1. Joshua Barczak CMSC 435 UMBC
Miscellaneous Math
Joshua Barczak
CMSC 435
UMBC

2. You had better know… Basic Algebra The Quadratic Formula
Geometry and Trigonometry
SOHCAHTOA or however you learned it
Pathagorean theorem
Angles and lines
What “the matrix” is…

3. Today Barrage of math facts useful to graphics Vectors Matrices
Lines/Planes
Triangles
Implicits

4. Vectors N-vector Tuple of N of scalars Arbitrary dimension
2, 3, 4 are ubiquitous in graphics

5. Vectors Vectors are arrows protruding from a point… Direction
Magnitude z x y

6. Points … and points are vectors from the origin Direction from origin
Magnitude=Distance z x y

7. Spherical Coordinates
2 angles + Distance
Wikipedia

8. Vectors Some definitions: Normalization Orthogonal  Perpendicular
Magnitude (Length):
Normalized  Length=1
Normalization
Divide by length

9. Surface Normals Surface normal =
Normalized vector perendicular to some surface
Perpendicular to polygon plane
Perpendicular to curve tangent at some point N N

10. Arithmetic
Commutative
Distributive
Associative

11. Vector Addition u v u+v u v u+v = v+u u v
Put them end to end, then connect the dots…
Order doesn’t matter….

12. Vector Subtraction
Subtracting vectors gives a vector between the two “tips”
xb-xa V
yb-ya A B

13. Point Subtraction
Subtracting points gives a vector from last to first, with first as its “origin”… P1 V P0
y1-y0
x1-x0

14. Vector Scaling au
axu
ayu u xu yu
Scales vector length by a

15. Dot Product q u,v are orthogonal  u,v are perpendicular
cos(theta) = 0  u.v=0

16. Dot Product Implications: Same direction Opposite direction
u.v = |u||v|
Length is sqrt(x.x)
Opposite direction
u.v = -|u||v|
Normalized vectors
Cosine of angle
Same sign, same side… N

17. Dot Product u v’ v
Dot products can be used to project onto other vectors

18. Dot Product More useful facts (a,b,c vectors. x,y scalars)
Just like multiplication…
If you’re queasy about dot products, go home and prove all these….

19. Basis Vectors
All vectors can be written as linear combinations of other vectors

20. Basis Vectors Basis: Orthogonal basis Orthonormal basis
“Linearly independent set of vectors that span a space”
One is not a weighted sum of the others
Orthogonal basis
Basis S.T. all vectors are perpendicular
Orthonormal basis
Orthogonal basis with unit-length vectors

21. Basis Vectors Vectors are always relative to some basis
The “Default” basis vectors are the coordinate axes… B A X+ Y+ D C

22. Basis Vectors
….and a point is relative to an origin AND some basis vectors B A X+ Y+ D C

23. Coordinate System Basis Vectors + Origin Cartesian Coordinates
Basis is a “span”
Sometimes basis is orthonormal
Cartesian Coordinates
Origin is 0,0,0
Basis Vectors
X+,Y+,Z+ X Y Z
Wikipedia

24. Change of Basis
To write a vector v in terms of new basis vectors Bx and By we…. xv v By yv Bx
This will become very important later…

25. Change of Basis …project it onto each basis vector.
Bx, By are the new coordinate axes… x’ v’ By y’ Bx
This will become very important later…

26. Change of Coordinate System
To switch coordinate systems we… By P O Bx
This will become very important later…

27. Change of Coordinate System
…write P in terms of new origin, then…. By P’ O Bx
This will become very important later…

28. Change of Coordinate System
…project onto new basis vectors By Q O Bx
This will become very important later…

29. Change of Coordinate System
To go back out, add scaled basis vectors to origin… By Q P O Bx
This will become very important later…

30. Handedness X Y Z Y X Y Z Z X X Y Z Y X Y Z Z X Wikipedia
Wikipedia, Reflected X Y Z Y X Y Z Z X

31. Cross Product
Wikipedia

32. Cross Product Perpendicular to inputs Length
Length is area of parallelogram q a b
a x b

33. Cross Product If a,b are orthonormal, axb is normalized
If a,b have same or opposite direction, axb is zero
Also:
Anticommutative
Distributive
NOT associative q a b
a x b

34. Cross Product
More potentially useful facts…

35. Building an Orthogonal Basis
(AxB)xA B B
(AxB)xA A A A A
AxB
AxB
AxB

36. 2D “Cross Product” r1 r0 Signed area of Parallelogram.
Negative Vectors are clockwise
Positive Vectors are counterclockwise

37. 2D “Cross Product” Application:
Given two points, which side of their line is a third point on? B C A

38. 2D “Sidedness” Use 2D Cross product: Positive Left Negative  Right
Zero  Co-linear B C A

39. Matrices N row vectors M column vectors NxM scalars…
If you have not taken your red pill, do so NOW…

40. Matrix Determinant(2x2)
Didn’t we see this already…? c1 c0
Signed area of Parallelogram.
Negative Vectors are clockwise
Positive Vectors are counterclockwise
Positive  Identity matrix

41. Matrix Determinant (3x3)
Signed volume of that thing
PositiveRight handed basis
NegativeLeft handed basis
Wikipedia

42. Matrix Multiplication

43. Matrix Multiplication
…is just a bunch of dot products

44. Matrix/Vector Multiplication

45. Matrix/Vector Multiplication
…. is also a bunch of dot products
This IS change of basis
Projection onto row/column vectors

46. Matrix Inverse
If rows form an orthonormal basis, Inverse equals transpose
Square matrix is invertible if it has non-zero determinant
Zero determinant  Zero area/volume
Vectors do not span the full space…
If det(M)=0, it goes flat…

47. Linear Interpolation In-betweening… s A A B t B C D t=0.25 t=0 t=0.75

48. Implicit 2D Lines You should know slope-intercept
Lines also have an implicit form
(x2,y2)
(x1,y1)
(A,B)

49. Implicit 2D Lines Point-Line Distance Plug point into line equation
Result is scaled by length of [A,B]
To “normalize” line, divide A,B,C by normal length
(x2,y2)
(x1,y1)
(A,B)
Negative Here
Positive Here

50. 3D Planes Planes are analogous to 2D Lines
Plug in points for signed distance
Positive  “in front”
Negative  “behind”
0  On plane
The rain in Spain…

51. 3D Planes
Plane from point and normal N P P1

52. 3D Planes
Plane from 3 Points
Cross Product N P3 P2 P1

53. Triangle Area This again… Sign indicates vertex order…
Positive: A,B,C counter-clockwise
In 3D: Use magnitude of cross product… C B A

54. Point-In-Triangle Test
Method 1:
Three determinant tests. Check signs… Y P
(blue) Z
(red) X
(green)

55. Point-In-Triangle Test
Method 2:
Extract line equations
See earlier slides
Test signed distances
Same sign Inside B P C A

56. Barycentric Coordinates
g=0
Ratios of areas
a=0 B
(0,1,0) a P g b C
b=0
(0,0,1) A
(1,0,0)
Inside Triangle

57. Barycentric Coordinates
Edge equations normalized to opposite vertex…
g=0
a=0
b=1
g=1
a=1
b=0

58. Barycentric Interpolation
Interpolation across triangle surface
Weight per-vertex values by barycentric coordinates

59. Implicit Surfaces Surface Surface normal
Set of all zeros of implicit function
“Inside”Negative
“Outside”Positive
Surface normal
Vector of partials
“Gradient”
Points in direction of greatest change in f

60. Implicit Surfaces
…mmmmmm…. Formulii……