Realtime 3D Computer Graphics & Virtual Reality. Viewing

Transcription

1 Realtime 3D Computer Graphics & Virtual Realit Viewing

2 Transformation Pol. Per Verte Pipeline CPU DL Piel Teture Raster Frag FB v e r t e object ee clip normalied device Modelview Matri Projection Matri Perspective Division Viewport Transform window Modelview Modelview Projection other calculations here material color shade model (flat) polgon rendering mode polgon culling clipping

3 Eample Frame Rendering Establish World Establish Viewpoint & Displa Plane Perform 3D Clipping Projection of World onto Displa Plane Perform Hidden Surface Removal Determine Displa Colors Rasteriation of 2D Displa *Rendering order ma be changed based on algorithms involved!

4 Classical and General Viewing

5 Viewing Process for Seeing a world Projection of a 3D world onto a 2D plane Snthetic Camera Model

6 Viewing Issues Location of viewer Location of view plane What can be seen (Culling and Clipping) How relationships are maintained Parallel Lines Angles Distances (Foreshortening) Relation to Viewer

7 Objects vs. Scenes Some viewing techniques better suited for viewing single objects rather than entire scenes Viewing an object from the outside (eternal viewing) Engineering, Eternal Buildings Viewing an object from within (internal viewing) Internal Buildings, Games

8 Definitions Projection: a transformation that maps from a higher dimensional space to a lower dimensional space (e.g. 3D->2D) Center of projection (CoP): the position of the ee or camera with respect to which the projection is performed (also ee point, camera point, proj. reference point) Projection plane: in a 3D->2D projection, the plane to which the projection is performed (also view plane)

9 Projectors Projectors: lines from coordinate in Original Space to coordinate in Projected Space A A Projectors A' B Projectors A' B Center of Projection B' Projection Plane B' Center of Projection at Infinit (Direction of Projection) Projection Plane Perspective: Distance to CoP is finite Parallel: Distance to CoP is infinite

10 Planar Geometric Projections Projection onto a plane Projectors are straight lines Alternatives: Some Cartographic Projections Omnima

11 Center of Projection Perspective View Parallel View

12 Projections Planar Geometric Projections Top (Plan) Front elevation Orthographic Parallel Oblique Cabinet Cavalier Perspective One-point Two-point Three-point Side elevation Aonometric Other Other

13 Parallel Projections Orthographic: Direction of projection is orthogonal to the projection plane Elevations: Projection plane is perpendicular to a principal ais Front Top (Plan) Side Aonometric: Projection plane is not orthogonal to a principal ais Isometric: Direction of projection makes equal angles with each principal ais. Oblique: Direction of projection is not orthogonal to the projection plane; projection plane is normal to a principal ais Cavalier: Direction of projection makes a 45 angle with the projection plane Cabinet: Direction of projection makes a 63.4 angle with the projection plane

14 Parallel Projections: Orthographic Projections Parallel Projectors Perpendicular to Projection Plane Special Case of Perspective Projection

15 Orthographic Projections: Multiview Classical Drafting Views Preserves both distance and angles Suitable to Object Views, not scenes Front-Elevation Projection Plane Parallel to Principle Faces Top or Plan-Elevation Side-Elevation

16 Orthographic Projections: Aonometric Projections Projection plane can have an orientation to object

17 Aonometric Projections Isometric Smmetric to three faces Dimetric Smmetric to two faces Trimetric General Aonometric case Foreshortening: Length is shorter in image space than in object space Uniform Foreshortening (As opposed to perspective projections where foreshortening is dependent on distance from object to COP) Parallel lines preserved Angles are not preserved

18 Oblique Projections Parallel Projections not perpendicular to projection plane Oblique projection tpes: Cavalier 45-degree Angles from Projection Plane Cabinet Arctan(2) or 63.4-degree Angles from Projection Plane

