Transformations. Prof. George Wolberg Dept. of Computer Science City College of New York

Transcription

1 Transforations Prof. George Wolberg Dept. of Coputer Science City College of New York

2 Objectives Introduce standard transforations - Rotations - Translation - Scaling - Shear Derive hoogeneous coordinate transforation atrices Learn to build arbitrary transforation atrices fro siple transforations Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 2

3 General Transforations A transforation aps points to other points and/or vectors to other vectors v=t(u) Q=T(P) Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 3

4 Pipeline Ipleentation T (fro application progra) u v transforation T(u) T(v) rasterizer frae buffer v T(v) u T(u) T(u) vertices vertices pixels T(v) Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 4

5 Translation Move (translate, displace) a point to a new location P P d Displaceent deterined by a vector d - Three degrees of freedo -P =P+d Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 5

6 Object Translation Every point in object is displaced by sae vector object Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 translation: every point is displaced by sae vector 6

7 Translation Using Representations Using the hoogeneous coordinate representation in soe frae p = [ x y z 1] T p = [x y z 1] T d = [dx dy dz ] T Hence p = p + d or x = x+dx y = y+dy z = z+dz note that this expression is in four diensions and expresses that point = vector + point Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 7

8 Translation Matrix We can also express translation using a 4 x 4 atrix T in hoogeneous coordinates p =Tp where T = T(d x, d y, d z ) = This for is better for ipleentation because all affine transforations can be expressed this way and ultiple transforations can be concatenated together d d d x y z 1 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 8

9 Rotation (2D) Consider rotation about the origin by degrees - radius stays the sae, angle increases by x = r cos ( y = r sin ( x = x cos y sin y = x sin + y cos x = r cos y = r sin Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 9

10 Rotation about the z-axis Rotation about z axis in three diensions leaves all points with the sae z - Equivalent to rotation in two diensions in planes of constant z x = x cos y sin y = x sin + y cos z = z - or in hoogeneous coordinates p =R z ()p Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 1

11 Rotation Matrix R = R z () = cos sin sin cos 1 1 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

12 Rotation about x and y axes Sae arguent as for rotation about z-axis - For rotation about x-axis, x is unchanged - For rotation about y-axis, y is unchanged 1 cos - sin R = R x () = sin cos 1 cos sin R = R y () = 1 - sin cos 1 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

13 Scaling Expand or contract along each axis (fixed point of origin) x =s x x y =s y y z =s z z p =Sp S = S(s x, s y, s z ) = sx s y s z 1 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

14 Reflection corresponds to negative scale factors s x = -1 s y = 1 original s x = -1 s y = -1 s x = 1 s y = -1 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

15 Inverses Although we could copute inverse atrices by general forulas, we can use siple geoetric 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 atrix 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 ) Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

16 Concatenation We can for arbitrary affine transforation atrices by ultiplying together rotation, translation, and scaling atrices Because the sae transforation is applied to any vertices, the cost of foring a atrix M=ABCD is not significant copared to the cost of coputing Mp for any vertices p The difficult part is how to for a desired transforation fro the specifications in the application Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

17 Order of Transforations Note that atrix on the right is the first applied Matheatically, the following are equivalent p = ABCp = A(B(Cp)) Note any references use colun atrices to present points. In ters of colun atrices p T = p T C T B T A T Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

18 General Rotation About the Origin A rotation by about an arbitrary axis can be decoposed 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 y v Note that rotations do not coute We can use rotations in another order but with different angles z Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 x 18

19 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 ) Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

20 Shear Helpful to add one ore basic transforation Equivalent to pulling faces in opposite directions Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 2

21 Shear Matrix Consider siple shear along x axis x = x + y cot y = y z = z 1 H() = cot Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

22 M A vertex is transfored by 4 4 atrices All atrices are stored colun-ajor in OpenGL - this is opposite of what C prograers expect Matrices are always post-ultiplied - product of atrix and vector is 3D Transforations Mv v v v v v

