1. Supriya H. Madane

2. 3D Transformation Supriya H. Madane Slide 2

3. Types of 3D reference system according to co-ordiante axes.
Left handed System
Right handed System
3D Transformation Supriya H. Madane Slide 3

4. 1) Left handed System
3D Transformation Supriya H. Madane Slide 4

5. 2) Right handed System
3D Transformation Supriya H. Madane Slide 5

6. 3D Transformation Supriya H. Madane Slide 6

7. Translation
3D
Transformation
Rotation
3D Transformation Supriya H. Madane Slide 7

8. Translation in 3D!
 Remembering 2D transformations -> 3x3 matrices,
take a wild guess what happens to 3D transformations.
1 0 tx
x tx
T tx , t y 0 1 ty T=(tx, ty, tz)
y ty
0 0 1
1 0 0 tx
x tx
0 1 0 ty
T tx , t y , tz y ty
0 0 1 tz
z tz
0 0 0 1
3D Transformation Supriya H. Madane Slide 8

9. Scaling, 3D Style
sx 0 0 S=(sx, sy, sz)
sx 0 x
S sx , s y * 0 sy 0
0 sy y
0 0 1
sx 0 0 0
sx 0 0 x
0 sy 0 0
S sx , s y , sz 0 sy 0 * y
0 0 sz 0
0 0 sz z
0 0 0 1
3D Transformation Supriya H. Madane 9

10. Rotations about the Z axis
R=(0,0,1, )
cos sin 0
cos sin
R sin cos 0
sin cos
0 0 1
cos sin 0 0
sin cos 0 0
R (0,0,1, )
0 0 1 0
0 0 0 1
3D Transformation Supriya H. Madane 10

11. Rotations about the X axis
Let’s look at the other axis rotations
R=(1,0,0, )
1 0 0 0
0 cos sin 0
R (1,0,0, )
0 sin cos 0
0 0 0 1
3D Transformation Supriya H. Madane 11

12. Rotations about the Y axis
R=(0,1,0, )
cos 0 sin 0
0 1 0 0
R (0,1,0, )
sin 0 cos 0
0 0 0 1
3D Transformation Supriya H. Madane 12

13. Viewing in 3D
3D Transformation Supriya H. Madane Slide 13

14.  Man-made objects often have “cube-like” shape.
These objects have 3 principal axes.
From www.loc.gov/ jefftour/cutaway.html
3D Transformation Supriya H. Madane Slide 14

15. Display device
(a screen) is • How do we map 3D objects to 2D space?
2D…
2D to 2D is • 2D window to world.. and a viewport on the 2D
surface.
straight • Clip what won't be shown in the 2D window, and
forward… map the remainder to the viewport.
3D to 2D is
• Solution : Transform 3D objects on
more to a 2D plane using projections
complicated…
3D Transformation Supriya H. Madane Slide 15

16. Rays converge on eye position Rays parallel to view plane
Perspective Parallel
Orthographic Oblique
Elevations Axonometric Cavalier Cabinet
Top Left Right Isometric Dimetric Trimetric

17.  In 3D…
 View volume in the world
 Projection onto the 2D projection plane
 A viewport to the view surface
 Process…
 1… clip against the view volume,
 2… project to 2D plane, or window,
 3… map to viewport.
3D Transformation Supriya H. Madane Slide 17

18.  Conceptual Model of the 3D viewing process
3D Transformation Supriya H. Madane 18

19.  2 types of projections
 perspective and parallel.
 Key factor is the center of projection.
 if distance to center of projection is finite : perspective
 if infinite : parallel
3D Transformation Supriya H. Madane Slide 19

20.  Perspective:
 visual effect is similar to human visual system...
 has 'perspective foreshortening'
 size of object varies inversely with distance from the center of
projection.
 angles only remain intact for faces parallel to projection
plane.
 Parallel:
 less realistic view because of no foreshortening
 however, parallel lines remain parallel.
 angles only remain intact for faces parallel to projection
plane.
3D Transformation Supriya H. Madane Slide

21.  Any parallel lines not parallel to the projection
plane, converge at a vanishing point.
 There are an infinite number of these, 1 for each of the
infinite amount of directions line can be oriented.
 If a set of lines are parallel to one of the three
principle axes, the vanishing point is called an axis
vanishing point.
 There are at most 3 such points, corresponding to the
number of axes cut by the projection plane.
3D Transformation Supriya H. Madane Slide 21

22. One point, two point, three point perspective
One point perspective: One principal axis intersects view plane
3D Transformation Supriya H. Madane Slide

23. One point, two point, three point perspective
Two point perspective: two principal axes intersect view plane
3D Transformation Supriya H. Madane 23

24. One point, two point, three point perspective
3D Transformation Supriya Three principal axes intersect view plane
Three point perspective: H. Madane 24

25. View Plane
Three point
Two point
3D Transformation Supriya H. Madane
One point 25

26.  2 principle types:
 orthographic and oblique.
 Orthographic :
 direction of projection = normal to the projection plane.
 Oblique :
 direction of projection != normal to the projection
plane.
3D Transformation Supriya H. Madane Slide

27.  Orthographic (or orthogonal) projections:
 front elevation, top-elevation and side-elevation.
 all have projection plane perpendicular to a principle axes.
 Useful because angle and distance measurements can be
made...
 However, As only one face of an object is shown, it can be
hard to create a mental image of the object, even when
several view are available.
3D Transformation Supriya H. Madane Slide

28.  Orthogonal projections:
3D Transformation Supriya H. Madane 28

29. 3D Transformation Supriya H. Madane Slide

30. 3D Transformation Supriya H. Madane 30

31. 3D Transformation Supriya H. Madane 31