MATHEMATICAL FOUNDATIONS Useful math for graphics and games

MATHEMATICAL FOUNDATIONS Useful math for graphics and games

Mathematical Topics � � Lots of math used in graphics and games Various concepts in this course: � aliasing � coordinate systems � homogeneous coordinates � matrices � quaternions � vectors

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

Vector magnitude � length of vector == magnitude p = ai + bj + ck |p| = sqrt(a*a + b*b + c*c) � Normalized vector: magnitude is 1 � �

Aside: Normal vector � � Perpendicular to surface Specified at vertices in usual geometric models

Vector Notation � � “How do you know if something is a vector? ” In code – look at the type! � XNA: � Vector 2, Vector 3, Vector 4 In math (or on slides) – may not have types

Vector Notation → � 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?

Vector Notation � � � 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

Vector Notation � 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

Vector Addition � � Add components p = ai + bj + ck q = si + tj + uk p + q = (a + s) i + (b + t) j + (c + u) k

Vector Subtraction � � 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

Vector Multiplication � 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

Dot product � p = ai + bj + ck q = si + tj + uk p ∙ q = a*s + b*t + c*u scalar quantity � p ∙ q = |p| |q| cos θpq � � �

Dot product applications � Basic use: projection of one vector onto another � Lighting: � classic � diffuse shading: N∙L Collisions: � amount � of velocity in arbitrary direction Vector reflection: R = V – 2 V ∙ N ( N/|N|)

Cross product � � � 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

Cross product applications � 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 =Fxd � F = q v x B for charged particle in magnetic field