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An introduction to trigonometry and vector algebra as covered in the cap4720 fall2008 course at ucf. Topics include angles, trigonometric functions, triangle laws, points, vectors, vector algebra, and geometric primitives. Learn about the unit circle, vector magnitudes, scalar and vector products, and more.
Typology: Exams
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Juraj [email protected] http://graphics.cs.ucf.edu/cap4720/fall2008/
Angles: φ 2π−φ
Length of the arc of the unit circlethat is cut by two directions. Which arc?Convention: The arc swept out counterclockwise from the first lineto the second line. unit: Radian conversion: * degrees radians 180
degrees^180 *radians =^ ππ
=
Trigonometric functions:
φ
φ φ^ p b
h
Triangle Laws:
trianglearea 41 ( )( )( )( )
(Lawoftangents) tan 2
tan 2
2 cos (Lawofcosines)
sin sin sin (Lawofsines) 2 2 2
a b c a b c a b c a b c
A B
A B a b
a b
c a b ab C
c
C b
B a
A
= + + − + + − + + −
^ −
= ^ +^ −
= + −
= =
c b
a C (^) A
B
Polygons circular sequence (cycle) of points (vertices) and segments (edges)
Segment intersection: Given two segments, do they intersect
Inclusion in polygon: Is a point inside or outside a polygon?
Simple closed path: intersecting polygon with vertices on the points. given a set of points, find a non-
Test whether segments (a,b) and (c,d) intersect. How do we do it? a
c b
d
given a polygon and a point is the point inside? or outside the polygon?
Magnitude? Q' = (x 1 - x, y 1 - y, z 1 - z) | V | = √√√√(( x 1 -x )^2 + ( y 1 -y )^2 + ( z 1 - z )^2 )
Z^ X
Y
P
V Q Q'
aV
Scalar Multiplication : for V = (V a V = x^ , V (aVy^ , Vx , aVz)y , aVz)
V How to make a Unit Vector?
Vector addition: V 1 +^ V 2 = ( V1x +^ V2x , V1y +^ V2y , V1z +^ V2z )
V 1
V 2 V^1
+ V^2
Scalar Product (dot product): V 1 • V 2 = | = V V 1x 1 | |. V V (^2) 2x | cos + V^ 1y θθθθ. V2y + V1z. V2z
V (^2) θ V 1
Vector product ( V cross product ): 1 ××××^ V 2 = (V1y.V2z - V1z.V2y , V1z.V2x - V1x.V2z , V1x.V2y - V1y.V2x)
x y z
x y z
2 2 2
1 1 1
Magnitude of the vector is the area of the shaded region.
Cross product is a Vector. Right-hand rule determines the direction of the productvector
V (^2) V 1
V^1 ××××^ V^2 θ
A set of axis vectors is called a basis.
Z^ X
Y
normal basis : A basis with unit axis vectors. orthogonal basis perpendicular axis vectors. : A basis with mutually Ortho-normal basis mutually perpendicular axis vectors. : A basis with unit and orthonormal basis for a 3D cartesian co-ordinatesystem is: i = (1,0,0), j = (0,1,0), k = (0,0,1)
Matrix is a rectangular array of numbers. example: a 4×3 matrix
41 42 43
31 32 33
21 22 23
11 12 13
:represents column.
:representsrowand
: theelementsofthematrix, j
i
aij
Square Matrix:
Row Matrix: a single rows of elements Column Matrix: a single column of elements
Diagonal Matrix a square matrix with all except diagonals are 0i.e. a ij = 0 if^ i^ ≠^ j.
44
33
22
11
0 0 0
a
a
a
a
Identity Matrix All diagonal elements are 1. denoted by symbol I example:
Multiplication (or product) of matrices: possible only if # columns in 1st^ matrix = # rows in 2nd^ matrix. i.e. if A is a m × n matrix and B is n × p matrix then matrices are conformable for multiplicationand the product matrix C is a m × p matrix
The elements of the product of two matrices A and B:
3111 32 21 3112 32 22
2111 22 21 2112 22 22
1111 12 21 1112 12 22 21 22
11 12 31 32
21 22
11 12
example :
a b a b a b a b
a b a b a b a b
a b a b a b a b b b
b b a a
a a
a a
example:
cij = (^) k aikbkj
3
2
1 1 2 3
ab ab a b b
b
b A B AB a a a
A a a a B b b b
T (^) = + +
example:
cij = (^) k aikbkj
3
2
1 2 1
3 1
3 2
1 2 3 1 2 3
ab ab ab ab ab a b
b
b
b a a
a a
a a A B
A a a a B b b b
