Linear Independence. , v 2. is linearly independent if. =0 iff 1

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1 Representation Objectives Introduce concepts such as dimension and basis Introduce coordinate systems for representing vectors spaces and frames for representing affine spaces Discuss change of frames and bases Introduce homogeneous coordinates 1

2 Linear Independence A set of vectors,,, v n is linearly independent if n v n =0 iff 1 = 2 = =0 If a set of vectors is linearly independent, we cannot represent one in terms of the others If a set of vectors is linearly dependent, as least one can be written in terms of the others 2

3 Dimension In a vector space, the maximum number of linearly independent vectors is fixed and is called the dimension of the space In an n-dimensional space, any set of n linearly independent vectors form a basis for the space Given a basis,,., v n, any vector v can be written as v= n v n where the { i } are unique 3

4 Representation Until now we have been able to work with geometric entities without using any frame of reference, such as a coordinate system Need a frame of reference to relate points and objects to our physical world. For example, where is a point? Can t answer without a reference system World coordinates Camera coordinates 4

5 Coordinate Systems Consider a basis,,., v n A vector is written v= n v n The list of scalars { 1, 2,. n }is the representation of v with respect to the given basis We can write the representation as a row or column array of scalars [ 1 a=[ 1 2. n ] T = 2 n] 5

6 Example v=2 +3-4v 3 a=[2 3 4] T Note that this representation is with respect to a particular basis For example, in OpenGL we start by representing vectors using the object basis but later the system needs a representation in terms of the camera or eye basis 6

7 Coordinate Systems Which is correct? v v Both are because vectors have no fixed location 7

8 Frames A coordinate system is insufficient to represent points If we work in an affine space we can add a single point, the origin, to the basis vectors to form a frame P 0 v 3 8

9 Representation in a Frame Frame determined by (P 0,,, v 3 ) Within this frame, every vector can be written as v= n v n Every point can be written as P = P n v n 9

10 Confusing Points and Vectors Consider the point and the vector P = P n v n v= n v n They appear to have the similar representations p=[ ] v=[ ] which confuses the point with the vector A vector has no position Vector can be placed anywhere v p v point: fixed 10

11 A Single Representation If we define 0 P = 0 and 1 P =P then we can write v= v 3 =[ v 3 P 0 ] [ ]T P = P v 3 = [ v 3 P 0 ] [ ] T Thus we obtain the four-dimensional homogeneous coordinate representation v = [ ] T P= [ ] T 11

12 Homogeneous Coordinates The homogeneous coordinates form for a three dimensional point [x y z] is given as p =[x y z 1] T, the scaling is unimportant p is the same as [wx wy wz w] T for w0 xwx/w ywy/w zwz/w If w=0, the representation is that of a vector 12

13 Homogeneous Coordinates and Graphics Homogeneous coordinates are key to all computer graphics systems All standard 3D transformations (rotation, translation, scaling) can be implemented with matrix multiplications using 4 x 4 matrices Hardware pipeline works with 4 dimensional representations For orthographic viewing, we can maintain w=0 for vectors and w=1 for points For perspective we need a perspective division 13

14 Change of Coordinate Systems Consider two representations of a the same vector with respect to two different bases. The representations are where a= [ ] T b= [ ] T w= v 3 = [ ] [ v 3 ] T = 1 u u u 3 = [ ] [u 1 u 2 u 3 ] T 14

15 Representing second basis in terms of first Each of the basis vectors, u 1,u 2, u 3, are vectors that can be represented in terms of the first basis u 1 = v 3 u 2 = v 3 u 3 = v 3 The coefficients define a 3 x 3 matrix M = v 15

16 Change of Frames We can apply a similar process in homogeneous coordinates to the representations of both points and vectors u u 2 Consider two frames: 2 (P 0,,, v 3 ) Q 0 (Q 0, u 1, u 2, u 3 ) P 0 v 3 Any point or vector can be represented in either frame u 3 We can represent Q 0, u 1, u 2, u 3 in terms of P 0,,, v 3 16

17 Frame to Frame Transform-Version 0 Extending what we did with change of bases u 1 = v 3 u 2 = v 3 u 3 = v 3 Q 0 = v 3 +P 0 defining a 4 x 4 matrix of coefficients M = and [u 1,u 2, u 3, Q 0 ]=[,, v 3, P 0 ] M T

18 Frame to Frame Transform-Version 0 ][ a w=[ u 1, u 2,u 3,Q 0 T[ a w=[,, v 3, P 0 ] M T[ ]=[ a u av b M u b v ] and c u c v w w [ u 1,u 2,u 3,Q 0 ]=[,, v 3,P 0 ] M T u b u [ av ]=[ b,,v 3, P 0 ] v ] and c u c v w w u b u c u w [ ][ ]=[,, v 3, P 0 u, u, u au b u c u w ]=( M T ) 1[ a v b v c v therefore w ] av b v ] c v w so 18

19 Working with Representations Within the two frames any point or vector has a representation of the same form a=[ ] in the first frame b=[ ] in the second frame where 4 4 1for points and 4 4 for vectors and a=m T b, (M T ) -1 a=b The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates 19

