Realtime 3D Computer Graphics Virtual Reality

Transcription

1 Realtime 3D Comuter Grahics Virtual Realit Viewing an rojection Classical an General Viewing

2 Transformation Pieline CPU CPU Pol. Pol. DL DL Piel Piel Per Per Verte Verte Teture Teture Raster Raster Frag Frag FB FB v e r t e object ee cli normalie evice Moelview Matri Projection Matri Persective Division Viewort Transform winow Moelview Moelview Projection other calculations here material color shae moel (flat) olgon renering moe olgon culling cliing Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Viewing Process for Seeing a worl Projection of a 3D worl onto a D lane Snthetic Camera Moel Location of viewer an view lane What can be seen (Culling an Cliing) How relationshis are maintaine Parallel Lines Angles Distances (Foreshortening) Relation to Viewer Objects vs. Scenes: Some viewing techniques better suite for viewing single objects rather than entire scenes Viewing an object from the outsie (eternal viewing) Engineering Eternal Builings Viewing an object from within (internal viewing) Internal Builings Games Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

3 Viewing an rojection Ma transforme worl from 3D to D. b aroriate rojection matrices. Several rojection tes eist: Persective rojection is of major interest for 3D-CG. Parallel rojection imortance Pieline often searates rojection normaliation from 3D-D rojection for aitional eth testing. To (Plan) Front Parallel Orthograhic Oblique Cabinet Sie Aonometric Calculation of the ersective rojection for VR islas: Disla often require off-ais rojection. COPs are constantl moving. Projection calculation has to be erforme for ees. Image lanes might not be flat. Planar Geometric Projections Cavalier Persective One-oint Two-oint Three-oint Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Definitions Projection: a transformation that mas from a higher imensional sace to a lower imensional sace (e.g. 3D->D) Center of rojection (COP): the osition of the ee or camera with resect to which the rojection is erforme (also ee oint camera oint roj. reference oint) Direction of rojection (DOP): the irection of an ee or camera assume to be infinite far awa. Projection lane: in a 3D->D rojection the lane to which the rojection is erforme (also view lane) Projectors: lines from coorinate in original sace to coorinate in rojecte sace Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

4 Projections Persective: Distance to COP is finite Parallel: Distance to COP is infinite A A Projectors Center of Projection A' B' Projection B Projectors A' B' Center of Projection at Infinit (Direction of Projection) B Projection Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Planar Geometric Projections Projection onto a lane Projectors are straight lines Alternatives: Some Cartograhic Projections Omnima Planar Geometric Projections Parallel Persective One-oint Orthograhic Oblique To Two-oint (Plan) Cabinet Front Three-oint Cavalier Sie Aonometric Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

5 Parallel Projections Orthograhic: Direction of rojection is orthogonal to the rojection lane Elevations: Projection lane is erenicular to a rincial ais Front To (Plan) Sie Aonometric: Projection lane is not orthogonal to a rincial ais Isometric: Direction of rojection makes equal angles with each rincial ais. Oblique: Direction of rojection is not orthogonal to the rojection lane; rojection lane is normal to a rincial ais Cavalier: Direction of rojection makes a 45 angle with the rojection lane Cabinet: Direction of rojection makes a 63.4 angle with the rojection lane Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Orthograhic Projections Secial case of ersective rojection Parallel rojectors erenicular to rojection lane Planar Geometric Projections Parallel Persective One-oint Orthograhic Oblique To Two-oint (Plan) Cabinet Front Three-oint Cavalier Sie Aonometric Multiview Projection Parallel to Princile Faces Classical Drafting Views Preserves both istance an angles Suitable to Object Views not scenes (a): Front-Elevation (b): To or Plan-Elevation (c): Sie-Elevation (b) (a) (c) Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

