Computer Graphics CSC 630 Lecture 2- Linear Algebra

2 What is Computer Graphics? Using a computer to generate an image from a representation. Model Image computer

3 Model Representations How do we represent an object? –Points –Mathematical Functions X 2 + Y 2 = R 2 –Polygons (most commonly used) Points Connectivity

4 Linear Algebra Why study Linear Algebra? –Deals with the representation and operations commonly used in Computer Graphics

5 What is a Matrix? A matrix is a set of elements, organized into rows and columns n columns m rows m×n matrix

6 What is a Vector? Vector: n × 1 matrix x y v

7 Representing Points and Vectors A 3D point –Represents a location with respect to some coordinate system A 3D vector –Represents a displacement from a position

8 Basic Operations Transpose: Swap rows with columns

9 Vector Addition Given v = [x y z] T and w = [a b c] T v + w = [x+a y+b z+c] T Properties of Vector addition –Commutative: v + w = w + v –Associative (u + v) + w = u + (v + w) –Additive Identity: v + 0 = v –Additive Inverse: v + (-v) = 0

10 Parallelogram Rule To visualize what a vector addition is doing, here is a 2D example: v w v+w

11 Vector Multiplication Given v = [x y z] T and a Scalar s and t –sv = [sx sy sz] T and tv = [tx ty tz] T Properties of Vector multiplication –Associative: (st)v = s(tv) –Multiplicative Identity: 1v = v –Scalar Distribution: (s+t)v = sv+tv –Vector Distribution: s (v+w) = sv+sw

12 Vector Spaces Consists of a set of elements, called vectors. The set is closed under vector addition and vector multiplication.

13 Dot Product and Distances Given u = [x y z] T and v = [a b c] T –vu = v T u = ax+by+cz The Euclidean distance of u from the origin is denoted by ||u|| and called norm or length –||u|| = sqrt(x 2 +y 2 +z 2 ) –Notice that ||u|| = sqrt(u u) –Unit vector ||u|| = 1, zero vector denoted 0 The Euclidean distance between u and v is sqrt((x-a) 2 +(y-b) 2 +(z-c) 2 )) and is denoted by ||u-v||

14 Properties of the Dot Product Given a vector u, v, w and scalar s –The result of a dot product is a SCALAR value –Commutative: vw = wv –Non-degenerate: vv=0 only when v=0

15 Angles and Projection Alternative view of the dot product vw=||v|| ||w|| cos(  ) where  is the angle between v and w v w v w v=w If v and w have length 1 … v·w = 0v·w = 1 v·w = cos θ θ

16 Angles and Projection If v is a unit vector (||v|| = 1) then if we perpendicularly project w onto v can call this newly projected vector u then ||u|| = vw w vu

17 Cross Product Properties The Cross Product c of v and w is denoted by v  w c Is a VECTOR, perpendicular to the plane defined by v and w The magnitude of c is proportional to the sin of the angle between v and w The direction of c follows the right hand rule. ||v  w||=||v|| ||w|| |sin  | –  is the angle between v and w v  w=-(w  v) v w vwvw

18 Cross Product Given 2 vectors v=[v 1 v 2 v 3 ], w=[w 1 w 2 w 3 ], the cross product is defined to be the determinant of where i,j,k are vectors

19 Matrices A compact way of representing operations on points and vectors 3x3 Matrix A looks like a ij refers to the element of matrix A in i th row and j th column

20 Matrix Addition Just add elements

21 Matrix Multiplication If A is an n  k matrix and B is a k  p then AB is a n  p matrix with entries c ij where c ij =  a is b sj Alternatively, if we took the rows of A and columns of B as individual vectors then c ij = A i B j where the subscript refers to the row and column, respectively Multiply each row by each column

22 Matrix Multiplication Properties Associative: (AB)C = A(BC) Distributive: A(B+C) = AB+AC Multiplicative Identity: I= diag(1) (defined only for square matrix) Identity matrix: AI = A NOT commutative: AB  BA

23 Matrix Inverse If A and B are nxn matrices and AB=BA=I then B is the inverse of A, denoted by A -1 (AB) -1 =B -1 A -1 same applies for transpose

24 Determinant of a Matrix Defined on a square matrix (nxn) Used for inversion If det A = 0, then A has no inverse

25 Determinant of a Matrix For a nxn matrix, where A 1i determinant of (n-1)x(n-1) submatrix A gotten by deleting the first row and the ith column

26 Transformations Why use transformations?  Create object in convenient coordinates  Reuse basic shape multiple times  Hierarchical modeling  System independent  Virtual cameras

27 Translation =+ =+

28 Properties of Translation = = = =

29 Rotations (2D)

30 Rotations 2D So in matrix notation

31 Rotations (3D)

32 Properties of Rotations order matters!

33 Combining Translation & Rotation

34 Combining Translation & Rotation

35 Scaling Uniform scaling iff

36 Homogeneous Coordinates can be represented as where

37 Translation Revisited

38 Rotation & Scaling Revisited

39 Combining Transformations where

40 Transforming Tangents

41 Transforming Normals