INF3320 Computer Graphics and Discrete Geometry

Transcription

1 INF3320 Computer Graphics and Discrete Geometry Transforms (part II) Christopher Dyken and Martin Reimers Page 1

2 Transforms (part II) Real Time Rendering book: Transforms (Chapter 4) The Red Book: Viewing (Chapter 3) Homogeneous Coordinates... (Appendix F) Page 2

3 Recap Page 3

4 Vertex transforms Page 4

5 Last time - highlights Coordinate systems Homogenous coordinates and transforms OpenGL matrix stacks Modelview transforms Page 5

6 Modelview transforms Transform vertex coordinates to camera coordinate system Page 6

7 Recap model transforms Two ways to think of model transforms: move local (drawing) coordinate system (read top-down) move objects in global coordinate-system (read bottom-up) glutwiresphere(1.0, 20, 16); /* draw sun */ glrotatef ( year, 0.0, 1.0, 0.0); gltranslatef (2.0, 0.0, 0.0); glrotatef ( day, 0.0, 1.0, 0.0); glutwiresphere(0.2, 10, 8); /* draw smaller planet */ Example on page 173 Page 7

8 Creating Transform Matrices Page 8

9 Designing transforms Example: rotation aound a point Page 9

10 OpenGL Mathematics - the glm templates C++ header-only templates for OpenGL mathematics Math utilities, Vectors, Matrices, Transformations Based on C/GLSL, can use in C++, shaders, CUDA etc Replacements for deprecated functionality (e.g. matrix stacks) E.g make and use a homogenous translation matrix: #include <glm/glm.hpp> //vec3, vec4, ivec4, mat4 glm::mat4 translation = glm::translate(glm::mat4(1.0f), glm::vec3(1.0f, 0.0f, 0.0f)); glm::vec4 point = glm::vec4(1.0f, 0.0f, 0.0f,1.0f)); glm::vec4 translated_point = translation*point; Page 10

11 Displace p with the vector t, p = p + t Translation p z Translation transform matrix x x T (x, y, z) = y z T 1 (x, y, z) = y z OpenGL gltranslate(x,y,z) multiplies the current matrix with T (x, y, z) glm::translate(mat,vec) multiplies mat (matrix) with T (vec) Page 11 t y p x

12 Rotation Rotation in the xy plane: R z (θ) Describe p in polar coordinates, p = [ x, y ] T = ρ [ cos(φ), sin(φ) ] T p is found by adding θ to the angle p = ρ [ cos(φ + θ), sin(φ + θ) ] T y p φ+θ and if we expand the expressions and substitute, we get x = x cos(θ) y sin(θ), y = x sin(θ) + y sin(θ) which in matrix form is [ ] [ ] [ ] x cos(θ) sin(θ) x y = sin(θ) cos(θ) y φ θ p x Page 12

13 Rotation about the x-axis (in the yz-plane) R x (θ) = 0 cos θ sin θ 0 0 sin θ cos θ 0 glrotate(θ, 1, 0, 0) Rotation about the y-axis (in the zx-plane) cos θ 0 sin θ 0 R y (θ) = sin θ 0 cos θ 0 glrotate(θ, 0, 1, 0) Rotation about the z-axis (in the xy-plane) cos θ sin θ 0 0 R z (θ) = sin θ cos θ glrotate(θ, 0, 0, 1) Page 13

14 Rotation θ radians about an arbitrary axis r = [a x, a y, a z ] One strategy is to decompose the operation to simpler operations: 1. Rotate about the x-axis s.t. r = R x (φ x )r is in the y =0-plane: φ x = ([0, a y, a z ], z) 2. rotate about the y-axis s.t. R y (φ y )r is in the x = 0 plane: φ y = ([a x, 0, [a y, a z ] ], z) 3. perform the intended rotation about the z-axis: R z (θ) 4. undo the rotation about the y-axis (R y ( φ y )) 5. undo the rotation about the x-axis (R x ( φ x )) which gives the rotation matrix R r (θ) = R x ( φ x )R y ( φ y )R z (θ)r y (φ y )R x (φ x ). Page 14

