CS 352: Computer Graphics

1. CS 352: Computer Graphics Chapter 5: Viewing

2. Overview • Specifying the viewpoint • Specifying the projection • Types of projections • Viewing APIs • Walking through a scene • Projections and shadows

3. How do cameras work?

4. Synthentic camera model • 1. Camera is placed at a location, pointed in a direction (modeling matrix) • 2. 3D points are flattened onto the viewing plane (projection matrix) • What do we need to know about the camera (real or synthetic)?

5. Synthetic camera parameters • Position of camera • Direction it is pointed [look vector] • Angle of film to look vector [view plane normal] • Rotation around viewing direction [up vector] • Height angle (zoom setting) [fovy] • Aspect ratio of "film" (width/height) • Front and back clipping planes • Focal length • Field of view • Shutter speed

7. Perspective distortion • How would you film dizziness? • Vertigo effect [2]

8. Projections • Basic Elements: • Objects, viewer • Projection plane • Projectors • Center of projection • Direction of projection (DOP) • Basic Types • Perspective • Parallel (COP at infinity)

9. Classical viewing

10. Orthographic projection • Orthographic: parallel projection with projectors perpendicular to the projection plane. • Often used as front, side, top views for 3Ddesign • Importance: preservationof distance and angle • Often used for top, front, and size views, e.g. in a modeling program or working drawing

11. Perspective projection • Perspective projections: projectors converge at COP • Classical perspective views: 1, 2, and 3-point (1, 2, or 3 vanishing points) • Difference: how many of the principle axes of the object are parallel to projection plane (I.e., depends on relationship of object to viewing frame)

12. 1. Position the camera • By default, camera is at origin, looking in –z dir • To “move the camera”, set up a modelview matrix that moves objects that are drawn • Ignore Z-coordinate when drawing • E.g. dimetric view? modelview = identity translate(0,0,-d) rotate(-45,<0,1,0>);

13. Exercise: look from +x axis • How would you change the camera to generate a view down the +x axis to origin? • Do this before displaying objects: modelview = identity; translate(0, 0, -d); rotate(-90, [0, 1, 0]);

14. Exercise: front/top view • How would you change the camera to generate a view from (0, 10, 10) to origin? • Do this before displaying objects: modelview = identity; translate(0,0,-14.14); rotate(45, [1, 0, 0]);

15. Helper function: lookAt • Most 3D toolkits let you position the camera by setting eyepoint,lookpoint, and up direction • lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup): • Effect: set themodelviewmatrix

16. Rolling your own lookAt • How could you write your own lookAt function? lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup):

17. Defining a lookAt function • lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup): • translate <Xat, Yat, Zat> to origin • rotate so that <Xeye, Yeye, Zeye>points in the Z direction • normalize <Xeye, Yeye, Zeye> • trackball-like rotation to <0,0,1> • rotate so <Xup, Yup, Zup> is <0,1,0> • trackball-like rotation

18. Camera API 2: uvn frame • Camera parameters: • VRP: view reference point, a point on the image plane • VPN: view plane normal (n) • VUP: vector in up direction • (also need viewing direction, if not VPN) • Result: viewing coordinate system, u-v-n. • v = projection of VUP onto image plane • u = v x n • u, v axes: coordinates in the image plane • n axis: normal to image plane

19. Camera API 3: roll, pitch, yaw • Specify location + orientation: roll, pitch, yaw

20. 2. Specify projection • Once we have located and pointed the camera along the –z axis, we still need to specify the lens (projection).

21. Parallel projection • We’re already looking along the –z axis • Set z=0 for all points (or ignore z coordinate when rendering)

22. Parallel projection • View volume is generally specified with clipping planes: e.g. glOrtho(xmin, xmax, ymin, ymax, near, far) • Z clipping planes at –near and –far

23. Perspective projection • Need to build appropriate perspective projection matrix into vertex shader • What kind of transformation would this be?

24. Perspective projections • COP at origin • Looking in –z direction • Projection plane in front of origin at z=d

25. Foreshortening • By similar triangles in previous image, we see that and similarly for y. • Using the perspective matrix we get p’ = • Adding divide-by-w to the graphics pipeline gives the correct result.

26. Perspective Frustum • Perspective viewing region is a “frustum”: Viewplane normally coincides with front clip plane

27. Camera APIs • In raw OpenGL ES, you “position the camera” by programming a vertex shader to apply a modelview matrix • Frameworks provide functions to build a viewing matrix for you, using a “camera API” • Example: perspectiveCamera(FOV, aspect, near, far)

28. Perspective projection • 3D graphics toolkits provide tools for specifying a perspective projection, e.g.

29. Shadows • How can one generate shadows in a scene using interactive graphics techniques? • In general it's hard, not supported in standard graphics pipeline—you need to know where everything is globally to render a point locally • Special techniques let you “fake it”

30. Projections and shadows • Projections can be used to generate simple shadow polygons • Light (xl, yl, zl) • Translate light to origin • Project down y axis [M] • Translate back

31. Simple Shadow in OpenGL GLfloat m[16]; //projection matrix for (int i=0; i<16; i++) m[i]=0; m[0]=m[5]=m[10]=1; m[7] = -1/yl; glBegin(); [draw polygon normally]; glEnd(); glMatrixMode(GL_MODELVIEW); glPushMatrix; //save state glTranslatef(xl, yl, zl); glMultMatrix(m); glTranslatef(-xl, -yl, -zl); glColor3fv(shadow_color); [draw polygon] glEnd(); glPopMatrix();

32. Stereo Viewing • In our stereo setup, we need two images (4x3 size ratio), side-by-side • We want to render perspective views from viewpoints (say) 3 inches apart • How to set up the views?* * “Simple, Low-Cost Stereographics: VR for Everyone,” J. Zelle & C. Figura, Proc. SIGCSE 2004 p. 348.

33. Direct both eyes at the same point?

34. Direct both eyes in parallel?

35. Parallel views and asymmetric frustum

36. Stereo viewing: // set up the projection transformation // focalLength is distance to screen (objects // closer will float in front of screen) top = eyeSeparation / 2.0 * (near / focalLength); glFrustum(-right+off, right+off, -top, top, near, far); // now set up the model-view transformation // right is a unit vector in right direction viewpoint = viewpoint – right * eyeOffset; center = center – right * eyeOffset; gluLookAt(viewpoint[X],viewpoint[Y],viewpoint[Z], center[X], center[Y], center[Z], up[x], up[y], up[z]);

37. Drawing left and right views //create window int width=400; int height=300; glutInitWindowSize(2*width, height); //------------------------------------- // draw left image glViewport (0, 0, width, height); // set up projection and modeling matrices // render image // draw right image glViewport (width, 0, width, height); // set up projection and modeling matrices // render image

38. Walking through a scene • How to animate viewer motion through a scene? [Demo] • Assume viewer’s height is fixed; looking direction is in y=6 plane • Store viewer’s location (x,6,z) and orientation (θ). Update appropriately with user commands • LookAt( x, y, z, x + cos(θ), y, z + sin(θ), 0, 1, 0);

39. Credits • 1. (Pinhole camera): Wikipedia. • 5. Synthetic camera parameters: Liz Marai, Pitt Demos • Musical solar system