02mathematics by srikanth sabbrapu

Graphics

Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Contents 

Coordinate-Reference Frames  



Vector Addition and Scalar Multiplication Scalar Product / Vector Product

Basis Vectors and the Metric Tensor  



2D Cartesian Reference Frames / Polar Coordinates 3D Cartesian Reference Frames / Curvilinear Coordinates

Points and Vectors 



CGVR

Orthonormal Basis Metric Tensor

Matrices   

Scalar Multiplication and Matrix Addition Matrix Multiplication / Transpose Determinant of a Matrix / Matrix Inverse

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Coordinate Reference Frames 

CGVR

Coordinate Reference Frames 

Cartesian coordinate system 



x, y, z 좌표축사용 , 전형적 좌표계

Non-Cartesian coordinate system  

특수한 경우의 object 표현에 사용 . Polar, Spherical, Cylindrical 좌표계 등

cgvr.korea.ac.kr

Graphics Lab @ Korea University

2D Cartesian Reference System ď Ž

CGVR

2D Cartesian Reference Frames

x y x

Coordinate origin at the lower-left screen corner

cgvr.korea.ac.kr

Coordinate origin in the upper-left screen corner

Graphics Lab @ Korea University

Polar Coordinates 

CGVR

가장 많이 쓰이는 Non-Cartesian System r θ

x = r cosθ , r = x2 + y2 ,

y = r sin θ  y θ = tan −1    x

s = rθ 

Elliptical Coordinates, Hyperbolic or Parabolic Plane Coordinates 등 원 이외에 Symmetry 를 가진 다른 2 차 곡선들로도 좌표계 표현 가능

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Why Polar Coordinates? 

CGVR

Circle 

2D Cartesian : 비균등 분포

 Polar Coordinate y

x2 + y2 = r 2

x = r cos θ,

y = r sin θ

dθ dθ x

dx dx 균등하게 분포되지 않은 점들

Cartesian Coordinates cgvr.korea.ac.kr

연속된 점들 사이에 일정간격유지

Polar Coordinates Graphics Lab @ Korea University

3D Cartesian Reference Frames

CGVR

Three Dimensional Point

cgvr.korea.ac.kr

Graphics Lab @ Korea University

3D Cartesian Reference Frames 

오른손 좌표계 



대부분의 Graphics Package 에서 표준

왼손 좌표계 



cgvr.korea.ac.kr

CGVR

관찰자로부터 얼마만큼 떨 어져 있는지 나타내기에 편 리함 Video Monitor 의 좌표계

Graphics Lab @ Korea University

3D Curvilinear Coordinate Systems 

CGVR

General Curvilinear Reference Frame 

Orthogonal coordinate system 

Each coordinate surfaces intersects at right angles

x2 axis

x1 = const1

x3 axis

x3 = const3

x2 = const2

x1 axis

A general Curvilinear coordinate reference frame cgvr.korea.ac.kr

Graphics Lab @ Korea University

3D Non-Cartesian System 

Cylindrical Coordinates z z axis



Spherical Coordinates z axis

P(ρ,θ,z)

θ ρ x axis

y axis

x = ρ cos θ y = ρ sin θ z =z cgvr.korea.ac.kr

CGVR

φ θ x axis

P(r,θ, ρ) r y axis

x = r cos θ sin φ y = r sin θ sin φ z = r cos φ Graphics Lab @ Korea University

Points and Vectors

CGVR



Point: 좌표계의 한 점을 차지 , 위치표시



Vector: 두 position 간의 차로 정의 V = P2 − P1 = ( x2 − x1 , y2 − y1 ) = (Vx , V y ) 

Magnitude 와 Direction 으로도 표기 P2

y2 y1

Vy α = tan   Vx

−1

x1 cgvr.korea.ac.kr

V = Vx2 +V y2    

x2 Graphics Lab @ Korea University

Vectors 

CGVR

3 차원에서의 Vector

V = Vx2 +V y2 +Vz2 Vy V V cos α = x , cos β = , cos γ = z |V | |V | |V |

cos 2 α + cos 2 β + cos 2 γ = 1



γ β α

V y

Vector Addition and Scalar Multiplication V1 + V2 = (V1x + V2 x , V1 y + V2 y , V1z + V2 z )

αV = (αVx ,αV y ,αVz ) cgvr.korea.ac.kr

Graphics Lab @ Korea University

Scalar Product 

Definition V2 θ |V2|cosθ



CGVR

V1 ⋅V2 =| V1 || V2 | cos θ , 0 ≤ θ ≤ π V1

Dot Product, Inner Product 라고도 함

For Cartesian Reference Frame V1 ⋅ V2 = V1xV2 x + V1 yV2 y + V1zV2 z



Properties 

Commutative V1 ⋅ V2 = V2 ⋅ V1



Distributive

cgvr.korea.ac.kr

V1 ⋅ (V2 + V3 ) = V1 ⋅ V2 + V1 ⋅ V3 Graphics Lab @ Korea University

Vector Product 

Definition V1 × V2 u



CGVR

V2 θ

V1 × V2 = u | V1 || V2 | sin θ , 0 ≤ θ ≤ π Cross Product, Outer Product 라고도 함

For Cartesian Reference Frame V1 × V2 = (V1 yV2 z − V1zV2 y ,V1zV2 x − V1xV2 z ,V1xV2 y − V1 yV2 x )