15 Rotation θ radians about an arbitrary axis r = [r x, r y, r z ] Another strategy is to change basis: 1. Find an orthonormal basis (r, s, t) for R 3 2. Change basis: M = [r, s, t] T 3. Rotate around x-axis: R x (θ) 4. Change back to cardinal basis: M T = M 1 which gives the rotation matrix R r (θ) = M T R x (θ)m Page 15

16 Direct expression for arbitrary rotation about unit vector r cos θ+(1 cos θ)r 2 x (1 cos θ)r x r y r x sin θ (1 cos θ)r x r z+r y sin θ R r(θ)= (1 cosθ)r x r y+r z sin θ cos θ+(1 cos θ)ry 2 (1 cos θ)r y r z r x sin θ (1 cos θ)r x r z r y sin θ (1 cos θ)r y r z+r x sin θ cos θ+(1 cos θ)rz 2 OpenGL glrotate*(θ, r x, r y, r z ) multiplies current matrix with R r (θ) glm function rotate(m,θ,v) does the same job on M Inverse of a rotation R(θ) 1 = R(θ) T = R( θ) Convention: rotation is clockwise wrt direction Page 16

17 Rigid body transforms Transforms that preserve lengths, angles and handedness M Consists of translations and rotations only Rigid body transforms can be written on the form [ ] R t M = 0 T 1 Simple inverse M 1 = [ R T R T ] t 0 T 1 Two rigid transforms = rigid transform Page 17

18 Scale Scale the coordinates by s = (s x, s y, s z ) Scale transformation matrices s x /s x S(s) = 0 s y s z 0, S(s) 1 = 0 1/s y /s z ( ) i.e. S 1 (s x, s y, s z ) = S( 1 s x, 1 s y, 1 s z ) OpenGL glscale(s x, s y, s z ) multiplies the current matrix with S(s x, s y, s z ) glm::scale(m,s) does the same job on the matrix M Page 18

19 Uniform scale If s x = s y = s z, we scale uniformly. On the right, s = (2, 2, 2) Non-uniform scale Otherwise, we scale non-uniformly. On the right, s = (1, 2, 1) Page 19

20 Scale along an arbitrary direction We can scale along any axis by combining scale and rotation: R z (45 ) S(1, 2, 1) R z ( 45 ) p = R z ( 45 )S(1, 2, 1)R z (45 )p Page 20

21 Shear We shear a shape by adding some of one component onto another. Shear of x by y We add sy to the x coordinate, x = x + sy, y = y, z = z. H xy In matrix form, this is 1 s 0 0 H xy (s) = , s 0 0 H 1 xy (s) = OpenGL Use e.g. glmultmatrix*() to set up shear Page 21

22 Euler transforms Page 22

23 Euler angles In R 2, we combine two rotations by adding the angles Euler transform: combine three rotations in R 3 head/yaw (around y axis) pitch (around x axis) roll (around z axis) Yields a composite rotation, E(h, p, r) = R z (r)r x (p)r y (h) Rotations are generally non-commutative! Other orderings of Euler angle rotations exists as well... Page 23

24 Gimbal lock happens when one axis is rotated to the direction of another = the rotation gets locked... (loose a degree of freedom) R z (r)r x (π/2)r y (h) = R x (π/2)r y (r + h) = R z (r + h)r x (π/2) Euler angles can go into Gimbal lock (so can aeroplane instruments!) Page 24

25 Quaternions Page 25

26 Quaternions The 4D generalization of complex numbers (2D). Complex numbers can describe rotations in R 2. Similarly, quaternions can describe rotations in R 3... and avoids the problems of Euler angles... Page 26

27 Quick re-cap on complex numbers A complex number=point in the complex plane: p = ρ(cos(θ), sin(θ)) = ρe iθ = ρ(cos θ + i sin θ) Im p φ θ+φ p The imaginary unit i has the property i 2 = 1 To rotate φ radians, simply multiply p with r = e iφ θ Re p = pr = ρe iθ e iφ = ρe i(θ+φ) = any rotation achieved with complex number r = 1 Page 27