19 Perspective Projections First discovered b Donatello, Brunelleschi, and DaVinci during Renaissance Objects closer to viewer look larger Parallel lines appear to converge to single point One-point: One principal ais cut b projection plane One ais vanishing point Two-point: Two principal aes cut b projection plane Two ais vanishing points Three-point: Three principal aes cut b projection plane Three ais vanishing points

20 Vanishing Points Perspective Projection of an set of parallel line (not perpendicular to the projection plane) converge to a vanishing point Infinit of vanishing points one of each set of parallel lines

21 Ais Vanishing Points Vanishing point of lines parallel to one of the three principal aes There is one ais vanishing point for each ais cut b the projection plane At most, 3 such points Perspective Projections are categoried b number of ais vanishing points

22 Perspective Projections

23 Perspective Projection n n In the real world, objects ehibit perspective foreshortening: distant objects appear smaller The basic situation:

24 Perspective Views Objects further awa look smaller Natural look Length is not preserved Foreshortening of lines depends on distance from viewer

25 Perspective Projection When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world: How tall should this bunn be?

26 Projection Transformations NOTE: Throughout the following discussions, we assume an OpenGL-like camera coordinate sstem (COP at the origin, DOP along the ais, vp < ). The concepts are the same for an arbitrar viewing configuration but the math is slightl more complicated.

27 Projecting Points onto a Displa Plane* Projection of world onto displa plane involves a perspective transformation: p =M per p Not affine (parallel lines do not remain parallel) Non-reversable ( p,d) =d ( p,d) =d *Note: Angels book has several errors in the mathematics in this area

28 Projection of point onto displa plane Observe: = (Same for ), = / d Results in non-uniform foreshortening d p p ( p,d) =d ( p,d) =d

29 One-Point Projection Center of Projection on the negative - ais with viewplane in the - plane. -Z +Z (,, -d) (,, ) ( proj, proj, ) + + = + = )) / ( /( )) / ( /( ) / ( / d d d d (,, ) projected = d/(d+) = /(+(/d)) projected = d/(d+) = /(+(/d))

30 Perspective Projection step b step The geometr of the situation is that of similar triangles. View from above: P (,, ) View plane Z What is? =? d (,,)

31 Perspective Projection Desired result for a point [,,, ] T projected onto the view plane: What could a matri look like to do this? d d d d d d d = = = = = = =, ', ' ', '

32 Use of Homogeneous Coordinates We can use homogeneous coordinates to make perspective transformation easier Homogeneous Representation of point: Suppose, instead: w w p = w w p = As long as w, we can recover original point

33 Use of w in homogeneous coordinates Let matri M transforms point p into point q: = = = / / d Mp d q

34 What is q? Then convert q back to a 3D point: Thus q is our projection of p onto the displa plane! = = = / / / / / p p p d d d d d d q

35 A Perspective Projection Matri Eample: Or, in 3-D coordinates: = d d d d d,,

36 Thus, M is M pers Thus, the matri M is used to project points onto a perspective displa plane = / d M

37 Simple Projection Pipeline Simple OpenGL-like Projection Matri: Model-View Matri Projection Matri Perspective Division

38 Orthogonal Projections, M ortho Special Case of Perspective Projection Displa Plane at =,, = = = p p p = p p p

39 Another One-Point Projection Center of Projection at the origin with view plane parallel to the - plane at distance d from the origin. M per Points plotted are /w, /w where w = /d Perspective division d d d d d ) / /( ) / /( / / = -Z +Z projected = d/ = /(/d) projected = d/ = /(/d)

40 Specifing An Arbitrar 3-D View Two coordinate sstems World reference coordinate sstem (WRC) Viewing reference coordinate sstem (VRC) First specif a view plane and coordinate sstem (WRC) View Reference Point (VRP) View Plane Normal (VPN) View Up Vector (VUP) Specif a window on the view plane (VRC) Ma and min u,v values (Center of the window (CW)) Projection Reference Point (PRP) Front (F) and back (B) clipping planes (hither and on)

