Transcript Useful math for graphics and games   Lots of math used in graphics and games Various concepts in this course: aliasing  coordinate systems 

Useful math for graphics and games


Lots of math used in graphics and games
Various concepts in this course:
aliasing
 coordinate systems
 homogeneous coordinates
 matrices
 quaternions


vectors


quantity expressing magnitude and direction
often written with respect to a basis
e.g., basis vectors i, j, k
p = ai + bj + ck


bj
j
i
ai

length of vector == magnitude
p = ai + bj + ck
|p| = sqrt(a*a + b*b + c*c)

Normalized vector: magnitude is 1




Perpendicular to surface
Specified at vertices in usual geometric models


“How do you know if something is a vector?”
In code – look at the type!


XNA: Vector2, Vector3, Vector4
In math (or on slides) – may not have types
→

Has an arrow: x

in bold (sometimes) ax + y


very hard to see, I hate this notation
use domain knowledge

F=ma: which quantities are vectors?





Often you want to extract components of a
vector
If written explicitly, p = ai + bj + ck, can refer to
pieces by symbol
If not…
Index like arrays: p[0]
Subscript to show what piece: px

Special symbol for a unit vector (length 1): ^

so if we have ŵ, by definition |ŵ| = 1

Used sometimes in addition to arrow,
sometimes alone




Add components
p = ai + bj + ck
q = si + tj + uk
p + q = (a + s) i + (b + t) j + (c + u) k


p – q = p + (-q)
vector inverse defined:





q = si + tj + uk
-q = (-s)i + (-t)j + (-u)k
p = ai + bj + ck
q = si + tj + uk
p – q = (a – s) i + (b – t) j + (c – u) k

Dot product


magnitude of projection of one vector onto the other
Cross product


only defined for 3-vectors!
vector perpendicular to arguments, with magnitude
equal to area of the spanned parallelogram

p = ai + bj + ck
q = si + tj + uk
p ∙ q = a*s + b*t + c*u
scalar quantity

p ∙ q = |p| |q| cos θpq




Basic use: projection of one vector onto another

Lighting:


Collisions:


classic diffuse shading: N∙L
amount of velocity in arbitrary direction
Vector reflection: R = V – 2 V ∙ N ( N/|N|)



p = ai + bj + ck
q = si + tj + uk
p x q = (b*u – c*t)i + (c*s – a*u)j + (a*t – b*s)k



new vector perpendicular to both starting vectors
p x q = |p| |q| sin θpq n
Conventionally, right handed coordinate
system used

Obtain surface normal for triangle




use two vectors in plane, cross product is
perpendicular
also note, can find winding order (front or back
facing)
Calculate area of triangle
In physics, cross product appears regularly


torque = F x d
F = q v x B for charged particle in magnetic field