May 19th, 2018
Math for Game Programmers – Talks from GDC – Notes
GDC 2015 – Math for Game Programmers: Fast and Funky 1D Nonlinear Transformations
Presenter: Squirrel Eiserloh
Youtube link to presentation
Notes:
- Other names used for these concepts in other fields:
- Easing functions
- Filter functions
- Lerping functions
- Tweening functions
- Implicit vs. Parametric Equations
- Implicit: Ex. x^2 + y^2 = R^2
- Parametric: Ex. Px = R * cos(2*pi*t); Py = R * sin(2*pi*t)
- Parametric Equations use a single variable (usually a float) as input
- Opportunities to Use Parametric Transformations
- Anywhere you have a single float to change
- Anywhere that could be expressed as a single float
- Anytime you use time
- The two most important number ranges in mathematics and computer science are:
- 0 to 1: Good for fractions and percentages
- -1 to 1: Good for deviations from some original value
- These functions are normalized
- Input of 0 has output of 0, and input of 1 has output of 1
- The in-between values are those that vary function to function
- Types of functions
- Smooth Start: exponential functions with various orders
- Ex: t^2
- Smooth Stop: gives opposite feel of smooth start
- Ex: 1 – (1-t)^2
- Mix functions
- Crossfade: mix but “blend weight” is the t parameter itself
- Scale: multiply something by t; can also multiply functions together
- Bezier Curves
Other suggested talks:
Juice it or Lose it – by: Martin Jonasson and Petri PurhoThe art of screen shake – by: jan willem Nijman vlambeer
Math for Game Programmers – Talks from GDC – Notes
GDC 2013 – Math for Game Programmers: Interaction With 3D Geometry
Presenter: Stan Melax
Youtube link to presentation
Notes:
- Basic Vector Math
- Dot product
- Cross product
- Outer product
- Jacobian
- Determinants
- Geometry Building Blocks
- Traingles and planes
- In games, triangles and planes need to know above and below
- Triangles use normals
- Planes use Ax + By + Cz + D == 0
- Ray-Triangle Intersections
- Ray-Mesh Intersection
- Could check against all triangles, but there are ways to remove some for efficiency
- Need to check if you’ve hit the nearest triangle
- Mesh must be intact, can have issues if there are holes
- Convex Mesh
- Neighboring face normals tilt away
- Volume bounded by number of planes
- Every vertex lies at/below every other face
- Convex used because it makes a lot of tests/calculations simpler
- If point lies under every plane, it is inside
- Can move rays as they intersect triangles
- Convex Hulls
- Techniques for converting meshes into convex meshes
- There are two main techniques:
- Expand Outward – start with small mesh and expand out until all vertices are accounted for
- Reduce Inward – start with a large mesh and shrink it down to fit all vertices
- Convex Decomposition: breaking down complex shapes into separate convex meshes
- Objects in Motion – Spatial Properties
- Find the center of triangles, then the center of tetrahedrons
- Find the area of triangles and the volumes of tetrahedrons
- These properties can be used for basic accurate physics simulations of objects
- Time Integration without Numerical Drift
- Forward Euler update versus Runge Kutta update
- Forward Euler applies derivative at every step, which can give inaccurate results in too large of time steps
- Runge Kutta looks at multiple time steps and their derivatives and takes a weighted average to make steps more consistent
- Soft Body Meshes
- Connect points/vertices of mesh with spring-like connections
- Two main techniques:
- Kinematic: can have issues with stressed state oscillating as it will not come to rest, forces continue to act
- Dynamic: has better resting stressed states that will settle down
- How to learn more
- Practice tinkering with your own physics engine code
- Write gjk algorithm
- Help understand physics engines and their limitations
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