41 Snthetic Camera Model

42 Snthetic Camera Model

43 Camera Analog 3D is just like taking a photograph (lots of photographs!) camera viewing volume tripod model

44 Camera

45 Camera Projection

46 Camera Analog and Transformations Projection transformations adjust the lens of the camera Viewing transformations tripod define position and orientation of the viewing volume in the world Modeling transformations moving the model Viewport transformations enlarge or reduce the phsical photograph

47 Coordinate Sstems and Transformations Steps in Forming an Image specif geometr (world coordinates) specif camera (camera coordinates) project (window coordinates) map to viewport (screen coordinates) Each step uses transformations Ever transformation is equivalent to a change in coordinate sstems (frames)

48 Projection Specification

49 Projection Characteristics Camera/Viewpoint Location Camera/Viewpoint Direction Camera/Viewpoint Orientation Camera/Viewpoint Lens (View Volume) Width/Height of Lens Front/Back Clipping Planes Parallel or Perspective Projections Parallel is special case of Perspective

50 OpenGL Camera or Projection Coordinates Visible Objects in the - direction Displa Plane centered at (,,)

51 Specifing Viewing Characteristics Fied Camera / Move World Look At View Volume / Displa Plane Specification Vector Specification

52 Camera Location Relative to World Coordinates Can be thought of in two was: Camera location is specified in world coordinates World coordinate frame is located in camera coordinates Camera transformations are reverse of World transformations

53 Fied Camera / Move World Fi the camera at a specific location/orientation Transform the world such that the camera sees the world the right wa I.e., move the world, not the camera Tpical approach for OpenGL

54 Look At Setup the camera to Look At the world Set the camera location Set the look at point Set the camera rotation angle Eample: glulookat

55 View Volume / Displa Plane Specification Specif View Volume and Displa Plane Viewing Volume Projection Plane Near Clipping Plane Far Clipping Plane

56 Fun with View Volumes All projection information is captured in view volume and projection plane. What happens if we pla with these values? Oblique Parallel Projections Non-Right Frustum Perspective Projections View Volume Values Bo Lower-Left Corner/Upper Right Corner of back and front planes Back Plane Angles/Front Plane Angles View Projection Plane Plane Lower-Left Corner/Upper-Right Corner Front Clipping Angle Plane Location Relative to View Volume Back Clipping Plane

57 Vector Specification Most fleible method of viewing specification View plane ma be anwhere with respect to the world Locate the View Plane (I.e. Projection Plane) b: View reference point (VRP) View-plane normal (VPN) (n ais) v VRP n VPN u

58 VRC Coordinate Sstem Viewing-Reference Coordinate (VRC) Sstem n-ais - along View-plane normal (VPN) v-ais projection of View-up Vector (VUP) u-ais Right hand coordinate sstem v VUP VRP n VPN u

59 Viewing Window Min/Ma u and v ranges Need not be smmetrical around VRP v (u-min, v-min) n VPN VRP (u-ma, v-ma) u

60 Center of Projection Projection-Reference Point (PRP) v VRP n VPN u Center of Projection (PRP)

61 Direction of Projection Projection-Reference Point (PRP) v PRP CW VRP u n VPN

62 Oblique Parallel Projection Vector from PRP to Center of the Window not parallel to VPN v PRP CW VRP u n VPN

63 Perspective View Volumes Back Clipping Plane View Plane Front Clipping Plane VRP VPN

64 Orthogonal View Volumes Specification of Viewing Volume Front Clipping Plane View Plane Back Clipping Plane VRP VPN DOP

65 Oblique Projections View Plane Back Clipping Plane VPN VRP DOP

66 Projection Specification in OpenGL

67 Projection Specification in OpenGL Locate World in Camera Coordinates Alternative: glulookat encapsulates world to camera coordinate transformations Select View Tpe (Pespective or Orthogonal) Set View Volume (glfrustum, glortho, gluperspective) (near clipping plane is projection plane) Conceptuall not as fleible as VRC sstem Although with world coordinate transformation, ou can obtain the same results

