

If we had a function

$$
g : [0,1] \mapsto [0,1]
$$

with

$$
g(0) = 0, g(1) = 1
$$

we can compose it with f to get another interpolation function:

$$
f(g(t))
$$

---

# Useful functions

How about a cosine?

<title>Sine_one_period.svg - a nice plot of the sine function</title>
  <desc>
     Sine(sin)-function
     from Wikimedia Commons
     plot-range: 0 to 2pi
     plotted with three different cubic bezier-curves
     the bezier-controll-points are calculated to give a very accurate result.
     symbols in "Computer Modern" (TeX) font embedded
     created with a plain text editor using GNU/Linux

about: http://commons.wikimedia.org/wiki/Image:Sine_one_period.svg
     source: http://commons.wikimedia.org/
     rights: GNU Free Documentation license,
             Creative Commons Attribution ShareAlike license
  </desc>

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But that's not mapping from [0,1] to [0,1]?!

We need to do two things:

* Map the input of the cosine from [0,1] to something else
   * Map the output of the cosine to [0,1]
   
---

# Useful functions

Which area do we want?

Map to the domain:
$$
c(t) = \cos(\pi + t*\pi)
$$

---

# Useful functions

Which area do we want?

But we also need to fix the result:
$$
g(t) = \cos(\pi + t*\pi)/2 + 0.5
$$

---

# Useful functions

Here's another useful one: A sigmoid function

Corrected:
$$
g(t) = \frac{1}{2\cdot(1+e^{-(12\cdot t - 6)})} + 0.5
$$

---

# Comparison

Unity gives you: `Mathf.SmoothStep`

---

# References
  
  * [Unity Documentation on SmoothStep](https://docs.unity3d.com/ScriptReference/Mathf.SmoothStep.html)
  
  * [How to Lerp like a Pro](https://chicounity3d.wordpress.com/2014/05/23/how-to-lerp-like-a-pro/)