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Three dimensional concepts - Computer Graphics
Three key points about advanced computer graphics and 3D viewing: 1. 3D viewing involves establishing a viewing coordinate system and transforming 3D world coordinates to 2D viewing coordinates using translations and rotations. Projections like parallel and perspective then project the viewing coordinates onto a 2D view plane. 2. Common projections used in 3D viewing are parallel projections, which project lines parallel to the view plane, and perspective projections, which simulate how the human eye sees and cause objects to appear smaller with distance. 3. Viewing pipelines involve modeling, transformations between coordinate systems, projections, clipping to a view volume, and normalization before rendering the 2D image. Technologies like OpenGL help specify common operations like projections, view
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Three dimensional concepts - Computer Graphics
- 1. Advanced Computer GraphicsAdvancedComputer Graphics Three Dimensional ViewingThree Dimensional Viewing
- 2. OverviewOverview • Viewing a3D scene • Projections – Parallel and perspective
- 3.
- 4. OverviewOverview • Exploded andcutaway views
- 5. OverviewOverview • 3D andstereoscopic viewing
- 6. 3D Viewing Pipeline3DViewing Pipeline Modeling Transformation Viewing Transformation Projectionn Transformation Normalization Transformation and Clipping Viewport Transformation MC WC VC PC NC DC
- 7. Viewing CoordinatesViewing Coordinates •Generating a view of an object in 3D is similar to photographing the object. • Whatever appears in the viewfinder is projected onto the flat film surface. • Depending on the position, orientation and aperture size of the camera corresponding views of the scene is obtained.
- 8. Specifying The ViewCoordinatesSpecifying The View Coordinates • For a particular view of a scene first we establish viewing- coordinate system. • A view-plane (or projection plane) is set up perpendicular to the viewing z-axis. • World coordinates are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane. xw zw yw xv zv yv P0=(x0 , y0 , z0)
- 9. Specifying The ViewCoordinatesSpecifying The View Coordinates • To establish the viewing reference frame, we first pick a world coordinate position called the view reference point. • This point is the origin of our viewing coordinate system. If we choose a point on an object we can think of this point as the position where we aim a camera to take a picture of the object.
- 10. Specifying The ViewCoordinatesSpecifying The View Coordinates • Next, we select the positive direction for the viewing z-axis, and the orientation of the view plane, by specifying the view- plane normal vector, N. • We choose a world coordinate position P and this point establishes the direction for N. • OpenGL establishes the direction for N using the point P as a look at point relative to the viewing coordinate origin. xw zw yw xv zv P0 P N xv yv
- 11. Specifying The ViewCoordinatesSpecifying The View Coordinates • Finally, we choose the up direction for the view by specifying view-up vector V. • This vector is used to establish the positive direction for the yv axis. • The vector V is perpendicular to N. • Using N and V, we can compute a third vector U, perpendicular to both N and V, to define the direction for the xv axis. xw zw yw xv zv yv P0 P N V
- 12. Specifying The ViewCoordinatesSpecifying The View Coordinates To obtain a series of views of a scene , we can keep the the view reference point fixed and change the direcion of N. This corresponds to generating views as we move around the viewing coordinate origin. P0 V N N
- 13. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates Conversion of object descriptions from world to viewing coordinates is equivalent to transformation that superimpoes the viewing reference frame onto the world frame using the translation and rotation. xw yw zw xv yv zv
- 14. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates First, we translate the view reference point to the origin of the world coordinate system xw yw zw xv yv zv
