03 - Basic Linear Algebra and 2D Transformations

Transcription

1 03 - Basic Linear Algebra and 2D Transformations (invited lecture by Dr. Marcel Campen)

2 Overview In this box, you will find references to Eigen We will briefly overview the basic linear algebra concepts that we will need in the class You will not be able to follow the next lectures without a clear understanding of this material

3 Vectors

4 Vectors Eigen::VectorXd A vector describes a direction and a length Do not confuse it with a location, which represent a position When you encode them in your program, they will both require 2 (or 3) numbers to be represented, but they are not the same object! Origin These two are identical! Vectors represent displacements. If you represent the displacement wrt the origin, then they encode a location.

5 Sum Operator + a + b = b + a a a + b b b a

6 Difference Operator - a b a a b a b a = a + b

7 Coordinates Operator [] c = c 1 a + c 2 b c = a +2b c c 2b b a a a and b form a 2D basis

8 Cartesian Coordinates c = c 1 x + c 2 y x and y form a canonical, Cartesian basis c y x

9 Length The length of a vector is denoted as a a.norm() If the vector is represented in cartesian coordinates, then it is the L2 norm of the vector: q a = a a2 2 A vector can be normalized, to change its length to 1, without affecting the direction: CAREFUL: b = a a b.normalize() < in place b.normalized() < returns the normalized vector

10 Dot Product a.dot(b) a.transpose()*b a b = a b cos The dot product is related to the length of vector and of the angle between them If both are normalized, it is directly the cosine of the angle between them a b

11 Dot Product - Projection The length of the projection of b onto a can be computed a using the dot product b b! a = b cos = b a a

12 Cross Product Eigen::Vector3d v(1, 2, 3); Eigen::Vector3d w(4, 5, 6); v.cross(w); a b = a b sin Defined only for 3D vectors The resulting vector is perpendicular to both a and b, the direction depends on the right hand rule a b b The magnitude is equal to the area of the parallelogram formed by a and a b

13 Coordinate Systems You will often need to manipulate coordinate systems (i.e. for finding the position of the pixels in Assignment 1) You will always use orthonormal bases, which are formed by pairwise orthogonal unit vectors : 2D u = v =1, u v =0 3D u = v = w =1, u v = v w = w u =0 Right-handed if: w = u v

14 Coordinate Frame e is the origin of the reference system p is the center of the pixel v w u v e e u u,v,w are the coordinates of p wrt the frame of reference or coordinate frame p (note that they depend also on the origin e) p = e + uu + vv + ww

15 Change of frame a v w If you have a vector a expressed in global u coordinates, and you want to convert it into a e vector expressed in a local orthonormal u-v-w coordinate system, you can do it using projections of a onto u, v, w (which we assume are expressed in global coordinates): a G =(a u, a v, a w)

16 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 2

17 Matrices

18 Overview Matrices will allow us to conveniently represent and ally transformations on vectors, such as translation, scaling and rotation Similarly to what we did for vectors, we will briefly overview their basic operations

19 Determinants Think of a determinant as an operation between vectors. ab abc b c a b a Area of the parallelogram Volume of the parallelepiped (positive since abc is a right-handed basis) By Startswithj - Own work, CC BY-SA 3.0, curid=

20 Matrices Eigen::MatrixXd A(2,2) apple x11 x 12 A matrix is an array of numeric elements x 21 x 22 Sum apple x11 x 12 x 21 x 22 + apple y11 y 12 y 21 y 22 = apple x11 + y 11 x 12 + y 12 x 21 + y 21 x 22 + y 22 apple x11 x 12 apple yx11 yx 12 A.array() + B.array() Scalar Product y x 21 x 22 = yx 21 yx 22 A.array() * y

21 Transpose B = A.transpose(); A.transposeInPlace(); The transpose of a matrix is a new matrix whose entries are reflected over the diagonal 1 2 T = apple 1 2 apple T = apple T = apple The transpose of a product is the product of the transposed, in reverse order (AB) T = B T A T

22 The entry i,j is given by multiplying the entries Matrix Product Eigen::MatrixXd A(4,2); Eigen::MatrixXd B(2,3); A*B; on the i-th row of A with the entries of the j-th column of B and summing up the results It is NOT commutative (in general): AB 6= BA

23 Intuition r 1 x 1 4y5 = 4 r 2 5 4x5 4y5 = 4c 1 c 2 c x 2 5 r 3 x 3 y i = r i x y = x 1 c 1 + x 2 c 2 + x 3 c 3 Dot product on each row Weighted sum of the columns

