Computer Graphics. Ch 6. 3D Viewing

Transcription

1 Computer Graphics Ch 6. 3D Viewing

2 3D Viewing Basic do you see this image as flat? 12 lines

3 3D Coordinate System 3D homogeneous coordinates: p = [x y z w] T Our textbook and OpenGL use a RIGHT-HANDED system y x note: z axis comes toward the viewer from the screen, viewer faces toward x-y plane viewer z

4 3D to 2D Projection display screen at z = 0 y x camera z note: z axis comes toward the viewer from the screen, viewer look straight down to x-y plane (toward z direction)

5 Projection conversion from 3D coordinates to 2D screen 3D scenes are projected to the screen 2 projection types perspective: good for games, movies, realistic direction of projection between an individual point in space and the projection plane depends on the location of the point in space parallel: z-coordinate is ignored, not realistic, good for technical design all points are projected onto 2D using the same projection direction shape is preserved, direct measurement is possible orthographic (or orthogonal): orthogonal to the projection plane oblique : at an angle with respect to the projection plane

6 Projection Orthgonal Oblique Perspective

7 Orthogonal Parallel Projection throw away z-coordinates from all points in space draw points on the X-Y projection plane (screen) preserve lengths and angles (shapes), direct measurement possible p0(-5, 10, 5) p0(-5, 10, 0) p2(5, 0, 0) p1(-2, 2, 0) p2(5, 0, 0) p1(-2, 2, 12)

8 Orthogonal Parallel Projection collapse the world coordinates along the direction of projection front view (camera on z-axis facing z direction) : x = x, y = y side view (camera on x-axis facing -x direction) : x = -z, y = y top view (camera on y-axis facing y direction) : x = x, y = -z Ortho(front) Ortho(side) Ortho(top)

9 Oblique Parallel Projection direction of projection is not orthogonal to the projection plane convert all points p(x, y, z) in the scene to p (x, y ) so that the scene is viewed in α degree p (x, y ) α p(x, y, z) Ф orthogonal

10 Oblique Parallel Projection convert all points p(x, y, z) in the scene to p (x, y ) α: angle between the direction of projection and the projection plane Ф: angle between the red line below and the x-axis x = x + z*cos(ф) / tan(α), let x inc = cos(ф) / tan(α) y = y + z*sin(ф) / tan(α), y inc = sin(ф) / tan(α) orthogonal p(x, y, z) p (x, y ) α Ф 1 0 x inc y inc

11 Perspective objects farther away from the viewer look smaller than objects closer to the viewer when they are of the same size far-away objects ultimately disappear into a vanishing point

12 Vanishing Point place where lines converge in perspective one point perspective: head on, allows viewers to see depth two point perspective: at an angle, three point perspective

13 Perspective in Art

14 Perspective in Art

15 Perspective in Art The School of Athens by Raphael

16 Perspective in Art Ideal City by Pierro della Francesca

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23 Simple Perspective Setup camera on z-axis facing +z direction center of projection (focal point): origin projection plane is between the focal point and 3D objects focal length: d from the x-y plane parallel to x-y plane y focal point (0, 0, 0) x z focal length: d

24 Simple Perspective Setup x = x * d / z y = y * d / z z = d y p (x, y, z ) focal point (0, 0, 0) p(x, y, z) x z focal length: d

25 Perspective Projection how about when the center of projection is NOT at the Origin? what happens if the projection screen is beyond the center of projection? what happens if the center of projection is ON the projection plane?

26 Camera Setup LookFrom Up LookAt

27 Camera Setup LookFrom Up LookAt

28 Viewing Setup vertical field of view (angle) far LookFrom center of projection projection (image) plane near viewing volume

29 Viewing Setup horizontal field of view (angle) far LookFrom near center of projection projection plane