Reading Required: Watt, Section 5.2.2 5.2.4. Further reading: 8. Projections Fole, et al, Chapter 5.6 and Chapter 6 David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed., McGra-Hill, Ne York, 990, Chapter 2. 3D Geometr Pipeline 3D Geometr Pipeline (cont d) Before being turned into piels, a piece of geometr goes through a number of transformations... Ee space (Vie space) Model space (Object space) scale, translate, rotate,... Projective transformation, scale, translate World space (Object space) Normalied projection space Project, scale, translate rotate, translate Normalied device space (Screen space) Ee space (Vie space) scale Image space (Windo space) (Raster space) (Screen space) (Device space)
Projections Projections transform points in n-space to m-space, here m<n. In 3-D, e map points from 3-space to the projection plane () along projectors emanating from the center of projection (): The center of projection is eactl the same as the pinhole in apinhole camera. There are to basic tpes of projections: Perspective distance from to finite Parallel distance from to infinite DOP Parallel projections For parallel projections, e specif a direction of projection (DOP) instead of a. There are to tpes of parallel projections: Orthographic projection DOP perpendicular to Oblique projection DOP not perpendicular to We can rite orthographic projection onto the 0 plane ith a simple matri. 0 0 0 0 0 0 0 0 0 Normall, e do not drop the value right aa. Wh not? Oblique parallel projections There are to standard kinds of oblique projections: Properties of parallel projection Properties of parallel projection: Cavalier projection DOP makes 45 degree angle ith Does not foreshorten lines perpendicular to Cabinet projection DOP makes a 63.4 degree angle ith Foreshortens lines perpendicular to b onehalf Not realistic looking Good for eact measurements Are actuall a kind of affine transformation Parallel lines remain parallel Angles not (in general) preserved Most often used in CAD, architectural draings, etc., here taking eact measurement is important /2 N V Cavalier (2D) Cabinet (2D) Oblique projection geometr (3D)
Derivation of perspective projection Consider the projection of a point onto the projection plane: Homogeneous coordinates revisited Remember ho e said that affine transformations ork ith the last coordinate alas set to one. What happens if the coordinate is not one? d (', ', -d) (,, ) B similar triangles, e can compute ho much the and coordinates are scaled: We divide all the coordinates b : / / / If, then nothing changes. Sometimes e call this division step the perspective divide. Note: Watt uses a left-handed coordinate sstem, and he looks don the + ais, and the is at +d. Homogeneous coordinates and perspective projection No e can re-rite the perspective projection as a matri equation: 0 0 0 0 0 0 0 0 / d 0 / d Projective normaliation After appling the perspective transformation and dividing b, e are free to do a simple parallel projection (drop the ) to get the 2D image. What does this impl about the shape of things after the perspective transformation + divide? After division b, e get: d d Again, projection implies dropping the coordinate to give a 2D image, but e usuall keep it around a little hile longer.
Vanishing points Vanishing points (cont d) What happens to to parallel lines that are not parallel to the projection plane? Think of train tracks receding into the horion... Dividing b : p + tv d p + tv p tv + d p + tv The equation for a line is: p v p v l p+ tv + t p v 0 After perspective transformation e get: p + tv p + tv ( p + tv)/ d Letting t go to infinit: We get a point! What happens to the line l q + tv? Each set of parallel lines intersect at a vanishing point on the. Q: Ho man vanishing points are there? Principal vanishing points If e define a set of principal aes in orld coordinates, i.e., the,, and aes, then it s possible to choose the viepoint such that these aes ill converge to different vanishing points. The vanishing points of the principal aes are called the principal vanishing points. Eample: vieer image 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. Properties of perspective projections The perspective projection is an eample of a projective transformation. Here are some properties of projective transformations: Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved One of the advantages of perspective projection is that sie varies inversel ith distance looks realistic. A disadvantage is that e can't judge distances as eactl as e can ith parallel projections. Q: Wh did nature give us ees that perform perspective projections? Q: Do our ees ``see in 3D''?
Clipping and the vieing frustum The center of projection and the portion of the projection plane that map to the final image form an infinite pramid. The sides of the pramid are clipping planes. Frequentl, additional clipping planes are inserted to restrict the range of depths. These clipping planes are called the near and far or the hither and on clipping planes. D Summar What to take aa from this lecture: All the boldfaced ords. An appreciation for the various coordinate sstems used in computer graphics. Ho the persepctive transformation orks. Ho e use homogeneous coordinates to represent perspective projections. The classification of different tpes of projections. The concepts of vanishing points and one-, to-, and three-point perspective. The mathematical properties of projective transformations. Near (Hither) Far (Yon) All of the clipping planes bound the the vieing frustum.