24 Inverse Matrix Eigen::MatrixXd A(4,4); A.inverse() < do not use this to solve a linear system! A A 1 AA 1 = I The inverse of a matrix is the matrix such that where I is the identity matrix I = The inverse of a product is the product of the inverse in opposite order: (AB) 1 = B 1 A 1

25 Diagonal Matrices Eigen::Vector3d v(1,2,3); They are zero everywhere except the diagonal: A = v.asdiagonal() D = 2 a b c Useful properties: D 1 = 2 a b c 1 D = D T

26 Orthogonal Matrices An orthogonal matrix is a matrix where each column is a vector of length 1 each column is orthogonal to all the others A useful property of orthogonal matrices that their inverse corresponds to their transpose: (R T R)=I =(RR T )

27 Linear Systems We will often encounter in this class linear systems with n linear equations that depend on n variables. For example: 5x +3y 7z =4 3x +5y + 12z =9 9x 2y 2z = x 4y5 = z To find x,y,z you have to solve the linear system. Do not use an inverse, but rely on a direct solver: Matrix3f A; Vector3f b; A << 5,3,-7, -3,5,12, 9,-2,-2; b << 4, 9, -3; cout << "Here is the matrix A:\n" << A << endl; cout << "Here is the vector b:\n" << b << endl; Vector3f x = A.colPivHouseholderQr().solve(b); cout << "The solution is:\n" << x << endl;

28 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 5

29 2D Transformations

30 2D Linear Transformations Each 2D linear map can be represented by a unique 2 2 matrix x y = a b c d x y Concatenation of mappings corresponds to multiplication of matrices L 2 (L 1 (x)) = L 2 L 1 x L2 * L1 * x; Linear transformations are very common in computer graphics!

31 2D Scaling Scaling x = s x 0 x y 0 s y y S(s x,s y ) S(0.5, 0.5) Image Copyright: Mark Pauly

32 2D Rotation x cos sin x Rotation y = sin cos y R( ) R(20 ) apple 0 1 Special case: R(90) = 1 0 Image Copyright: Mark Pauly

33 2D Shearing Shear along x-axis x y = 1 a 0 1 x y H x (0.5) H x (a) Shear along y-axis x = 1 0 x y b 1 H y (b) y H y (0.5) Image Copyright: Mark Pauly

34 2D Translation Translation x y = x y + t x t y x x Matrix representation? = T(t x,t y ) y y Image Copyright: Mark Pauly

35 Affine Transformations Translation is not linear, but it is affine Origin is no longer a fixed point Affine map = linear map + translation x y = a b c d x y + t x t y = Lx + t Is there a matrix representation for affine transformations? We would like to handle all transformations in a unified framework -> simpler to code and easier to optimize!

36 Homogenous Coordinates Add a third coordinate (w-coordinate) 2D point = (x, y, 1) T 2D vector = (x, y, 0) T x y w 1 0 t x x x + t x = 0 1 t y y = y + t y Matrix representation of translations

37 Homogenous Coordinates Valid operation if results w-coord. is 1 or 0 vector + vector = vector point - point = vector point + vector = point point + point =???

38 Homogenous Coordinates Geometric interpretation: 2 hyperplanes in R 3 vectors points Image Copyright: Mark Pauly

39 Affine Transformations Affine map = linear map + translation x y = a b c d x y + t x t y Using homogenous coordinates: x y 1 a b t x x = c d t y y

40 2D Transformations Scale s x 0 0 S(s x,s y ) = 0 s y Rotation cos sin 0 R( ) = sin cos t x Translation T(t x,t y ) = 0 1 t y 0 0 1

41 Concatenation of Transformations Sequence of affine maps A1, A2, A3,... Concatenation by matrix multiplication x A n (...A 2 (A 1 (x))) = A n A 2 A 1 y 1 Very important for performance! Matrix multiplication not commutative, ordering is important!

42 Rotation and Translation Matrix multiplication is not commutative! First rotation, then translation R(45 ) T(t x, 0) First translation, then rotation T(t x, 0) R(45 ) Image Copyright: Mark Pauly

43 2D Rotation How to rotate around a given point c? 1. Translate center to origin 2. Rotate 3. Translate back T( c) R( ) T(c) Matrix representation? T(c) R( ) T( c) Image Copyright: Mark Pauly

44 Transform Object or Camera? T(-1,-1) S(0.5,0.5) R(45 o ) T(1,1) T(1,1) S(2,2) R(-45 o ) T(-1,-1) Image Copyright: Mark Pauly

45 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 6