6 Aonometric Projections Projection lane can have an orientation to object Parallel lines reserve Angles are not reserve Foreshortening: Length is shorter in image sace than in object sace Uniform Foreshortening (Persective rojections: foreshortening is eenent on istance from object to COP) Planar Geometric Projections Parallel Persective One-oint Orthograhic Oblique To Two-oint (Plan) Cabinet Front Three-oint Cavalier Sie Aonometric Isometric: Smmetric to three faces Dimetric: Smmetric to two faces Trimetric: General Aonometric case Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Oblique Projections To Parallel Projections not erenicular (Plan) Front to rojection lane Oblique rojection tes: Cavalier: 45-egree Angles from Projection Cabinet: Arctan() or 63.4-egree Angles from Projection Planar Geometric Projections Parallel Persective One-oint Orthograhic Oblique Two-oint Cabinet Three-oint Cavalier Sie Aonometric Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

7 Persective Projections First iscovere b Donatello Brunelleschi an DaVinci uring Renaissance Parallel lines aear to converge to single oint Foreshortening: Objects closer to viewer look larger Length is not reserve Deens on istance from viewer Planar Geometric Projections Parallel Persective One-oint Orthograhic Oblique To Two-oint (Plan) Cabinet Front Three-oint Cavalier Sie Aonometric Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik (a) Three-oint: Three rincial aes cut b rojection lane Three ais vanishing oints (b) Two-oint: Two rincial aes cut b rojection lane Two ais vanishing oints (c) One-oint: One rincial ais cut b rojection lane One ais vanishing oint Vanishing Points Persective Projection of an set of arallel line (not erenicular to the rojection lane) converge to a vanishing oint Infinit of vanishing oints one of each set of arallel lines Ais Vanishing Points Vanishing oint of lines arallel to one of the three rincial aes There is one ais vanishing oint for each ais cut b the rojection lane At most 3 such oints Persective Projections are categorie b number of ais vanishing oints Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

8 Persective Foreshortening How tall shoul this bunn be Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Transformations NOTE: Throughout the following iscussions we assume an OenGL- like camera coorinate sstem (COP at the origin DOP along the ais v < ). The concets are the same for an arbitrar viewing configuration.

9 Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Orthogonal Projections DOP arallel to -ais Looking own negative Disla at Secial Case of Persective Projection M orth orth Z -Z Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Orthogonal Projections DOP arallel to -ais Looking own negative Disla at Secial Case of Persective Projection M orth orth Z -Z

10 Persective Projection: Points onto a Projection of worl onto isla lane involves a ersective transformation: M er Not affine (arallel lines o not remain arallel) Not reversible Observe: ( b) : ( c) : Results in non-uniform foreshortening Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Persective Projection: Points onto a Projection of worl onto isla lane involves a ersective transformation: M er Not affine (arallel lines o not remain arallel) Not reversible Observe: ( b) : ( c) : Results in non-uniform foreshortening Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

11 Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik One-oint ersective rojection Center of Projection on the negative -ais Viewlane in the - lane. Geometr of similar triangles. To view: )) ( ( )) ( ( ) ( M er er (+) (+()) (+) (+()) Z -Z View lane -Z : Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik One-oint ersective rojection Center of Projection on the negative -ais Viewlane in the - lane. Geometr of similar triangles. To view: )) ( ( )) ( ( ) ( M er er (+) (+()) (+) (+()) Z -Z View lane -Z :

12 Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik One-oint ersective rojection Center of Projection on the negative -ais Viewlane not in the - lane. Point in 3D become lines through origin in 4D: As long as w we can recover original oint View lane -Z : ) ( ) ( M er er w w w w Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik One-oint ersective rojection Center of Projection on the negative -ais Viewlane not in the - lane. Point in 3D become lines through origin in 4D: As long as w we can recover original oint View lane -Z : ) ( ) ( M er er w w w w Persective ivision

13 3D Viewing Transformation Projection Matries Inut 3D Worl Coorinates Outut 3D Normalie Device Coorinates (a.k.a. Winow Coorinates) Data in 3D Worl Coorinates Data in 3D Camera Coorinates Entire Worl in Normalie Device Coorinates (NDC) Viewable Worl in Normalie Device Coorinates (NDC) NDC or Winow Coorinates Transform Worl into Camera Coorinates Al Normaliing Transformation Cli Against View Volume Project onto Projection cam M cam roj roj M ers cam Just set to ero or (or ignore) M ortho cam Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection normaliation Several issues are not aress with the simle rojection matrices we have eveloe: 3D Cliing Efficienc in a Frustum Viewing Volume Hien Surface Efficienc Solution: Use Projection Normaliation Get ri of ersective an other roblem rojections! Everthing is easier in an canonical orthogonal worl! Distort worl until viewing volume in worl fits into a arallel canonical view volume. Fin a transformation that istorts the vertices in a wa that we can use a simle canonical rojection. + Front Cliing Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik + - Back Cliing View Volume -