28 Quaternion definition and properties A quaternion can be regarded a 4-dimensional complex number ˆp = q x i + q y j + q z k + q w (1) where the i, j, and k have the following properties: i 2 = j 2 = k 2 = 1, and ij = k = ji, jk = i = kj, ki = j = ik (2) We often use vector notation for quaternions, ˆp = [ ] T [ q x, q y, q z, q w = q T ] T, q w where q and q w represents the imaginary and real parts respectively Page 28

29 Quaternion properties Quaternions can be added and subtracted The product of two quaternions can be found by writing them on the form (1) and simplify using (2). ˆpˆq = [ The norm of ˆp, p w q + q w p + p q }{{} imaginary part n(ˆp) = ˆp =, p w q w p q }{{} real part q 2 w + q q. ] Quaternion multiplicative identity [0, 1] The conjugate of a quaternion: [q, q w ] = [ q, q w ] The inverse of a quaternion is (from ˆpˆp 1 = [0, 1]) ˆp 1 = 1 ˆp 2 ˆp Page 29

30 Unit quaternions and rotations A unit quaternion is a quaternion for which ˆq = 1 ˆq 1 = ˆq for unit quaternions Unit quaternions can represent arbitrary R 3 rotations: Rotation of 2φ around the unit vector v can be represented by ˆr = [ sin(φ)v, cos(φ) ] T with a homogeneous point p transformed by R v (2φ)p = ˆrpˆr The product of two unit quaternions is a unit quaternion To concatinate rotations ˆqˆrpˆr ˆq = ĉpĉ, ĉ = ˆqˆr The simplicity of the latter is a big advantage! Page 30

31 Rotation matrix from a unit quaternion Given the unit quaternion ˆq = (q x, q y, q z, q w ), the corresponding rotation matrix is given by 1 2(q y q y +q z q z ) 2(q x q y q z q w ) 2(q x q z +q y q w ) R = 2(q x q y +q z q w ) 1 2(q x q x +q z q z ) 2(q y q z q x q w ) 2(q x q z q y q w ) 2(q y q z +q x q w ) 1 2(q x q x +q y q y ) Finding a quaternion from a rotation matrix is slightly more complicated A rotation matrix is orthogonal: R 1 = R T Recall: homogeneous rotation matrix: [ ] R 0 M = 0 T. 1 Page 31

32 Concatenating rotations Two (or more) consecutive rotations can be combined (concatenated) into one new rotation. We can find this new rotation in several ways: 1. Multiply the rotation matrices for each rotation. Because of round-off errors, the matrices become more and more non-orthogonal. Can be fixed by Gram-Schmidt orthogonalization. 2. Multiply the quaternions for each rotation. Round-off errors make the quaternions become less and less unitary. Can be fixed by normalization, i.e. ˆq = 1 ˆq ˆq Normalizing a quaternion is a lot cheaper than orthogonalizing a matrix! Page 32

33 Other uses of quaternions Interpolation between directions / rotations, spherical linear interpolation Rotation that takes one vector s to another vector t 1 2(1 + e) ˆq = [ (s t), ] 2(1 + e) 2 where e = s t Page 33

34 Transforms in glm glm provides quaternions (and Euler angles) All operations like length, normalize, cross, dot, etc, lots of utilities glm::gtc::quaternions glm::gtx::quaternions. Example: glm::quat Q = glm::angleaxis(90.0f, glm::vec3(0, 0, 1)); glm::vec3 euler = glm::gtx::quaternion::eularangles(q); Page 34

35 Projective transforms Page 35

36 Map view volume into canonical view volume Page 36

37 Projection transform Map view volume into canonical view volume [ 1, 1] 3 View volumes are defined by the modelview and projection transforms Objects outside the view volume will be invisible Mapped (x, y, z) represent screen (x, y) and depth (z) coordinates respectively For this we need projections... Page 37

38 Projection types Two types of projections are used in graphics: Orthographic (Orthographic camera model) Perspective (Pinhole camera model) Page 38

39 Orthographic projections Orthographic projections works like an inifinite tele-lens To project a point onto the z = 0 plane: (x, y, z ) = (x, y, 0), Can be performed with 4 4 matrix x x y z = y BUT: Maps all points to the z = 0 plane, also negative z Need to instead map points in a view volume to [ 1, 1] 3 Page 39

