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1Computer Graphics Homogeneous Coordinates & Transformations Lecture 11/12 John Shearer Culture Lab – space 2

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1 1Computer Graphics Homogeneous Coordinates & Transformations Lecture 11/12 John Shearer Culture Lab – space 2

2 2Computer Graphics Homogeneous Coordinates The homogeneous coordinates form for a three dimensional point [x y z] is given as p =[x’ y’ z’ w] T =[wx wy wz w] T We return to a three dimensional point (for w  0 ) by x  x’ /w y  y’/w z  z’/w If w=0, the representation is that of a vector Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions For w=1, the representation of a point is [x y z 1]

3 3Computer Graphics Homogeneous Coordinates and Computer Graphics Homogeneous coordinates are key to all computer graphics systems –All standard 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

4 4Computer Graphics Affine Transformations Every linear transformation is equivalent to a change in reference frame 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

5 5Computer Graphics General Transformations A transformation maps points to other points and/or vectors to other vectors

6 6Computer Graphics Affine Transformations Line preserving Characteristic of many physically important transformations –Rigid body transformations: rotation, translation –Scaling, shear Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

7 7Computer Graphics Pipeline Implementation transformationrasterizer u v u v T T(u) T(v) T(u) T(v) vertices pixels frame buffer (from application program)

8 8Computer Graphics Translation Move (translate, displace) a point to a new location Displacement determined by a vector d –Three degrees of freedom –P’=P+d P P’ d

9 9Computer Graphics How many ways? Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way objecttranslation: every point displaced by same vector

10 10Computer Graphics Translation Using Representations Using the homogeneous coordinate representation in some frame p=[ x y z 1] T p’=[x’ y’ z’ 1] T d=[dx dy dz 0] T Hence p’ = p + d or x’=x+d x y’=y+d y z’=z+d z note that this expression is in four dimensions and expresses point = vector + point

11 11Computer Graphics Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p ’= Tp where T = T (d x, d y, d z ) = This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

12 12Computer Graphics Translation Matrix glTranslate produces a translation by (x,y,z) The current matrix is multiplied by this translation matrix, with the product replacing the current matrix, as if glMultMatrix were called with that same matrix for its argument void glTranslatef (GLfloat x, GLfloat y, GLfloat z ); 100x 010y 001z 0001

13 13Computer Graphics Rotation (2D) Consider rotation about the origin by  degrees –radius stays the same, angle increases by  x’=x cos  –y sin  y’ = x sin  + y cos  x = r cos  y = r sin  x = r cos (  y = r sin ( 

14 14Computer Graphics Rotation about the z axis Rotation about z axis in three dimensions leaves all points with the same z –Equivalent to rotation in two dimensions in planes of constant z x’=x cos  –y sin  y’ = x sin  + y cos  z’ =z

15 15Computer Graphics Rotation Matrix R = R z (  ) =

16 16Computer Graphics Rotation about x and y axes Same argument as for rotation about z axis –For rotation about x axis, x is unchanged –For rotation about y axis, y is unchanged R = R x (  ) = R = R y (  ) =

17 17Computer Graphics General Rotation About the Origin  x z y v A rotation by  about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes R(  ) = R z (  z ) R y (  y ) R x (  x )  x  y  z are called the Euler angles Note that rotations do not commute We can use rotations in another order but with different angles

18 18Computer Graphics Rotation Matrix glRotate produces a rotation of angle degrees around the vector (xyz) The current matrix is multiplied by this rotation matrix with the product replacing the current matrix, as if glMultMatrix were called with that same matrix for its argument void glRotatef (GLfloat angle, GLfloat x, GLfloat y, GLfloat z ); Where c=cos(angle), s=sin(angle) and ||(xyz)||=1 (if not, the GL will normalize this vector). x 2 (1−c)+cxy(1−c)−zsxz(1−c)+ysx yx(1−c)+zsy 2 (1−c)+cyz(1−c)−xsy xz(1−c)−ysyz(1−c)+xsz 2 (1−c)+cz 0001

19 19Computer Graphics Rotation About a Fixed Point other than the Origin Move fixed point to origin Rotate Move fixed point back M = T(p f ) R(  ) T(-p f )

20 20Computer Graphics Scaling S = S(s x, s y, s z ) = x’=s x x y’=s y x z’=s z x p’=Sp Expand or contract along each axis (fixed point of origin)

21 21Computer Graphics Scale Matrix glScale produces a nonuniform scaling along the x, y, and z axes. The three parameters indicate the desired scale factor along each of the three axes. The current matrix is multiplied by this scale matrix, with the product replacing the current matrix as if glScale were called with that same matrix as its argument void glScalef (GLfloat x, GLfloat y, GLfloat z ); x000 0y00 00z0 0001

22 22Computer Graphics Reflection corresponds to negative scale factors original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

23 23Computer Graphics Inverses Although we could compute inverse matrices by general formulas, we can use simple geometric observations –Translation: T -1 (d x, d y, d z ) = T (-d x, -d y, -d z ) –Rotation: R -1 (  ) = R(-  ) Holds for any rotation matrix Note that since cos(-  ) = cos(  ) and sin(-  )=-sin(  ) R -1 (  ) = R T (  ) –Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z )

24 24Computer Graphics Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The difficult part is how to form a desired transformation from the specifications in the application

25 25Computer Graphics Order of Transformations Note that matrix on the right is the first applied Mathematically, the following are equivalent p’ = ABCp = A(B(Cp)) Note many references use column matrices to represent points. In terms of column matrices p ’T = p T C T B T A T


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