370 likes | 849 Views
Projection. Types of Projection Simple Projections Generalized Projection. Viewing In 3D. Conceptual model of the 3D viewing process. Projections.
E N D
Projection Types of Projection Simple Projections Generalized Projection RAC/RA
Viewing In 3D Conceptual model of the 3D viewing process RAC/RA
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
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
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
Perspective Projection-Anomalies 1. Perspective foreshortening The farther an object is from COP the smaller it appears Perspective foreshortening RAC/RA
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
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
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
Planner Geometric Projections N V y Projection Plane x z Subclasses of planar geometric projections RAC/RA
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
Orthographic Projections Orthographic parallel projections RAC/RA
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
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
Cavalier & Cabinet Cavalier projection b = 45°; l¢ = l Cabinet projection b = 63.4°; l¢ = l/2 RAC/RA
Projective Projection RAC/RA
Perspective Projections 1-Vanishing Point 2-Vanishing Point 3-Vanishing Point RAC/RA
Projective Transformations Settings for perspective projection RAC/RA
Projective Transformations z RAC/RA
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
Orthogonal Projection Matrix 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 RAC/RA