40 Specify a view volume by x min, x max, y min, y max, z min, z max Ortographic (affine) transformation by two steps: 1. Translate the center of the view volume to the origin, 2. Scale to cube [ 1, 1] 3, T ( x min+x max 2, y min+y max 2, z min+z max 2 S ( 2 x max x min, 2 y max y min, 2 z max z min ), ), The mapping is thus 2 x max x min 0 0 x min+x max x max x min 2 0 P = ST = y max y min 0 y min+y max y max y min z max z min z min+z max z max z min Page 40

41 Orthogonal projections in OpenGL OpenGL: glortho(xmin,xmax,ymin,ymax,near,far) sets up the GL_PROJECTION matrix P Also, the utility library provides gluortho2d(xmin,xmax,ymin,ymax) which is equivalent with glortho(xmin,xmax,ymin,ymax,-1,1) call glmatrixmode(gl_projection) before projections!!!! Page 41

42 Perspective projection Assume that eye-point at the origin, looking along the negative z-axis, y is up and the projection plane is parallel to the xy-plane positioned at z = d for some d > 0. The projection of p = (x, y, z) is q = (x, y, z ) = ( dx z, dy z, d) Perspective projection is clearly non-affine Page 42

43 Homogeneous matrices can represent perspective projections: A homogeneous point can be arbitrary scaled for any w 0, (x, y, z, 1) is the same point as (wx, wy, wz, w) We can recover the representation (x, y, z, 1) by perspective division, i.e. dividing the coordinates by w We can therefore write the projection as x x y z = y z 0 } 0 1/d {{ 0 1 }}{{} z/d P p division dx/z dy/z dz/z } 1 {{ } p Provides the correct perspective projection but sends all points into the plane z = d. Need to map the view volume injectively to [ 1, 1] 3 Page 43

44 Perspective projection matrix The scalars (l, r, b, t, n, f ) specify a view frustum, with front face corners at (l, b, n) and (r, t, n) and back-face at depth z = f The perspective projection matrix reads P p = 2n r+l r l 0 r l 0 2n t+b 0 t b t b f +n f n 2fn f n With perspective division, maps near/far planes to z = 1 and z = 1 Page 44

45 Perspective projections in OpenGL OpenGL: glfrustum(l,r,b,t,n,f) determines the view frustum and perspective projection matrix P Notice that n and f are positive, however, the near and far planes are at z = n and z = f. An alternative is a function in the utility library, gluperspective(fovy,aspect,near,far) fovy is the field of view in the y direction, aspect is the aspect ratio (width/height) of the window. call glmatrixmode(gl_projection) before projections!!!! glm has similar functions: glm::frustum, glm::perspective etc Page 45

46 Viewport transformation After transformations given by modelview matrix projection matrix perspective division visible geometry has normalized device coordinates in [ 1, 1] 3 Use glviewport(x,y,w,h) to set pixel rectangle to draw in. Use the same aspect-ratio as the frustum to avoid distortion, i.e. w h = r l t b The viewport transformation yields window coordinates and depth (user determined interval e.g. [0, 1]) Page 46

47 Vertex-processing by glm mat4 Projection = glm::perspective(45.0f, 1.33f, 0.1f, 100.f); glm::mat4 ViewTranslate = glm::translate(glm::mat4(1.0f), trans); glm::mat4 ViewRotateX = glm::rotate(viewtranslate, rotate.x, glm::vec3(1.0f, 0.0f, 0.0f)); glm::mat4 View = glm::rotate(viewrotatex, rotate.y, glm::vec3(0.0f, 1.0f, 0.0f)); glm::mat4 Model = glm::scale(glm::mat4(1.0f),glm::vec3(0.5f)); glm::mat4 MVP = Projection * View * Model; glloadmatrixf(value_ptr(mvp)); Page 47

48 The last slide Today: more transformations, Euler angles, quaternions, projections Page 48

49 The last slide Today: more transformations, Euler angles, quaternions, projections Transformations are easy to get wrong... Many ways to get a black screen Remember glmatrixmode Check obvious bugs, e.g. drawing black on black Visualize the model and camera Understand modelview transforms now (or tomorrow)! Next time: Visual appearance Page 49