Reading Fole et al. Chapter 6 Optional Projections David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition, McGra-Hill, Ne York, 990, Chapter 3. Projections Projections transform points in n-space to m-space, here m < n. In 3D, e map points from 3-space to the projection plane (PP) along projectors emanating from the center of projection (COP). Perspective vs. parallel projections Perspective projections pros and cons: + Sie varies inversel ith distance - looks realistic Distance and angles are not (in general) preserved Parallel lines do not (in general) remain parallel PP COP There are to basic tpes of projections: Perspective - distance from COP to PP finite Parallel - distance from COP to PP infinite Parallel projection pros and cons: Less realistic looking + Good for eact measurements + Parallel lines remain parallel Angles not (in general) preserved
Parallel projections Orthographic Projections For parallel projections, e specif a direction of projection (DOP) instead of a COP. There are to tpes of parallel projections: Orthographic projection DOP perpendicular to PP Oblique projection DOP not perpendicular to PP There are to especiall useful kinds of oblique projections: Cavalier projection DOP makes 45 angle ith PP Does not foreshorten lines perpendicular to PP Cabinet projection DOP makes 63.4 angle ith PP Foreshortens lines perpendicular to PP b one-half Oblique projections Oblique Projections To standard oblique projections: Cavalier projection DOP makes 45 angle ith PP Does not foreshorten lines perpendicular to PP Cabinet projection DOP makes 63.4 angle ith PP Foreshortens lines perpendicular to PP b one-half 2
Projection taonom Properties of projections The perspective projection is an eample of a projective transformation. Here are some properties of projective transformations: Lines map to lines Parallel lines don t necessaril remain parallel Ratios are not preserved Coordinate sstems for CG Model space for describing the objections (aka object space, orld space ) World space for assembling collections of objects (aka object space, problem space, application space ) Ee space a canonical space for vieing (aka camera space ) Screen space the result of perspective transformation (aka normalied device coordinate space, normalied projection space ) Image space a 2D space that uses device coordinates (aka indo space, screen space, normalied device coordinate space, raster space ) Ee Acts as the COP Placed at the origin Looks don the -ais Screen Lies in the PP Perpendicular to -ais At distance d from the ee Centered on -ais, ith radius s A tpical ee space Q: Which objects are visible? 3
Ee space Í screen space Q: Ho do e perform the perspective projection from ee space into screen space? P(,,) d Using similar triangles gives: Pd( p, p, d) d p P(,,) Ee space Í screen space, cont. We can rite this transformation in matri form: X 0 0 0 Y 0 0 0 = MP = = Z 0 0 0 W 0 0 / d 0 / d Perspective divide: X / W / d Y/ W = Z/ W / d W / W d Projective Normaliation After perspective transformation and perspective divide, e appl parallel projection (drop the ) to get a 2D image. Perspective depth Q: What did our perspective projection do to? Often, it s useful to have a around e.g., for hidden surface calculations. 4
Vanishing points Under perspective projections, an set of parallel lines that are not parallel to the PP ill converge to a vanishing point. A line Vanishing points p v p v P+ tv = + t p v 0 After perspective transformation e get: Vanishing points of lines parallel to a principal ais,, or are called principal vanishing points. Ho man of these can there be? p tv l + p + tv l = p + tv d Vanishing points, cont d p + tv Dividing b : d l p tv p tv l + = d p tv Letting t go to infinit: v d l v ( ) lim p tv lim p + tv v + v l d = d = d = d t p t tv ( p ) v tv v We get a point! This point does not depend on P so an line in the direction v ill go to the same point. Tpes of perspective draing Perspective draings are often classified b the number of principal vanishing points. One-point perspective simplest to dra To-point perspective gives better impression of depth Three-point perspective most difficult to dra All three tpes are equall simple ith computer graphics. 5
General perspective projection World Space Camera In general, the matri p q r s performs a perspective projection into the plane p + q + r + s =. Q: Suppose e have a cube C hose edges are aligned ith the principal aes. Which matrices give draings of C ith one-point perspective? to-point perspective? three-point perspective? Hither and on planes Projection taonom In order to preserve depth, e set up to planes: The hither (near) plane The on (far) plane 6
Summar Here s hat ou should take home from this lecture: The classification of different tpes of projections. The concepts of vanishing points and one-, to-, and three-point perspective. An appreciation for the various coordinate sstems used in computer graphics. Ho the perspective transformation orks. 7