3D transformation and Curves in Computer Graphics
- 1. Module 5 Three Dimensional Geometric Transformations, Curves and Fractal Generation
- 5. 3D Scaling 5
- 11. General 3D Rotation( Rotation axis not parallel to coordinate Rotation about an axis that is parallel to one of the Coordinate Axes • Translate the object so that the rotation axis passes through the coordinate origin • Rotate the object so that axes of rotation coincides with one of the coordinate axes • Perform the specified rotation about the coordinate axis • Perform inverse rotations to bring the rotation axis back to its original orientation • Translate the object so that the rotation axis is moved back too its original position. 11
- 20. 3D Viewing • Viewing in 3D involves the following considerations: - • We can view an object from any spatial position, eg. In front of an object, Behind the object, In the middle of a group of objects, Inside an object, etc. • 3D descriptions of objects must be projected onto the flat viewing surface of the output device.
- 21. Viewing Coordinates • Generating a view of an object in 3D is similar to photographing the object. • Whatever appears in the view finder is projected onto the flat film surface. • Depending on the position, orientation and aperture size of the camera corresponding views of the scene is obtained.
- 22. Specifying The View Coordinates • For a particular view of a scene first we establish viewing-coordinate system. • A view-plane (or projection plane) is set up perpendicular to the viewing z- axis. • World coordinates are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane. xw zw yw xv zv yv P0=(x0 , y0 , z0)
- 23. Specifying The View Coordinates • To establish the viewing reference frame, we first pick a world coordinate position called the view reference point. • This point is the origin of our viewing coordinate system. If we choose a point on an object we can think of this point as the position where we aim a camera to take a picture of the object.
- 24. Projection • General definition • Transform points in n-space to m-space(m<n) • In computer graphics • Map viewing coordinates to 2D screen coordinates
- 25. Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier One-point Two-point Three-point
- 26. Parallel & Perspective • Parallel Projection • Perspective Projection
- 27. Parallel Projection • Center of projection is at infinity • Direction of projection (DOP) same for all points View Plane DOP
- 28. Example of Perspective Projection
- 30. Orthographic & Oblique • Oblique parallel projection • The projectors are inclined with respect to the view plane
- 33. Orthographic Projections • DOP perpendicular to view plane Front Top Side
- 38. Perspective Projection • Map points onto “view plane” along “projectors” emanating from “center of projection”(cop) View Plane Center of Projection Projectors
- 39. Vanishing Point
- 41. Types of Perspective Projection
- 48. Perspective Projection • How many vanishing point?
- 49. Perspective Projection • How many vanishing point? Three-point perspective
- 50. Perspective Projection • How many vanishing point? Three-point perspective Two-point perspective
- 51. Perspective Projection • How many vanishing point? Three-point perspective Two-point perspective One-point perspective
- 52. Curves
- 53. Spline
- 55. Convex Hull
- 56. Control Graph
- 57. Bézier curves
- 58. Bézier curves
- 59. Bézier curves 2
- 60. Bézier curves 3
- 61. Bézier curves 4
- 62. Bézier curves
- 64. Disadvantages
- 65. Numerical
- 67. B-Spline Curve • Each control point is associated with a unique basis function • Each point affects the shape of the curve over a range of parameter values where the basis function is non-zero local control
- 68. B-spline Curve
- 72. B-spline Curve
- 73. B-spline Curve
- 74. B-spline Curve
- 77. Koch Fractals
- 78. Thank You Click icon to add picture Koch Fractals
- 79. Click icon to add picture