68 Projections in OpenGL Camera located at (,,) pointing in the direction. Up is positive direction. GL_Modelview Matri controls the world GL_Projection Matri controls the camera Vertices in Object or World Coordinates Model-View CTM Projection Vertices in Viewpoint Coordinates

69 OpenGL Move the world relative to the camera For eample: Moving the camera + along the Z ais is equivalent to moving the world along the Z ais Is rotating the camera positivel around the ais the same as rotating the world negativel around the ais?

70 OpenGL Implementation of View Volume / Displa Plane glfrustum Displa Plane is same as Near Clipping Plane Specif Lower-Left and Upper Right Corners of Near Clipping Plane Specif Near and Far Distances =-far -Z ( min, min,-near) CW ( ma, ma,-near) COP Projection Plane Near Clipping Plane

71 glfrustum Variations Displa Ais (-) does not have to go through the center of the window =-far -Z COP ( min, min,-near) CW Projection Plane ( ma, ma,-near) Near Clipping Plane

72 OpenGL Alternatives Alternative: gluperspective Specifies Frustum via viewing angles (ala Camera Lens) glortho Orthogonal Projection

73 Now What? Several issues are not address with the simple projection matries we have developed: 3D Clipping Efficienc in a Frustum Viewing Volume Hidden Surface Efficienc Solution: Use Projection Normaliation Get rid of perspective and other problem projections! Everthing is easier in an canonical, orthogonal world!

74 3D Viewing Transformation Projection Matries Input 3D World Coordinates Output 3D Normalied Device Coordinates (a.k.a. Window Coordinates) Data in 3D World Coordinates Data in 3D Camera Coordinates Entire World in Normalied Device Coordinates (NDC) Viewable World in Normalied Device Coordinates (NDC) NDC or Window Coordinates Transform World into Camera Coordinates Appl Normaliing Transformation Clip Against View Volume Project onto Projection Plane p cam = M cam p p p proj proj P = pers cam Just set to ero or (or ignore) = P ortho p p cam

75 Idea: Canonical View Volume Define Viewing Volume via Canonical View Volumes Plus: Easier Clipping Minus: Another Transformation View Volume Back Clipping Plane OpenGL s canonical view volume (other APIs ma be different): + Front Clipping Plane,,- Back Clipping Plane - DOP Back Clipping Plane + Front Clipping Plane View Volume View Plane - Front Clipping Plane - -,-,-

76 Projection Normaliation Distort world until viewing volume in world fits into a parallel canonical view volume. Find a transformation that distorts the vertices in a wa that we can use a simple canonical projection. +,,- Back Clipping Plane - + Front Clipping Plane View Volume - - -,-,-

77 Camera Transformation and Projection Normaliation for Orthogonal Views Camera Transformation in Orthogonal Views (M cam ). Convert World to Camera Coordinates Camera at origin, looking in the direction Displa plane center along the ais 2. Combinations of translate, scale, and rotate transformations Can be accomplished through camera location specification Y X Z Y (min,min,-near) X (ma, ma, -far) Z -Z Projection Normaliation for Orthographic Views (P ortho ). Translate along the ais until the front clipping plane is at the origin 2. Scale in all three dimensions until the viewing volume is in canonical form Y (min,min,-near) (ma, ma, -far) (,,) Y -Z X X

78 Projection Normaliation for Orthogonal Parallel Projections Mapping a orthogonal view volume to a canonical view volume requires two affine transformations: T ma + min ) ( + ) ( +, ma min, ma ( min ) S ( ma 2 2 2,, min ) ( ma min ) ( ma min ) these can be concatenated to The camera is pointing in the negative direction. All projectors are from infinit towards the origin. Hence P can be written as: P = ST = P = ma ma 2 2 OpenGL note: OpenGL offers just an interface for the orthogonal case using glortho(min, ma, ma, min, near, far) min min ma ma 2 2 min min ma 2 min 2 far near ma + min ma min ma + min ma min ma + min ma min ma + min ma min ma + min ma min far + near far near