- 15. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates Second, we apply rotations to align the xv,, yv and zv axes with the world xw, yw and zw axes, respectively. xw yw zw xv yv zv xv yv zv
- 16. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates If the view reference point is specified at word position (x0, y0, z0), this point is translated to the world origin with the translation matrix T. − − − = 1000 100 010 001 0 0 0 z y x T
- 17. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates • The rotation sequence requires 3 coordinate-axis transformation depending on the direction of N. • First we rotate around xw-axis to bring zv into the xw -zw plane. − = 1000 00 00 0001 θθ θθ CosSin SinCos xR
- 18. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates Then, we rotate around the world yw axis to align the zw and zv axes. − = 1000 00 0010 00 αα αα CosSin SinCos yR
- 19. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates The final rotation is about the world zw axis to align the yw and yv axes. − = 1000 0100 00 00 ββ ββ CosSin SinCos zR
- 20. Transformation From WorldToTransformation From World To Viewing CoordinatesViewing Coordinates The complete transformation from world to viewing coordinate transformation matrix is obtaine as the matrix product TRRRM ⋅⋅⋅= xyzvcwc,
- 21. Transformation From WorldTo ViewingTransformation From World To Viewing Coordinates:Coordinates: An Example For 2d SystemAn Example For 2d System y x x′ y′ Θ=300 0 2 4 6 0246 2 2 P=(5,5) P0=(4,3)
- 22. y x x′y′ Θ=300 2 4 6 0246 2 2 P P0 Translation: − − = 100 310 401 T TransformationFrom World To ViewingTransformation From World To Viewing Coordinates:Coordinates: An Example For 2d SystemAn Example For 2d System
- 23. Rotation 0246 y xx′ y′ 2 4 6 P P0 −= 100 0866.05.0 05.0866.0 R TransformationFrom World To ViewingTransformation From World To Viewing Coordinates:Coordinates: An Example For 2d SystemAn Example For 2d System
- 24. New coordinates − − ⋅ −= 100 310 401 100 0866.05.0 05.0866.0 .vcwcM = −− − = 1 232.1 866.1 1 5 5 100 598.0866.0500.0 964.4500.0866.0 1 ' ' y x Transformation FromWorld To ViewingTransformation From World To Viewing Coordinates:Coordinates: An Example For 2d SystemAn Example For 2d System
- 25. Alternative Method xTRx R v n ⋅⋅= −= −= = ' 100 0866.0500.0 0500.0866.0 )866.0500.0( )500.0866.0( y x x′y′ Θ=300 0123 1 1 P P0 12 3 n v Transformation From World To ViewingTransformation From World To Viewing Coordinates:Coordinates: An Example For 2d SystemAn Example For 2d System
- 26. ProjectionsProjections • Once WCdescription of the objects in a scene are converted to VC we can project the 3D objects onto 2D view-plane. • Two types of projections: -Parallel Projection -Perspective Projection
- 27. Classical ViewingsClassical Viewings •Hand drawings : Determined by a specific relationship between the object and the viewer.
- 28. Parallel ProjectionsParallel Projections CoordinatePositions are transformed to the view plane along parallel lines. P1 P2 P′2 P′1 View Plane
- 29. Parallel ProjectionsParallel Projections •Orthographic parallel projection The projection is perpendicular to the view plane. • Oblique parallel projecion The parallel projection is not perpendicular to the view plane.
- 30. Orthographic Parallel ProjectionOrthographicParallel Projection
- 31. Oblique Parallel ProjectionObliqueParallel Projection – The projectors are still ortogonal to the projection plane – But the projection plane can have any orientation with respect to the object. – It is used extensively in architectural and mechanical design.
- 32. Oblique Parallel ProjectionObliqueParallel Projection • Preserve parallel lines but not angles – Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object. – Dimetric view : Symmetric with two faces. – Trimetric view : General case.
- 33. Oblique Parallel ProjectionObliqueParallel Projection • Preserve parallel lines but not angles – Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object. – Dimetric view : Symmetric with two faces. – Trimetric view : General case.