23 23 Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 Affine Transforations Characteristic of any iportant transforations - Translation - Rotation - Scaling - Shear Line preserving ' ' ' z y x z y x

24 OpenGL Transforations Prof. George Wolberg Dept. of Coputer Science City College of New York

25 Objectives Learn how to carry out transforations in OpenGL - Rotation - Translation - Scaling Introduce QMatrix4x4 and QVector3D transforations - Model-view - Projection Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

26 Current Transforation Matrix (CTM) Conceptually there is a 4 x 4 hoogeneous coordinate atrix, the current transforation atrix (CTM) that is part of the state and is applied to all vertices that pass down the pipeline The CTM is defined in the user progra and loaded into a transforation unit vertices p C CTM p =Cp vertices Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

27 CTM operations The CTM can be altered either by loading a new CTM or by postutiplication Load an identity atrix: C I Load an arbitrary atrix: C M Load a translation atrix: C T Load a rotation atrix: C R Load a scaling atrix: C S Postultiply by an arbitrary atrix: C CM Postultiply by a translation atrix: C CT Postultiply by a rotation atrix: C CR Postultiply by a scaling atrix: C CS Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

28 Rotation about a Fixed Point Start with identity atrix: C I Move fixed point to origin: C CT Rotate: C CR Move fixed point back: C CT -1 Result: C = TR T 1 which is backwards. This result is a consequence of doing postultiplications. Let s try again. Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

29 Reversing the Order We want C = T 1 R T so we ust do the operations in the following order C I C CT -1 C CR C CT Each operation corresponds to one function call in the progra. The last operation specified is the first executed in the progra! Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

30 Rotation, Translation, Scaling Create an identity atrix: QMatrix4x4 ;.settoidentity(); Multiply on right by rotation atrix of theta in degrees where (vx, vy, vz) define axis of rotation.rotate(theta, QVector3D(vx, vy, vz)); Do sae with translation and scaling:.scale(sx, sy, sz);.translate(dx, dy, dz); Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley 212 3

31 Exaple Rotation about z axis by 3 degrees with a fixed point of (1., 2., 3.) QMatrix4x4 ;.settoidentity();.translate( 1., 2., 3.);.rotate (3., QVector3D(.,., 1.));.translate(-1.,-2.,-3.); Reeber that the last atrix specified is the first applied Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

32 Arbitrary Matrices Can load and ultiply by atrices defined in the application progra Matrices are stored as one diensional array of 16 eleents which are the coponents of the desired 4 x 4 atrix stored by coluns OpenGL functions that have atrices as paraeters allow the application to send the atrix or its transpose Angel/Shreiner: Interactive Coputer Graphics 6E Addison-Wesley

33 Vertex Shader for Rotation of Cube (1) in vec4 vposition; in vec4 vcolor; out vec4 color; unifor vec3 theta; void ain() { // Copute the sines and cosines of theta for // each of the three axes in one coputation. vec3 angles = radians( theta ); vec3 c = cos( angles ); vec3 s = sin( angles );

34 Vertex Shader for Rotation of Cube (2) // Reeber: these atrices are colun ajor at4 rx = at4( 1.,.,.,.,., c.x, s.x,.,., s.x, c.x,.,.,.,., 1. ); at4 ry = at4( c.y,., s.y,.,., 1.,.,., s.y,., c.y,.,.,.,., 1. );

35 Vertex Shader for Rotation of Cube (3) at4 rz = at4( c.z, s.z,.,., s.z, c.z,.,.,.,., 1.,.,.,.,., 1. ); } color = vcolor; gl_position = rz * ry * rx * vposition;

36 Sending Angles fro Application GLuint thetaid; // theta unifor location vec3 theta; // axis angles void display( void ) { glclear( GL_COLOR_BUFFER_BIT GL_DEPTH_BUFFER_BIT ); glunifor3fv( thetaid, 1, theta ); gldrawarrays( GL_TRIANGLES,, NuVertices ); }