14 Iea: Canonical View Volume Define Viewing Volume via Canonical View Volumes Plus: Easier Cliing Minus: Another Transformation Front Cliing View Volume Back Cliing OenGL s canonical view volume (other APIs ma be ifferent): + - Back Cliing DOP Back Cliing + Front Cliing View Volume - View - Front Cliing Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation Distort worl until viewing volume in worl fits into a arallel canonical view volume. Fin a transformation that istorts the vertices in a wa that we can use a simle canonical rojection. + Front Cliing Back Cliing View Volume - - Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

15 Camera Transformation an Projection Normaliation for Orthogonal Views Z Camera Transformation in Orthogonal Views (M cam ). Convert Worl to Camera Coorinates Camera at origin looking in the irection Disla lane center along the ais. Combinations of translate scale an rotate transformations Can be accomlishe through camera location secification Y X Z Y (-near) X (ma ma -far) -Z Projection Normaliation for Orthograhic Views (M ortho ). Translate along the ais until the front cliing lane is at the origin. Scale in all three imensions until the viewing volume is in canonical form Y (ma ma -far) Y -Z X (-near) () X Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Orthogonal Parallel Projections Maing a orthogonal view volume to a canonical view volume requires two affine transformations: T ma + ) ( ma + ) ( ma + ( ) these can be concatenate to M ST The camera is ointing in the negative irection. All rojectors are from infinit towars the origin. Hence M can be written as: M OenGL note: OenGL offers just an interface for the orthogonal case using glortho( ma ma near far) S ma ma ( ma ) ( ma ) ( ma ) ma ma ma far near ma + ma ma + ma ma + ma ma + ma ma + ma far + near far near Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

16 Projection Normaliation for Orthogonal Parallel Projections Finall the resulting matri has to be ost multilie b a simle orthogonal arallel rojection M M orth M ST ma ma ma ma ma ma ma ma + + far near ma + ma ma + ma far + near far near Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Oblique Parallel Projections Orthogonal arallel rojection can be seen as just a secial case of an oblique arallel rojection. An oblique rojection can be characterie b the angle of the rojectors with the VP. θ φ To an sie views (see left) of a rojector an the VP. ( θ φ ) characterie the egree of obliqueness. Consiering the to view (a) can be foun b tan θ an likewise following (b): cot θ cot φ For VP this results to P: M cot θ cot φ After etracting the orthogonal rojection from M we erive an aitional shear: M M orth H ( θ φ ) cot θ cot φ Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

17 Projection Normaliation for Oblique Parallel Projections M M orth H ( θ φ ) cot θ cot φ The same translation an scaling use for the orthograhic case has to be inserte between the shear an the rojection: ma M M orth STH θ φ ) ma P ma M is not in canonical form! It is a simle shear followe b an orthograhic rojection. cot θ ma cot φ ma far near ma + ma ma + ma ma + ma ma + ma far + near far near ( ma cot θ cot φ Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Camera Transformation (M cam ) an Projection Normaliation (M ers ) for Persective Views Camera Transformation for Persective Views (M cam ). Convert Worl to Camera Coorinates Camera (COP) at origin looking in the irection Disla lane center along the ais. Combinations of translate scale an rotate transformations Can be accomlishe through camera location secification Y X Z Y (ma ma -far) (-near) X Z Z Projection Normaliation for Persective Views (M ers ). Convert viewing bo to right frustum (on ais) This is because man APIs incluing OenGL allow non-right viewing volumes. Scale the right frustum into canonical form 3. Convert viewing bo (right frustum) to a right arallelie Shrinking objects that are further awa Y (ma ma -far) Y (-near) () X Z X Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

