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Viewing and Projection. The topics. Interior parameters Projection type Field of view Clipping Frustum… Exterior parameters Camera position Camera orientation. Transformation Pipeline. Local coordinate. Local->World. ModelView Matrix. World coordinate. World->Eye. Eye coordinate.

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Viewing and Projection


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    1. Viewing and Projection

    2. The topics • Interior parameters • Projection type • Field of view • Clipping • Frustum… • Exterior parameters • Camera position • Camera orientation

    3. Transformation Pipeline Local coordinate Local->World ModelView Matrix World coordinate World->Eye Eye coordinate Projection Matrix Clip coordinate others Screen coordinate

    4. Projection • The projection transforms a point from a high-dimensional space to a low-dimensional space. • In 3D, the projection means mapping a 3D point onto a 2D projection plane (or called image plane). • There are two basic projection types: • Parallel: orthographic, oblique • Perspective

    5. Orthographic Projection Image Plane Direction of Projection z-axis z=k

    6. Orthographic Projection

    7. Oblique Projection Image Plane Direction of Projection

    8. Properties of Parallel Projection • Definition: projection directions are parallel. • Doesn’t look real. • Can preserve parallel lines Projection Parallel in 3D Parallel in 2D

    9. Properties of Parallel Projection • Definition: projection directions are parallel. • Doesn’t look real. • Can preserve parallel lines • Can preserve ratios Projection

    10. Properties of Parallel Projection • Definition: projection directions are parallel. • Doesn’t look real. • Can preserve parallel lines • Can preserve ratios • CANNOT preserve angles Projection

    11. Properties of Parallel Projection • Definition: projection directions are parallel. • Doesn’t look real. • Can preserve parallel lines • Can preserve ratios • CANNOT preserve angles • Often used in CAD, architecture drawings, when images can be used for measurement.

    12. Properties of Parallel Projection • No foreshortening Image Plane

    13. Perspective Projection • Perspective projection has foreshortening: Center of Projection Image Plane

    14. Perspective Projection • Images are mapped onto the image plane in different ways: An image (640*640) Center of Projection Image Plane

    15. Perspective Projection • Images are mapped onto the image plane in different ways: An image (640*640) Center of Projection Image Plane

    16. Perspective Projection • Angle of view tells us the mapping from the image to the image plane: Field of view Angle of view Center of Projection Image Plane

    17. Perspective Projection • Not everything will be displayed. Frustum Center of Projection Image Plane Near Plane Far Plane

    18. Perspective Projection • The frustum of perspective projection looks like: Center of Projection Image Plane Near Plane

    19. Ortho Projection • What’s the frustum of orthographic projection? Image Plane Near Far

    20. Homogenous Coordinates • In general, the homogeneous coordinate system is define as: • For example, In homogeneous space In 3D

    21. Homogenous Coordinates • Both homogenous vectors and matrices are scalable:

    22. Perspective Projection • Derivation: Z Axis Image Plane XY Plane

    23. Perspective Projection • Derivation: Z Axis Image Plane XY Plane

    24. Perspective Matrix • Derivation: How? The z axis now becomes useless…

    25. OpenGL Perspective Matrix • In practice, OpenGL uses the z axis for depth test. Its matrix looks like this: The depth value now can be used for depth test. We will discuss this in more details later...

    26. Vanishing Point • Given a ray: • Its projection:

    27. Vanishing Point • When t goes to infinity: • What if there is another ray: Parallel lines meet at the vanishing point.

    28. Properties of Perspective Projection • Lines are mapped to lines. • Parallel lines may not remain parallel. Instead, they may meet at the vanishing point. • Ratios are not preserved. • It has foreshortening effects. So it looks real. • Distances cannot be directly measured, as in parallel projection.

    29. Basic OpenGL Projection • Everything will be considered in the eye space: • Geometry objects have been transformed into the eye coordinate system using the GL_MODELVIEW matrix. • You define the projection matrix in GL_PROJECTION, also in the eye space. • OpenGL always assume that the viewing direction is the –z direction. • OpenGL automatically processes each vertex using GL_PROJECTION: • After projection, the frustum is converted into a canonical view volume ( [-1, 1] in all coordinates)

    30. OpenGL Orthographic Projection glOrtho(left, right, bottom, top, near, far) X range Y range Z range (right, top, far) (1, 1, 1) (left, bottom, near) (-1, -1, -1)

    31. OpenGL Orthographic Projection glOrtho(l, r, b, t, n, f) • Translation so that the center is the origin. • Scaling so that the size becomes (2, 2, 2).

    32. OpenGL Perspective Projection glFrustum(left, right, bottom, top, near, far) Always positive, although it’s facing the –z direction (left, right, bottom, top) may not be centered along the axis Frustum Center of Projection Near Plane (Image Plane) Far Plane

    33. OpenGL Perspective Projection gluPerspective(fov, aspect_ratio, near, far) Always positive, although it’s facing the –z direction ratio=image_width/image_height top: near*ctan(fov/2) right: near*ratio*ctan(fov/2) Frustum fov Center of Projection Near Plane (Image Plane) Far Plane

    34. OpenGL Perspective Projection glFrustum(…) is less useful than gluPerspective(...). But we can still use it for demonstration purpose next.

    35. OpenGL Perspective Projection Projection

    36. OpenGL Perspective Projection Projection in (-1, 1) in (-n, -f), why???

    37. OpenGL Perspective Projection

    38. OpenGL Perspective Projection (r/n, t/n, 1) Projection (l/n, b/n, -1)

    39. OpenGL Perspective Projection (r/n, t/n, 1) (1, 1, 1) Transform (l/n, b/n, -1) (-1, -1, -1) Scaling Translation

    40. OpenGL Perspective Projection (1, 1, 1) Projection (-1, -1, -1) Transformation Original Projection

    41. What happens after projection? • Clipping • Viewport transformation • Rasterization clipping (800, 800) (1, 1) viewport (-1, -1) (0, 0) rasterize

    42. Depth Test • Before rasterization, all processes are done based on vertices. • The z coordinate at each vertex is transformed into a new z value (or called the depth value). • During rasterization, the z value of each pixel is interpolated from vertices. • The z value then stored in the depth buffer, for occlusion tests. (smaller z means closer).

    43. Depth Test • The depth buffer is part of the frame buffer: • To enable or disable the depth buffer: • Without the depth test, the occlusion is determined by the drawing order. glutInitDisplayMode(GLUT_DOUBLE|GLUT_RGB|GLUT_DEPTH); glEnable(GL_DEPTH_TEST); glDisable(GL_DEPTH_TEST);

    44. Common Issues • When you set up the perspective matrix: • Near (n) cannot be zero! • f: ncannot be too large! (far cannot be too large, or near cannot be too small.) Why?