79 Projection Normaliation for Orthogonal Parallel Projections = = min ma min ma min ma min ma min ma min ma near far near far near far ST M P orth Finall, the resulting matri has to be post multiplied b a simple orthogonal parallel projection + + = 2 2 min ma min ma min ma min ma min ma min ma P

80 Projection Normaliation for Oblique Parallel Projections Orthogonal parallel projection can be seen as just a special case of an oblique parallel projection. An oblique projection can be characteried b the angle of the projectors with the VP. θ φ Top and side views (see left) of a projector and the VP =. ( θ, φ) characterie the degree of obliqueness. Considering the top view (a), tanθ = p and likewise following (b): p p p can be found b = cotθ = cotφ For VP = this results to P: cotθ cotφ P = After etracting the orthogonal projection we derive an additional shear: P = M orth H( θ, φ) = cotθ cotφ

81 Projection Normaliation for Oblique Parallel Projections = = cot cot ), ( φ θ φ θ H M P orth P is not in canonical form! It is a simple shear followed b an orthographic projection. The same translation and scaling used for the orthographic case has to be inserted between the shear and the projection: = = cot cot ), ( min ma min ma min ma min ma min ma min ma φ θ φ θ near far near far near far STH M P orth + + = 2cot 2 2cot 2 min ma min ma min ma min ma min ma min ma min ma min ma P φ θ

82 Camera Transformation (M cam ) and Projection Normaliation (P pers ) for Perspective Views Camera Transformation for Perspective Views (M cam ). Convert World to Camera Coordinates Camera (COP) at origin, looking in the direction Displa plane center along the ais 2. Combinations of translate, scale, and rotate transformations Can be accomplished through camera location specification Y X Z Y (min,min,-near) Z X (ma, ma, -far) Z Projection Normaliation for Perspective Views (P pers ). Convert viewing bo to right frustum (on ais) This is because man APIs including OpenGL allow non-right viewing volumes 2. Scale the right frustum into canonical form 3. Convert viewing bo (right frustum) to a right parallelpiped Shrinking objects that are further awa Y (min,min,-near) X (ma, ma, -far) (,,) Y Z X

83 Projection Normaliation for Perspective Projections Again, find a transformation that distorts the vertices in a wa that we can use a simple canonical projection: perspective-normaliation transformation Given ) a simple perspective projection M = with VP =- and COP persp at origin: Given 2) a perspective view volume with the angle of view being 9 => frustum sides intersect VP at 45 angle. View volume is a semi infinite view pramid with: = +/-, = +/- View volume is finite b specifing the near plane = ma and the far plane = min with ma > min

84 Projection Normaliation for Perspective Projections = β α N Let N be a nonsingular matri similar to M with:, β α Appling N to a homogeneous-coordinate point: = + + = ' ' ' ' ' ' a division perspectiv e β β α β α Appling an orthographic projection along the -ais to N: = = β α M orth N = = ' Np M division perspectiv e orth Appling the result to an arbitrar point p : If we appl N followed b an orthogonal projection to a point we achieve the same result for and as appling a perspective projection to the same point!

85 Projection Normaliation for Perspective 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 (clipping) volume. Given the sides = +/- transformed b results to = +/- and = +/- transformed b results to = +/- and The front of the view volume = ma is transformed to: The back of the view volume = min is transformed to: Now we choose: α = ma ma + min min 2 β = ma Then the plane = is mapped to the plane =- and the plane = is mapped to the plane =, hence we achieve the canonical volume: ma min min β ' ' = a + β ' ' = a + ma min N transforms the viewing volume to a right parallelepiped, a following orthographic projection is the same as a perspective transformation. N is called the perspective normaliation matri.