- 34. Perspective ProjectionsPerspective Projections •First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance • Objects closer to viewer look larger • Parallel lines appear to converge to single point
- 35. Perspective ProjectionsPerspective Projections Inperspective projection object positions are transformed to the view plane along lines that converge to a point called the projection reference point (or center of projection)
- 36. Perspective ProjectionsPerspective Projections •In the real world, objects exhibit perspective foreshortening: distant objects appear smaller • The basic situation:
- 37. When we do3-D graphics, we think of the screen as a 2-D window onto the 3-D world: How tall should this bunny be? Perspective ProjectionsPerspective Projections
- 38. The geometry ofthe situation is that of similar triangles. View from above: d P (x, y, z) X Z (0,0,0)x′ = ? Perspective ProjectionsPerspective Projections View plane (xp, yp)
- 39. Desired result fora point [x, y, z, 1]T projected onto the view plane: dz dz y z yd y dz x z xd x z y d y z x d x == ⋅ == ⋅ = == ',',' ' , ' Perspective ProjectionsPerspective Projections
- 40. • When acamera used to take a picture, the type of lens used determines how much of the scene is caught on the film. • In 3D viewing, a rectangular view window in the view plane is used to the same effect. Edges of the view window are parallel to the xv-yv axes and window boundary positions are specified in viewing coordinates. View VolumesView Volumes
- 41. View VolumesView Volumes zv window FrontPlane Back Plane View volume Parallel ProjectionParallel Projection Perspective Projection Front Plane Back Plane View volume (frustum) window Projection Reference Point
- 42. ClippingClipping • An algorithmfor 3D clipping identifies and saves all surface segments within the view volume for display. • All parts of object that are outside the view volume are discarded.
- 43. Clipping LinesClipping Lines Justlike the case in two dimensions, clipping removes objects that will not be visible from the scene The point of this is to remove computational effort 3-D clipping is achieved in two basic steps – Discard objects that can’t be viewed • i.e. objects that are behind the camera, outside the field of view, or too far away – Clip objects that intersect with any clipping plane
- 44. Discard ObjectsDiscard Objects Discardingobjects that cannot possibly be seen involves comparing an objects bounding box/sphere against the dimensions of the view volume – Can be done before or after projection
- 45. Clipping Polygon SurfaceClippingPolygon Surface • To clip a polygon surface, we can clip the individual polygon edges. • First we test the coordinate extends against each boundary of the view volume to determine whether the object is completely inside or completely outside of that boundary. • If the object has intersection with the boundary then we apply intersection calculations.
- 46. Clipping Polygon SurfaceClippingPolygon Surface • The projection operation can take place before the view- volume clipping or after clipping. • All objects within the view volume map to the interior of the specified projection window. • The last step is to transform the window contents to a 2D view port.
- 47. NormalisationNormalisation The transformed volumeis then normalised around position (0, 0, 0) and the z axis is reversed
- 48. Dividing Up TheWorldDividing Up The World Similar to the case in two dimensions, we divide the world into regions This time we use a 6-bit region code to give us 27 different region codes The bits in these regions codes are as follows: bit 6 Far bit 5 Near bit 4 Top bit 3 Bottom bit 2 Right bit 1 Left
- 49.
- 50. 3D Line ClippingExample (cont…)3D Line Clipping Example (cont…) When then simply continue as per the two dimensional line clipping algorithm
- 51. Clipping Polygon SurfaceClippingPolygon Surface Viev volume
- 52. 3D Polygon Clipping(cont…)3D Polygon Clipping (cont…) In this case we first try to eliminate the entire object using its bounding volume Next we perform clipping on the individual polygons using the Sutherland-Hodgman algorithm we studied previously
- 53. OpenGL Projection CommandsOpenGLProjection Commands
- 54. OpenGLOpenGL Look-At FunctionLook-AtFunction • OpenGL utility function – VRP: eyePoint (eyex, eyey, eyez) – VPN: – ( atPoint – eyePoint ) (atx, aty, atz) – (eyex, eyey, eyez) – VUP: upPoint – eyePoint (upx, upy, upz) gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz); look-at positioning
- 55. Projections in OpenGLProjectionsin OpenGL • Angle of view, field of view : – Only objects that fit within the angle of view of the camera appear in the image • View volume, view frustum : – Be clipped out of scene – Frustum – truncated pyramid