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Projection

Projection. Types of Projection Simple Projections Generalized Projection. Viewing In 3D. Conceptual model of the 3D viewing process. Projections.

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Projection

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

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

  3. 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. We will deal with planar geometric projections where: • The projection is onto a plane rather than a curved surface • The projectors are straight lines rather than curves RAC/RA

  4. Projections 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. RAC/RA

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

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

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

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

  9. Perspective Projection-Anomalies P' P 3 1 Plane containing Center of Projection (C) Y P 3 C View Plane P 2 P' 2 P' 1 X Z 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. RAC/RA

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

  11. Parallel Projections Two types (on the basis of the relation between the directionof projection Vand thenormal to the projection plane N) : • orthographic: V and N are the same or the reverse of each other. • oblique : V and N are neither same nor reverse. RAC/RA

  12. Orthographic Projections Orthographic parallel projections RAC/RA

  13. Axonometric Projections Axonometric Projections use projection planes that are not normal to a principal axis.On the basis of projection planenormal N = (dx, dy, dz) subclasses are: • Isometric: | dx | = | dy | = | dz |i.e. N makes equal angles with all principal axes. • others : N makes unequal angles with one or more principal axes. RAC/RA

  14. 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 RAC/RA

  15. Cavalier & Cabinet Cavalier projection b = 45°; l¢ = l Cabinet projection b = 63.4°; l¢ = l/2 RAC/RA

  16. Projective Projection RAC/RA

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  18. Perspective Projections 1-Vanishing Point 2-Vanishing Point 3-Vanishing Point RAC/RA

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  20. Projective Transformations Settings for perspective projection RAC/RA

  21. Projective Transformations z RAC/RA

  22. Matrices for Projective Trans. RAC/RA

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

  24. Orthogonal Projection Matrix RAC/RA

  25. 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 RAC/RA

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