Must first specify the type of projection desired. When use parallel projections? For technical drawings, etc. Specify the viewing parameters

Transcription

1 CSE 480/580 Lecture 4 Slide 3-D Viewing Continued Eamples of 3-D Viewing Must first specif the tpe of projection desired When use parallel projections? For technical drawings, etc. When distance between points in the scene are small compared with distance between "camera" and scene Perspective Projections Specif the viewing parameters Viewing Parameter Value VRP(WC) (0, 0, 0) (defines one point of the VP) VPN(WC) (0, 0, ) (now VP is defined) VUP(WC) (0,, 0) (now u,v,n (VRC) coordinate sstem defined) PRP(VRC) (0.5, 0.5,.0) (defines the center of projection) window(vrc) (-, 3,-,3) (umin, uma, vmin, vma).0 2.0

2 CSE 480/580 Slide 2 How get a two- point perspective view of this scene? 2 Need to have view plane cut two aes (sa and ) VRP(WC) (2, 0.5, 0) (to get view plane in "front" of scene) VPN(WC) (, 0, ) (to get view plane to cut and aes) VUP(WC) (0,, 0) (to specifi the window orientation) PRP(VRC) (0, 0, 0) (to get center of projection) window(vrc) (-0, 20, -0, 20) (to specif view volume) What does projection look like? How to get a larger image? How know how to specif window? Can do it interactivel. CDQ: What if cut and aes?

3 CSE 480/580 Slide 3 Eample Parallel Projection 2 VRP(WC) (, 0.5, 0) (to get view plane reference point) VPN(WC) (0, 0, ) (to get view plane to cut aes) VUP(WC) (0,, 0) (to specifi the window orientation) PRP(VRC) (0, 0, 0) (to projection direction parallel to ais) window(vrc) (-2, 2, -2, 2) (to specif view volume)

4 CSE 480/580 Slide 4 How do we implement projections in our graphics sstems? We'll see we can do it with a 44 projection matri Look at the basic mathematics of planar projections Parallel projection start b assuming VP lies in plane if VRP = (0, 0, 0) VPN = (0, 0, ) VUP = (0,, 0) given (,, ) --> (p, p) p =? p =? Perspective projection start b assuming VP is normal to ais at = d and center of projection = (0, 0, 0) VP p d p = / ( / d) d p =

5 CSE 480/580 Slide 5 One-point perspective projection p = / ( /d) p = / (/d) Thus d is simpl a scale factor, and the division b gives the foreshortening How epress this as a transformation matri? X Y Z W = /d 0 X =, Y =, Z =, W = /d As homogeneous coordinates, divide b W to get another equal homogeneous point p = X/W = / (/d) p = / (/d) p = X Y Z W = p p p Mper = /d 0

6 CSE 480/580 Slide 6 Parallel Projection Matri p VP p = p = p = p p p = Mort = Mper onl good for center of projection at origin Mort onl good for VP in plane at origin

7 CSE 480/580 Slide 7 Class Discussion Question: How to get two-point perspective from Mper? Need to have VP cut two aes (preferabl and ) How do we need to transform VP? General Projection Matri 0 -(d/d) p(d/d) Mgen = 0 -(d/d) p(d/d) 0 0 -p/(q d) p + (p 2 /(Q d) /(Q d) + (p/(q d) where Q is the distance from the center of the projection to the point (0, 0, p), which is the intersection of the ais and the projection plane (which is normal to the ais) and the direction from (0, 0, p) to the center of projection is given b the normalied direction vector, (d, d, d) or COP Q (d, d, d) VP Pp = (p, p, p) (0, 0, p) P = (,,)

8 CSE 480/580 Slide 8 Implementation of Planar Geometric Projections What needs to be done: Clip scene b view volume Transform 3-D world coordinates to 2-D device coordinates First step is hard, so transform it into an easier problem Clip scene b eas canonical view volume Si eas planes Parallel projections: = -, =, = -, =, = 0, = - Perspective Projection =, = -, =, = -, = -min, = - or

9 CSE 480/580 Slide 9 Find normaliing transformations, Npar and Nper, that transform arbitrar view volumes into the canonical ones So normalie in 3-D Then clip in 3-D Then appl simple projection matrices to get 2-D Then tranform into device coordinates Which can be composed? Trade-off of clipping more and not composing all versus simple clipping Derive Npar Translate VRP to origin Rotate VRC such that the n ais (VPN) lies on the ais, the u ais lies on the ais and the v ais lies on the ais Shear such that the DOP is parallel to the ais Translate and scale into the canonical view volume

10 CSE 480/580 Slide 0 Derive Nper Translate VRP to the origin Rotate VRC such that n ais lies on ais, u ais lies on, and v ais becomes the ais Translate such that PRP is at the origin Shear such that the center line of the view volume becomes the ais Scale such that the view volume becomes the canonical perspective view volume Clipping against the canonical view volumes Modif the Cohen-Sutherland Clipping algorithm si bit outcode for each end of a line eg for parallel cononical view volume bit = if > bit 2 = if <- bit 3 = if > bit 4 = if < - bit 5 = if < - bit 6 = if > 0 Triviall accept if both outcodes all eros Triviall reject if logical and of the codes is not all eros Else subdivide line and retest Use parametric representation of line to compute intersections

11 CSE 480/580 Slide Wh go through all this complication to get projections? Can get neat effects b changing just a few parameters: Move VRP to acheive a "fl-b" or a "walk-through" "Look around" b changing VPN to face diferent directions "Tilt our head" b changing VUP VP projectors PA DOP PRP VPN