86 Projection Normaliation for Perspective Projections β ' ' = a + is nonlinear but preserves depth-ordering, hence > 2 ' ' > ' ' 2 Notes: Hidden surface removal works in the normalied volume. Nonlinearit can cause numerical problems due to limited resolution in the depth buffer. Onl one viewing pipeline is required b carefull choosing a projection matri to insert into the pipeline. Perspective-Normaliation Matri (N per ) converts frustum view volume into canonical orthogonal view volume: N per = far + near far near 2 far near far near

87 Projection Normaliation for nonright Perspective Projections The right (smmetric) perspective projection is a special case for an arbitrar perspective projection like the orthographic projection was for the parallel oblique case. An arbitrar perspective projection is required, e.g., for driving several large-screen projection-based VR displa tpes which. use head tracking and 2. fi the VPs w.r.t. the moving COP and which hence require dnamic frustum calculation (responsive workbenches, Holoscreens, CAVEs, ) This tpe of projection is a.k.a. off-ais projection! Do derive the projection matri for off-ais set-ups we follow the same path as we did for the parallel projection case: Insertion of a shear transformation into the projection pipeline.

88 Projection Normaliation for nonright Perspective Projections Find H which satisfies: ( ( H min min min ma ma H ( θ, φ) = H cot ) ) = min min + 2ma ma,cot ma + 2ma The resulting frustum is described b the planes: ma min ma min, = ±, = min, 2ma 2ma min ( ) ( ) = ± = ma

89 Projection Normaliation for nonright Perspective Projections Now scale the sides to achieve without changing near/far planes. = 2ma 2ma S S,, ma min ma min = ±, = ± S is without reference to since it is uniquel determined b its results on four points, here the intersection points of the near plane and the sides. Now N = α β gets the far plane to - and the near plane to with the alread chosen α = ma ma + min min 2 β = ma ma min min

90 P = Projection Normaliation for Perspective Views This results to the projection matri: pers N per SH 2( near) ma min = 2( near) ma min ma + min ma min ma + min ma min far + near far near 2 far near far near Where H converts a non-right frustum to a right frustum Where S scales the frustum into a canonical perspective view volume Where N is the Perspective-Normaliation Matri

91 Project onto Projection Plane Since normaliation changed all projections into an orthogonal projection: Just ignore the value! In effect, a non-event! In realit, we retain the -value for hiddensurface removal and shading effects. Viewable world now in Normalied Device Coordinates (NDC) or Window Coordinates

92 3D Viewing Summar Data in local Object Coordinates Data in 3D World Coordinates Data in 3D Camera Coordinates Entire World in Normalied Device Coordinates (NDC) Viewable World in Normalied Device Coordinates (NDC) (aka World Coordinates) Transform Object to World Coordinates Transform World into Camera Coordinates Appl Normaliing Transformation Clip Against View Volume CTM Model-View Projection

93 Matri Operations Specif Current Matri Stack glmatrimode( GL_MODELVIEW or GL_PROJECTION ) Other Matri or Stack Operations glloadidentit() glpushmatri() glpopmatri() Viewport usuall same as window sie viewport aspect ratio should be same as projection transformation or resulting image ma be distorted glviewport(,, width, height )

94 Projection Transformation Shape of viewing frustum Perspective projection gluperspective( fov, aspect, Near, Far ) glfrustum( left, right, bottom, top, Near, Far ) Orthographic parallel projection glortho( left, right, bottom, top, Near, Far ) gluortho2d( left, right, bottom, top ) calls glortho with values near ero

95 Appling Projection Transformations Tpical use (orthographic projection) glmatrimode( GL_PROJECTION ); glloadidentit(); glortho( left, right, bottom, top, Near, Far );

