13 Example Viewplane origin (1,2,0) u-axis (3,4,0) v-axis (-4,3,0)
14 Orthographic Projection Def: direction of projection viewplane v n … is a parallel projection
15 Definitions Direction cosine (ref)ref Foreshortening ratio = (length of projected segment)/(length of original segment)
16 Theorem If the direction cosines of the plane normal (in world coordinate system) are n 1, n 2, and n 3, the foreshortening ratios in the x-, y-, and z- directions are (n 2 2 + n 3 2 ) 1/2, (n 1 2 + n 3 2 ) 1/2, and (n 1 2 + n 2 2 ) 1/2, respectively. Front, side, top views: n = (1,0,0,0), (0,1,0,0), or (0,0,1,0) as in engineering drawings
17 Types of Orthographic Projections Axonometric projections: attempts to portray general 3D shape –Isometric projection: all foreshortening ratio are the same –Dimetric projection: exactly two are the same –Trimetric projection: all foreshortening ratio are different
26 Vanishing Point Suppose (x i, y i, z i ) i =1,2,3 are a set of mutually perpendicular vectors. The viewplane normal (n 1, n 2, n 3 ) of a perspective projection can be perpendicular to (a) none (b) one (c) two of the vectors. (a) (b)(c) n n n
27 Vanishing Point If a perspective projection maps a point-at- infinity (x,y,z,0) to a finite point (x’,y’,z’,1) on the viewplane, the lines in the direction (x,y,z) appear as lines converging to point on the (Cartesian) viewplane. The point (x’,y’,z’) is called the vanishing point in the direction (x,y,z).
28 Three-point perspective Two-point perspective One-point perspective Vanishing point
29 IMAGE FORMATION – Perspective Imaging Image courtesy of C. Taylor “The Scholar of Athens,” Raphael, 1518
30 Example Determine (and verify it is indeed so) the vanishing point of an OpenGL setting. Eye = [15,0,0] Eye = [15,0,15]
31 Numeric Example How about (1,0,1,0)? Viewpoint (15,0,15,1) Viewplane: x + z + 1 = 0
32 Summary Projection –Parallel projection –Perspective projection Parallel projection –Orthographic Isometric Dimetric Trimetric –Oblique Cavalier Cabinet Perspective projection –Three-point perspective –Two-point perspective –One-point perspective Understand how they are differentiated
33 Fig. 8. Constructing a perspective image of a house. (a) Drawing the floor plan and defining the viewing conditions (observer position and image plane). (b) Constructing a perspective view of the floor. (c) A reference height (in this case the height of an external wall) is drawn from the ground line and the first wall is constructed in perspective by joining the reference end points to the horizontal vanishing point v2. (d) All four external walls are constructed. (e) The elevations of all other objects (the door, windows and roofs) are first defined on the reference segment and then constructed in the rendered perspective view.
34 Exercise Hand sketch a perspective drawing of a house Use Maxima to compute 2-point perspective projection, setting viewplane coordinate system
35 Cross Ratio The cross-ratio of every set of four collinear points shown in this figure has the same value Cross ratio is preserved in projective geometry (ratio is NOT preserved) z1z1 z2z2 z3z3 z4z4