Geometry Primer
Lines and rays
 Planes
 Spheres
 Frustums
 Triangles
 Polygon
 Polyhedron

Lines and Rays

Given 2 points, we can parametrically define
a line which passes through these two points
as
 P(t)


= (1-t)P1 + tP2
The line segment between these two points
corresponds to all values of t between zero
and one
A ray is defined parametrically as
 P(t) = Po + tV
 Where t is greater than or equal to zero
 Allowing
negative values of t will give a valid
equation for a line
Planes

Plane equation
 Ax


+ By + Cz + D = 0
A, B, and C are the respective x, y, and z coordinates of a
vector normal to the plane
D = -N.Po



If N is a unit vector, the equation


N is the plane normal vector
Po is an arbitrary point on the plane
d = N.Q + D
Gives the distance from the plane to an arbitrary point Q
If d > 0, Q lies on the positive side of the plane. Otherwise Q
lies on the negative side of the plane.

Use this to test for path-plane collision
A Different Plane Equation

Point – normal form
 N.(P

– Po) = 0
P and Po are arbitrary points on the plane
 Stored
as a point and a normal in many applications,
or calculated on the fly from three points.

Given by equation

 Line

N.Q + D = 0
– Plane intersection
Example
Spheres

Sphere equation
- x0)2 + (y - y0)2 + (z - z0)2 = r2
 r is the radius, and is centered at (x0,y0,z0)
 (x

In an application, store the center point
and the radius.
Frustums


A frustum as a volume of space containing
everything visible in a scene.
A frustum can be defined by a:
 Near
plane (or near plane distance)
 Far plane (or far plane distance)
 Field of view

FOVx or FOVy?
 Aspect

ratio
Alternatively, it can be described by the six
planes which bound it.
Frustum Illus.
Triangle
Polygon
Polyhedron
Convex polyhedrons are also known as
polytopes, and are often used to
approximate non-convex meshes where
speed is an issue.
 Also known as brushes.

Applied Geometry

Collision models for games can be
approximated using fundamental shapes
 E.g.
Use boxes for limbs of characters, 3
spheres for an ant…
Visibility Determination
*Bounding box test
 Bounding sphere test

Bounding Box

Axis aligned, or AABB


Find minimum and maximum points and use these to construct
box
Oriented, or OBB
Find natural axes for points
 Compute dot product of each vertex position with unit vectors R,
S, T of the natural axis and take the minimum and maximum
values
Planes are given by:
<R, -min{Pi.R}>
<-R, max{Pi.R}>

Collision Detection
Sphere – Sphere
 Sphere - Plane
 *AABB – AABB
 *OBB – OBB
 *Box – Plane
 Ray – Plane
 Ray – Triangle

Ray – Triangle Intersection
Example
 Investigate other methods

 Badoul’s
algorithm
 Moller-Haines’ algorithm

www.realtimerendering.com
 D3DX

algorithm
D3DX common source
Usage / Examples
Detecting when a user has clicked an
object
 Determining if, when, and where objects
have collided

 Decide
how to respond to collision
Reading

Chapters 4, 5, 7, 8 in Mathematics for 3D
Game Programming and Computer
Graphics cover the material presented
here