20 Example Find the 4x4 matrix T that converts points in the standard frame to points in the frame given by {u 1, u 2, u 3, P 0 } where u 1 = 2 2 [1 0] 1, u 2 = 2 2 [0 1] 1, u 3 = [ ] =[ 1 1, P ] 20

21 Example M=[ 1] =[ ,M T ], T =M T 1 21

22 Inverting Transform Matrices (optional) M=[ ]=[ m v 1] O where m=[ v=[ O=[0,0,0] ], 43], If W=[ w 1 w 2 w 3 1 Setting W 1=[W w 2 w w4]. 3 ], then W= [W 1 w 4 ] 22

23 Inverting Modelview Matrices(optional) M W =[ m v O 1][ W 1 [ W= m 1 X 1 x 4 m 1 v x 4 ] w = [ mw 1w 4 v ] 4 OW 1 w 4 mw 1 w 4 v=x 1, w 4 =x 4 W 1 =m 1 X 1 m 1 w 4 v = [ mw 1w 4 v ] w 4 ] = [ m 1 m 1 v][ X ] 1 O 1 x 4 = [ X ] 1 x =X 4 =M 1 X If m is an orthogonal matrix (e.g. A transformation between orthonormal basis) then =[ M 1 mt m T v] O 1 23

24 Inverting Modelview Matrices Transposes(optional) M=[ m=[ ], 0= [ 0 0 ]=[ m 0 1] v where T 0], vt =[ 14, 24, 34 ] If W=[w 1 w 2 w 3 1 Setting W 1=[W w 2 w w 3 4]. ], then W = [ W 1 w 4 ] 24

25 Inverting Modelview Matrix Transposes(optional) M W =[ m 0 v T 1][ W ] 1 w 4 [ W= m 1 X 1 T] x 4 m 1 v = [ = [ mw w 0 ] 1 4 v t W 1 w = [ 4 mw 1 = X 1, v T W 1 w 4 =x 4 W 1 =m 1 X 1 1 m 1][ m 1 v T mw 1 v T W 1 w 4] = [ X 1 0 X ] 1 x 4 = M 1 X ] x = X 4 If m is an orthogonal matrix (e.g. A transformation between orthonormal basis) then M 1 =[ m T 0 m T v T 1] 25

26 Affine Transformations Every linear transformation is equivalent to a change in frames Every affine transformation preserves lines However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations 26

27 The World and Camera Frames When we work with representations, we work with n-tuples or arrays of scalars Changes in frame are then defined by 4 x 4 matrices In OpenGL and WebGL, the base frame that we start with is the world frame Eventually we represent entities in the camera frame by changing the world representation using the model-view matrix Initially these frames are the same (M=I) 27

28 Other Coordinates Clip coordinates projected to clip coordinates cube centered around origin used for clipping Normalized device coordinates produced by division by w, called perspective division Window coordinates using viewport produce 3 dimensional representation in pixel units 28

29 Moving the Camera If objects are on both sides of z=0, we must move camera frame M = d 1 29

30 Moving the Camera X=cross(-P1,U) 30

31 Frame to Frame Transform (optional) Frame 1: [,, v 3, Q 0 ] where {,, v 3 } are orthonormal Frame 2: [u 1, u 2, u 3, P 0 ]where {u 1, u 2, u 3 } are orthonormal Let w be a vector T a T] u1 T ; ; u 2 u 3 w=[ w=[ v 3 ][ v b v c v] w=[ [ au b u]=[ T T] u1 T u u 2 c u [ 3 au b u]=[ u1 u c w=[u 1 u 2 u 3 ][ T u1 T u 2 u 3 T] au b u c u] [ v 3 ][ T u 1 T u 1 T v 3 u 2 T u 2 T u 2 T v 3 u 3 T u 3T u 3 T v 3] [ av b v c v]= M[ av b v] v c [ av b v c v] au b u c u] 31

32 Frame to Frame Transform(optional) Frame 1: [,, v 3, Q 0 ] where {,, v 3 } are orthonormal Frame 2: [u 1, u 2, u 3, P 0 ] where {u 1, u 2, u 3 } are orthonormal Q 0 P 0 =d 1 u 1 +d 2 u 2 +d 3 u 3 where d 1 =u T 1 (Q 0 P 0 );d 2 =u T 2 (Q 0 P 0 );d 3 =u T 3 (Q 0 P 0 ). Let S be a point, then S Q 0 =S u1 u 1 +S u2 u 2 +S u3 u 3 =S +S +S v 3 v 3. [ S [S u1 Therefore from above, it follows that S u2 S Then S u3]=m S v 3]. S P 0 =R u1 u 1 +R u2 u 2 +R u3 u 3 where {R u1, R u2, R u3 } are the coordinate of S in Frame 2 and {S, S, S v 3 } are the coordinates of S in Frame 1. But as S P 0 =(S Q 0 )+(Q 0 P 0 )=S u1 u 1 +S u2 u 2 +S u3 u 3 +d 1 u 1 +d 2 u 2 +d 3 u 3 =(S u1 +d 1 )u 1 +(S u2 +d 2 )u 2 +(S u3 +d 3 )u [ 3, R [S u1 it follows that R u2 S d 2 R u3]=m S v 3]+[d1 3] for any point S. d 32