Properties   

AntiCommutative V1 × V2 = −(V2 × V1 ) Not Associative V1 × (V2 × V3 ) ≠ (V1 × V2 ) × V3 V1 × (V2 + V3 ) = (V1 × V2 ) + (V1 × V3 ) Distributive

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Examples 

CGVR

Scalar Product



Vector Product

(x2,y2) V2

θ (x0,y0)

(x1,y1)

Angle between Two Edges

cgvr.korea.ac.kr

Normal Vector of the Plane

Graphics Lab @ Korea University

Basis Vectors 

CGVR

Basis (or a Set of Base Vectors) Specify the coordinate axes in any reference frame  Linearly independent set of vectors  Any other vector in that space can be written as linear combination of them 



Vector Space  

Contains scalars and vectors Dimension: the number of base vectors

cgvr.korea.ac.kr

u1 u3 Curvilinear coordinateaxis vectors Graphics Lab @ Korea University

Orthonormal Basis 

CGVR

Normal Basis + Orthogonal Basis

u k ⋅ u k = 1,

for all k

u j ⋅ u k = 0, for all j ≠ k 

Example 

Orthonormal basis for 2D Cartesian reference frame



Orthonormal basis for 3D Cartesian reference frame

u x = (1, 0) u y = ( 0, 1)

u x = (1, 0, 0 ) u y = ( 0, 1, 0 ) u z = ( 0, 0, 1)

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Metric Tensor 

Tensor  



CGVR

Quantity having a number of components, depending on the tensor rank and the dimension of the space Vector – tensor of rank 1, scalar – tensor of rank 0

Metric Tensor for any General Coordinate System   

Rank 2 Elements: g jk = u j ⋅ u k Symmetric: g jk = g kj

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Properties of Metric Tensors 

CGVR

The Elements of a Metric Tensor can be used to Determine   

Distance between two points in that space Transformation equations for conversion to another space Components of various differential vector operators (such as gradient, divergence, and curl) within that space

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Examples of Metric Tensors 

CGVR

Cartesian Coordinate System u x = (1, 0) u y = ( 0, 1)

1, if j = k g jk =  0, otherwise  Polar Coordinates

u r = u x cos θ + u y sin θ ,

uθ = −u x r sin θ + u y r cos θ 1 0  g= 2 0 r   cgvr.korea.ac.kr

Graphics Lab @ Korea University

Matrices 

CGVR

Definition 

A rectangular array of quantities

 a11 a A =  21  :  am1



a12 ... a1n  a22 ... a2 n  : :   am 2 ... amn 

Scalar Multiplication and Matrix Addition

 a11 A =   a21

cgvr.korea.ac.kr

a12   b11 b12   , B =   a22   b21 b22 

 a11 + b11 A + B =   a21 + b21  ka11 kA =   ka21

a12 + b12   a22 + b22 

ka12   ka22  Graphics Lab @ Korea University

Matrix Multiplication 

Definition

j-th column

C = AB n

cij = ∑ aik bkj

i-th row

k =1



CGVR

Properties

(i,j)

l n

AB ≠ BA  Associative ( AB )C = A( BC )  Distributive A( B + C ) = AB + BC  Scalar Multiplication ( kA) B = A( kB ) = k ( AB ) 

Not Commutative

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Matrix Transpose 

Definition 

Interchanging rows and columns

1 4  1 2 3 2 5, =  4 5 6     3 6 T



CGVR

b c]

a  = b   c 

Transpose of Matrix Product

( AB ) T cgvr.korea.ac.kr

= BT A T

Graphics Lab @ Korea University

Determinant of Matrix 

CGVR

Definition 

For a square matrix, combining the matrix elements to product a single number



2 × 2 matrix



a11 a12 = a11a22 − a12 a21 a21 a22 Determinant of n×n Matrix A (n ≥2) n

det A = ∑ (−1) j + k a jk det A jk j =1

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Inverse Matrix 

Definition 



−1

AA = I

A A= I

Non-singular matrix 



CGVR

If and only if the determinant of the matrix is non-zero

2 × 2 matrix a b  A=   c d  

1  d − b A = ad − bc  − c a  −1

Properties

( A − 1 )− 1 = A cgvr.korea.ac.kr

( AB )−1 = B −1 A−1

( AT )−1 = ( A−1 )T Graphics Lab @ Korea University