3-Dimensional Viewing

Transcription

1 CHAPTER 6 3-Dimensional Vieing Vieing and projection Objects in orld coordinates are projected on to the vie plane, hich is defined perpendicular to the vieing direction along the v -ais. The to main tpes of projection in Computer Graphics are: parallel projection perspective projection 2 CG - Chapter - 6

2 Projection illustrations Parallel projection All projection lines are crossing the vie plane in parallel; preserve relative proportions Perspective projection Projection lines are crossing the vie plane and converge in a projection reference point (PRP) 3 Overvie of projections 4 CG - Chapter - 6 2

3 Parallel projection To tpes of Parallel projection are: Orthographic (aonometric,isometric) most common projection perpendicular to vie plane Oblique (cabinet and cavalier) projection not perpendicular to vie plane less common 5 Orthographic projection Assume vie plane at vp (perpendicular to the v -ais) and ( v, v, v ) an arbitrar point in VC Then p = v p = v p = vp ( v is kept for depth purposes onl) 6 CG - Chapter - 6 3

4 Oblique projection When the projection path is not perpendicular to the vie plane. A vector direction is defining the projection lines can improve the vie of an object 7 Oblique projection (cont...) An oblique parallel projection is often specified ith to angles, (-9 ) och (-36 ), as shon belo 8 CG - Chapter - 6 4

5 Oblique formula (from fig.) Assume (,,) an point in VC (cp. v, v, v ) cos =( p -)/L => p =+L. cos sin =( p -)/L => p =+L. sin tan =( vp -)/L, thus L=( vp -)/tan = =L ( vp -), here L =cot Hence, p = + L ( vp - ). cos p = + L ( vp - ). sin Observe: if orthographic projection, then L = 9 When Cavalier and Cabinet tan = then the projection is called Cavalier ( = 45 ) tan = 2 then the projection is called Cabinet ( 63 ) usuall takes the value 3 or 45 CG - Chapter - 6 5

6 Cavalier - an eample Perspective projection 2 CG - Chapter - 6 6

7 A general approach p prp vp prp prp vp prp p prp vp prp prp vp prp 3 Special cases Various restrictions are often used, such as: PRP on the v -ais => prp = prp = PRP in the VC origin => prp = prp = prp = vie plane in the v v -plane => vp = vie plane in the v v -plane and PRP on the v -ais => prp = prp = vp = 4 CG - Chapter - 6 7

8 Special case: PRP on the v -ais Similarit properties give: p prp vp prp > p d p prp p prp vp prp > p d p prp 5 Windo and clipping in 3D 6 CG - Chapter - 6 8

9 Windo in 3D => Vie Volume A rectangular indo on the vie plane corresponds to a vie volume of tpe: infinite parallelepiped (parallel projection) half-infinite pramid ith ape at PRP (perspective projection) 7 Vie volumes 8 CG - Chapter - 6 9

10 Finite vie volumes To get a finite volume (one or) to etra v - boundar planes, parallel to the vie plane, are added: the front (near) plane and the back (far) plane resulting in: a rectangular parallelepiped (parallel projection) a pramidal frustum (perspective projection) 9 Finite vie volumes 2 CG - Chapter - 6

11 Camera properties The to ne planes are mainl used as far and near clipping planes to eliminate objects close to and far from PRP (camera) Other camera similarities: PRP close to the vie plane => ide angle lens PRP far from the vie plane => tele photo lens Matri representations for both parallel and perspective projections are possible 2 3D Clipping A 3D algorithm for clipping identifies and saves those surface parts that are ithin the vie volume Etended 2D algorithms are ell suited also in 3D; instead of clipping against straight boundar edges, clipping in 3D is against boundar planes, i.e. testing lines/surfaces against plane equations 22 CG - Chapter - 6

12 Clipping planes Testing a point against the front and back clipping planes are eas; onl the -coordinate has to be checked Testing against the other vie volume sides are more comple hen perspective projection (pramid), but still eas hen parallel projection, since the clipping sides are then parallel to the - and -aes 23 Clipping hen perspective projection Before clipping, convert the vie volume, a pramidal frustum, to a rectangular parallelepiped (see net figure) Clipping can then be performed as in the case of parallel projection, hich means much less processing From no on, all vie volumes are assumed to be rectangular parallelepipeds (either including the special transformation or not) 24 CG - Chapter - 6 2

13 The perspective transformation The perspective transformation ill transform the object A to A so that the parallel projection of A ill be identical to the perspective projection of A 25 Normalied coordinates A possible (and usual!) further transformation is to a unit cube; a normalied coordinate sstem (NC) is then introduced, ith either,, or -,, Since screen coordinates are often specified in a left-handed reference sstem, also normalied coordinates are often specified in a left-handed sstem, hich means, for instance, vieing in the positive -direction 26 CG - Chapter - 6 3

14 Left-handed screen coordinates 27 Parallel projection vie volume to normalied vie volume 28 CG - Chapter - 6 4

15 Perspective projection vie volume to normalied vie volume 29 Advantages ith the parallelepiped / unit cube all vie volumes have a standard shape and corresponds to common output devices simplified and standardied clipping depth determinations are simplified hen it comes to Visible Surface Detection 3 CG - Chapter - 6 5

16 CG - Chapter D Transformations Same idea as 2D transformations Homogeneous coordinates: (,,,) 44 transformation matrices p o n m l k j i h g f e d c b a 3 Basic 3D Transformations t t t s s s Identit Scale Translation Mirror about Y/Z plane 32

17 CG - Chapter Basic 3D Transformations cos sin sin cos Rotate around Z ais: cos sin sin cos Rotate around Y ais: cos sin sin cos Rotate around X ais: 33 Summar Coordinate sstems World vs. modeling coordinates 2-D and 3-D transformations Trigonometr and geometr Matri representations Linear vs. affine transformations Matri operations Matri composition 34