Reading. Projections. Projections. Perspective vs. parallel projections. Foley et al. Chapter 6. Optional. Perspective projections pros and cons:

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1 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

2 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

3 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

4 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 Y = MP = = Z 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

5 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

6 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

7 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

Reading. Projections. The 3D synthetic camera model. Imaging with the synthetic camera. Angel. Chapter 5. Optional

Reading. Projections. The 3D synthetic camera model. Imaging with the synthetic camera. Angel. Chapter 5. Optional Reading Angel. Chapter 5 Optional Projections David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition, McGraw-Hill, New York, 1990, Chapter 3. The 3D snthetic camera