96 Viewing Transformations Position the camera/ee in the scene place the tripod down; aim camera To fl through a scene change viewing transformation and redraw scene glulookat( ee, ee, ee, aim, aim, aim, up, up, up ) up vector determines unique orientation careful of degenerate positions tripod

97 Projection Tutorial

98 Modeling Transformations Move object gltranslate{fd}(,, ) Rotate object around arbitrar ais ( ) glrotate{fd}( angle,,, ) angle is in degrees Dilate (stretch or shrink) or mirror object glscale{fd}(,, )

99 Transformation Tutorial

100 Connection: Viewing and Modeling Moving camera is equivalent to moving ever object in the world towards a stationar camera Viewing transformations are equivalent to several modeling transformations glulookat() has its own command can make our own polar view or pilot view

101 Projection is left handed Projection transformations (gluperspective, glortho) are left handed think of Near and Far as distance from view point Everthing else is right handed, including the vertees to be rendered + left handed right handed +

102 Common Transformation Usage 3 eamples of resie() routine restate projection & viewing transformations Usuall called when window resied Registered as callback for glutreshapefunc()

103 resie(): Perspective & LookAt void resie( int w, int h ) { glviewport(,, (GLsiei) w, (GLsiei) h ); glmatrimode( GL_PROJECTION ); glloadidentit(); gluperspective( 65., (GLfloat) w / h,.,. ); glmatrimode( GL_MODELVIEW ); glloadidentit(); glulookat(.,., 5.,.,.,.,.,.,. ); }

104 resie(): Perspective & Translate Same effect as previous LookAt void resie( int w, int h ) { glviewport(,, (GLsiei) w, (GLsiei) h ); glmatrimode( GL_PROJECTION ); glloadidentit(); gluperspective( 65., (GLfloat) w/h,.,. ); glmatrimode( GL_MODELVIEW ); glloadidentit(); gltranslatef(.,., -5. ); } 4

105 resie(): Ortho (part ) void resie( int width, int height ) { GLdouble aspect = (GLdouble) width / height; GLdouble left = -2.5, right = 2.5; GLdouble bottom = -2.5, top = 2.5; glviewport(,, (GLsiei) w, (GLsiei) h ); glmatrimode( GL_PROJECTION ); glloadidentit(); continued

106 resie(): Ortho (part 2) } if ( aspect <. ) { left /= aspect; right /= aspect; } else { bottom *= aspect; top *= aspect; } glortho( left, right, bottom, top, near, far ); glmatrimode( GL_MODELVIEW ); glloadidentit();

107 Compositing Modeling Transformations Problem : hierarchical objects one position depends upon a previous position robot arm or hand; sub-assemblies Solution : moving local coordinate sstem modeling transformations move coordinate sstem post-multipl column-major matrices OpenGL post-multiplies matrices

108 Compositing Modeling Transformations Problem 2: objects move relative to absolute world origin m object rotates around the wrong origin make it spin around its center or something else Solution 2: fied coordinate sstem modeling transformations move objects around fied coordinate sstem pre-multipl column-major matrices OpenGL post-multiplies matrices must reverse order of operations to achieve desired effect

109 Additional Clipping Planes At least 6 more clipping planes available Good for cross-sections Modelview matri moves clipping plane A + B + C + D < clipped glenable( GL_CLIP_PLANEi ) glclipplane( GL_CLIP_PLANEi, GLdouble* coeff )

110 Reversing Coordinate Projection Screen space back to world space glgetintegerv( GL_VIEWPORT, GLint viewport[4] ) glgetdoublev( GL_MODELVIEW_MATRIX, GLdouble mvmatri[6] ) glgetdoublev( GL_PROJECTION_MATRIX, GLdouble projmatri[6] ) gluunproject( GLdouble win, win, win, mvmatri[6], projmatri[6], GLint viewport[4], GLdouble *obj, *obj, *obj ) gluproject goes from world to screen space