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 Purho
The 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