18 Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Persective Projections Again fin a transformation that istorts the vertices in a wa that we can use a simle canonical rojection: ersective-normaliation transformation M ers Given ) a simle ersective rojection with VP - an COP at origin: Given ) a ersective view volume with the angle of view being 9 > frustum sies intersect VP at 45 angle. View volume is a semi infinite view rami with: View volume is finite b secifing the near lane ma an the far lane with ma > Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Persective Projections β α N Let N be a nonsingular matri similar to M with: β α Aling N to a homogeneous-coorinate oint: + + ' ' ' ' ' ' a ivision e ersectiv β β α β α Aling an orthograhic rojection along the -ais to N: β α N M orth ' N M ivision e ersectiv orth Aling the result to an arbitrar oint : If we al N followe b an orthogonal rojection to a oint we achieve the same result for an as aling a ersective rojection to the same oint!

19 Projection Normaliation for Persective Projections Nonsingular matri N transforms the original viewing volume into a new volume. Now choosing α β such that the new volume is the canonical view (cliing) volume. Given the sies +- transforme b results to +- an +- transforme b results to +- an The front of the view volume ma is transforme to: β ' ' a + ma The back of the view volume is transforme to: β ' ' a + Now we choose: ma + ma α β ma Then the lane is mae to the lane - an the lane is mae to the lane hence we achieve the canonical volume: ma N transforms the viewing volume to a right aralleleie a following orthograhic rojection is the same as a ersective transformation. N is calle the ersective normaliation matri. Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Persective Projections β ' ' a + is nonlinear but reserves eth-orering hence > ' ' > ' ' Notes: Hien surface removal works in the normalie volume. Nonlinearit can cause numerical roblems ue to limite resolution in the eth buffer. Onl one viewing ieline is require b carefull choosing a rojection matri to insert into the ieline. Persective-Normaliation Matri (N er ) converts frustum view volume into canonical orthogonal view volume: N er far + near far near far near far near Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

20 Projection Normaliation for non-right Persective Projections The right (smmetric) ersective rojection is a secial case for an arbitrar ersective rojection like the orthograhic rojection was for the arallel oblique case. An arbitrar ersective rojection is require e.g. for riving several large-screen rojection-base VR isla tes which. use hea tracking an. fi the VPs w.r.t. the moving COP an which hence require namic frustum calculation (resonsive workbenches Holoscreens CAVEs ) This te of rojection is a.k.a. off-ais rojection! Do erive the rojection matri for off-ais set-us we follow the same ath as we i for the arallel rojection case: Insertion of a shear transformation into the rojection ieline. Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for non-right Persective Projections Fin H which satisfies: ( + ma ) ( + ma ) H + H ( θ φ ) H cot ma ma cot ma + ma The resulting frustum is escribe b the lanes: ( ma ) ( ma ) ± ± ma ma ma Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

21 Projection Normaliation for non-right Persective Projections Now scale the sies to achieve ± ± without changing nearfar lanes. ma ma S S ma ma S is without reference to since it is uniquel etere b its results on four oints here the intersection oints of the near lane an the sies. Now N α β gets the far lane to - an the near lane to with the alrea chosen α ma ma + β ma ma Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik Projection Normaliation for Persective Views This results to the rojection matri: P ers N er ( near ) ma SH ( near ) ma ma + ma ma + ma far + near far near far near far near Where H converts a non- right frustum to a right frustum Where S scales the frustum into a canonical ersective view volume Where N is the Persective- Normaliation Matri Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik

22 Project onto Projection Since normaliation change all rojections into an orthogonal rojection: Just ignore the value! In effect a non- event! In realit we retain the -value for hien-surface removal an shaing effects. Viewable worl now in Normalie Device Coorinates (NDC) or Winow Coorinates Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik 3D Viewing Summar Data in local Object Coorinates Data in 3D Worl Coorinates Data in 3D Camera Coorinates Entire Worl in Normalie Device Coorinates (NDC) Viewable Worl in Normalie Device Coorinates (NDC) (aka Worl Coorinates) Transform Object to Worl Coorinates Transform Worl into Camera Coorinates Al Normaliing Transformation Cli Against View Volume CTM Moel-View Projection Realtime 3D Comuter Grahics Virtual Realit WS 56 Marc Erich Latoschik