33 Frame to Frame Transform Frame 1: [,, v 3, Q 0 ] where {,, v 3 } are orthonormal Frame 2: [u 1, u 2, u 3, P 0 ] where {u 1, u 2, u 3 } are orthonormal Q 0 P 0 =d 1 u 1 +d 2 u 2 +d 3 u 3 where d 1 =u T 1 (Q 0 P 0 ); d 2 =u T 2 (Q 0 P 0 ); d 3 =u T 3 (Q 0 P 0 ). Let S be a point where {R u1, R u2, R u3 } are the coordinate of S in Frame 2 and {S v1, S, S v 3 } are the coordinates of S in Frame 1. u 1 Define M=[ T u T 1 u T 1 v 3 d 1 u T 2 u T 2 u T 2 v 3 d 2. Then u T 3 u 3 T u T 3 v 3 d [ [ Ru1 Sv1 R u2 S ]=M v ] 2 for any point or vector S(w=0,1). R u3 S v 3 w w ] 33

34 Octave Frame to Frame Transform #Frame 1: [v_1,v_2, v_3, Q_0] where { v_1,v_2,v_3 } are orthonormal #Frame 2: [u_1,u_2, u_3, P_0] where { u_1,u_2,u_3} are orthonormal Q_0=[0,0,0]' P_0=[-1-,-1,-1]' u_3=p_0/norm(p_0) up=[0,1,0]' u_1=cross(u_3,up) u_1=u_1/norm(u_1) u_2 = cross(u_1,u_3) v_1=[1,0,0]' v_2=[0,1,0]' v_3=[0,0,1]' d_1=u_1'* (Q_0-P_0) d_2=u_2' * (Q_0-P_0) d_3=u_3' *(Q_0-P_0) M=[u_1'* v_1, u_1'* v_2, u_1'* v_3, d_1 ; u_2'* v_1, u_2'* v_2, u_2'* v_3,d_2; u_3'* v_1, u_3'* v_2, u_3'* v_3,d_3 ; 0, 0, 0, 1] 34

35 Octave Frame to Frame Transform Version 1a #Frame 1: [v_1,v_2, v_3, Q_0] where { v_1,v_2,v_3 } are orthonormal #Frame 2: [u_1,u_2, u_3, P_0] where { u_1,u_2,u_3} are orthonormal P_0=[-3,20,-1]' Q_0=[1,1,1]' #Compute Frame 1 u_3=p_0/norm(p_0) up=[0,1,0]' u_1=cross(u_3,up) u_1=u_1/norm(u_1) u_2 = cross(u_1,u_3) u=[u_1,u_2,u_3 ] #Compute Frame 2 v_3=q_0/norm(q_0) v_1=cross(v_3,up) v_1=v_1/norm(v_1) v_2=cross(v_1,v_3) v=[v_1,v_2,v_3] #compute transform d=u'*(q_0-p_0) m=u'*v #Define Transform M=[m,d;0,0,0,1] #verify right hand frames #Compute a the coordinate of a point in both frames S=[3,-5,-2]' SQv=v'*(S-Q_0) #coefficients of S in frame 1 SQu=u'*(S-Q_0) SPv=v'*(S-P_0) SPu=u'*(S-P_0) #coefficients of S in frame 2 Squ-m*SQv #should be 0 #Check Transform (S-Q_0)+(Q_0-P_0)-(u*SQu+u*d) (S-Q_0)+(Q_0-P_0)-u*(m*SQv+d) u'*(s-p_0) -(m*sqv+d) SPu-(m*SQv+d) [SPu;1] -M*[SQv;1] #should be 0 35

36 Octave: determine if points are in a plane v1 = [ ]' v2 = [ ]' v3 = [ ]' v4 = [ ]' v5 = [ ]' disp ("Is v1 in the same plan as v2,v3,v4?") nv=cross(v3-v2,v4-v2) nv=nv/norm(nv) if abs(nv'*(v1-v2)) > 1.0e-5 disp ("no") else disp ("yes") endif disp ("Is v5 in the same plan as v2,v3,v4?") if abs(nv'*(v5-v2)) > 1.0e-5 disp ("no") else disp ("yes") endif # Note that as we are using 5 significant figures, # any value less than 1.0e-5 could be 0. 36

37 Exercise a) Read Chapter 4,5. b) We want to move the camera to P1=[1,1,1] T,point it at [0,0,0] T, and have up in the general direction of [0,1,0] T. Determine the new frame we should use, and determine the resulting 4x4 transformation matrix. c) At what point does the line though the points P0=[0,1,1] T and P1=[0,0,0] T, intersect the plane through the point P2=[1,1,1] T with normal n=2/2[1,0,-1] T? Is the point of intersection between points P0 and P1? d) Which of the 5 points P0=[0,4,3] T, P1=[-1,3,2] T, P2=[0,3,2] T, P3=[-1,3,3] T, P4=[0,4,2] T lie on the same plane. 37