1. RAC/RA Projection Types of Projection Simple Projections Generalized Projection

2. RAC/RA Conceptual model of the 3D viewing process Viewing In 3D

3. RAC/RA Projections projections In general, projections transform points in a coordinate system of dimension n into points in a coordinate system of dimension less than n. We shall limit ourselves to the projection from 3D to 2D. The projection is onto a plane rather than a curved surface The projectors are straight lines rather than curves planar geometric projections We will deal with planar geometric projections where:

4. RAC/RA projection projectors center of projection projection plane The projection of a 3D object is defined by straight projection rays (called projectors) emanating from a center of projection, passing through each point of the object, and intersecting a projection plane to form the projection. Projections

5. RAC/RA Planer Geometric Projections center of projectionprojection plane Two basic classes (on the basis of the distance of the center of projection from the projection plane) : Perspective projection perspective projection perspective projection : the distance is finite Parallel projection Parallel projectionParallel projection : the distance is infinite

6. RAC/RA 1. Perspective foreshortening 1. Perspective foreshortening The farther an object is from COP the smaller it appears Perspective foreshortening Perspective Projection- Anomalies

7. RAC/RA 2. Vanishing Points: 2. Vanishing Points: Any set of parallel lines not parallel to view plane appear to meet at some point. Perspective Projection- Anomalies Vanishing point

8. RAC/RA 3. View Confusion: 3. View Confusion: Objects behind the center of projection are projected upside down and backward onto the view-plane. Perspective Projection- Anomalies x y z P1 P2 P3 P1` P2` P3` C O

9. RAC/RA Perspective Projection- Anomalies 4. Topological distortion: 4. Topological distortion: A line segment joining a point which lies in front of the viewer to a point in back of the viewer is projected to a broken line of infinite extent. P 1 P 3 P' 3 C Y X Z P 2 2 1 View Plane Plane containing Center of Projection (C)

10. RAC/RA Subclasses of planar geometric projections Planner Geometric Projections N V z x y Projection Plane

11. RAC/RA direction of projection Vnormal to the projection plane N Two types (on the basis of the relation between the direction of projection V and the normal to the projection plane N) : orthographic VNorthographic : V and N are the same or the reverse of each other. obliqueVN oblique : V and N are neither same nor reverse. Parallel Projections

12. RAC/RA Orthographic parallel projections Orthographic Projections

13. RAC/RA Axonometric Projections projection plane normal N = (d x, d y, d z Axonometric Projections use projection planes that are not normal to a principal axis.On the basis of projection plane normal N = (d x, d y, d z ) subclasses are: Isometric d x | = d y | = d z | NIsometric : | d x | = | d y | = | d z | i.e. N makes equal angles with all principal axes. othersN others : N makes unequal angles with one or more principal axes. Axonometric Projections

14. RAC/RA Oblique Projections  : is the angle the projection makes with x-axis  : angle between view plane and direction of projection l : original length of a line perpendicular to view plane l : projected length of a line perpendicular to view plane

15. RAC/RA Cavalier & Cabinet Cavalier projection  = 45  l = l Cabinet projection  = 63.4  l = l/2

16. RAC/RA Projective Projection

17. RAC/RA

18. Perspective Projections 3-Vanishing Point 1-Vanishing Point 2-Vanishing Point

19. RAC/RA

20. Settings for perspective projection Projective Transformations

21. RAC/RA Projective Transformations z

22. RAC/RA Matrices for Projective Trans.

23. RAC/RA Projective Transformations C(0,0,-d) y x z xpxp P(x,y,z) P(x p,y p,0) x Alternative approach, Projection plane at Z=0 And Center at Z=-d

24. RAC/RA Orthogonal Projection Matrix

25. RAC/RA Projection To study Foley: 6.1, 6.1.1, 6.1.2 Schaum: 7.1, 7.2, 7.3 Problems: 7.3, 7.4, 7.5, 7.10, 7.11, 7.12

26. RAC/RA