Modeling and The Viewing PipelineThe Rendering EquationModelingPolygon MeshRepresenting Polygon MeshArriving at a MeshExample: The Utah TeapotPatch Representation vs. Polygon MeshCommon Polyhedral Shape Construction OperationsSweep (Revolve and Extrude)Constructive Solid Geometry (CSG)A CSG TreeExample Modeling Package: Alias StudioPowerPoint PresentationSlide 15Viewing PipelineSlide 17Slide 18Slide 192D TransformationHomogeneous CoordinatesTranslation in Homogenous CoordinatesWhy these properties are importantRotation in Homogeneous SpacePutting Translation and Rotation TogetherAffine TransformationAffine TransformationsHow to determine an Affine 2D Transformation?Affine Transformation in 3DMore RotationViewingSlide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38View VolumeDifficultyCanonical View VolumePerspective TransformAbout Perspective TransformSlide 44Perspective + Projection MatrixCamera Control and ViewingComplete PerspectiveSlide 48Implementation … Full BlownViewport mappingTrue Or FalseSlide 52Modeling and The Viewing PipelineJian HuangCS594The material in this set of slides have referenced slides from Ohio State, Minnesota and MIT.The Rendering Equation•I(x,x’) = intensity passing from x’ to x•g(x,x’) = geometry term (1, or 1/r2, if x visible from x’, 0 otherwise)• (x,x’) = intensity emitted from x’ in the direction of x• (x,x’,x’’) = scattering term for x’ (fraction of intensity arriving at x’ from the direction of x’’ scattered in the direction of x)•S = union of all surfacesModeling•Types: –Polygon surfaces–Curved surfaces•Generating models: –Interactive –ProceduralPolygon Mesh•Set of surface polygons that enclose an object interior, polygon mesh•De facto: triangles, triangle mesh.Representing Polygon Mesh•Vertex coordinates list, polygon table and (maybe) edge table•Auxiliary:–Per vertex normal–Neighborhood information, arranged with regard to vertices and edgesArriving at a Mesh•Use patches model as implicit or parametric surfaces •Beziér Patches : control polyhedron with 16 points and the resulting bicubic patch:Example: The Utah Teapot•32 patchessingle shaded patchwireframe of the control pointsPatch edgesPatch Representation vs. Polygon Mesh•Polygons are simple and flexible building block. •But, a parametric representation has advantages:–Conciseness•A parametric representation is exact and analytical. –Deformation and shape change•Deformations appear smooth, which is not generally the case with a polygonal object.Common Polyhedral Shape Construction Operations•Extrude: add a height to a flat polygon•Revolve: Rotate a polygon around a certain axis•Sweep: sweep a shape along a certain curve (a generalization of the above two)•Loft: shape from contours (usually in parallel slices)•Set operations (intersection, union, difference), CSG (constructive solid geometry)•Rounding operations: round a sharp cornerSweep (Revolve and Extrude)Constructive Solid Geometry (CSG)•To combine the volumes occupied by overlapping 3D shapes using set operations. unionintersection differenceA CSG TreeExample Modeling Package: Alias StudioPinhole Model•Visibility Cone with apex at observer•Reduce hole to a point - the cone becomes a ray•Pin hole - focal point, eye point or center of projection. PPFTransformations – Need ?•Modeling transformations•build complex models by positioning simple components•Viewing transformations•placing virtual camera in the world•transformation from world coordinates to eye coordinates•Side note: animation:vary transformations over time to create motionWORLDOBJECTEYEViewing Pipeline•Object space: coordinate space where each component is defined•World space: all components put together into the same 3D scene via affine transformation. (camera, lighting defined in this space)•Eye space: camera at the origin, view direction coincides with the z axis. Hither and Yon planes perpendicular to the z axis•Clipping space: do clipping here. All point is in homogeneous coordinate, i.e., each point is represented by (x,y,z,w)•3D image space (Canonical view volume): a parallelpipied shape defined by (-1:1,-1:1,0,1). Objects in this space is distorted•Screen space: x and y coordinates are screen pixel coordinatesObject SpaceWorld SpaceEye SpaceClipping SpaceCanonical view volume Screen SpaceSpaces: ExampleObject Space and World Space:Eye-Space:eye3.Spaces: ExampleClip Space:Image Space:1.2.3.4.5.6.2D Transformation•Translation•RotationHomogeneous Coordinates•Matrix/Vector format for translation:Translation in Homogenous Coordinates•There exists an inverse mapping for each function•There exists an identity mappingWhy these properties are important•when these conditions are shown for any class of functions it can be proven that such a class is closed under composition•i. e. any series of translations can be composed to a single translation.Rotation in Homogeneous SpaceThe two properties still apply.Putting Translation and Rotation Together•Order matters !!Affine Transformation•Property: preserving parallel lines•The coordinates of three corresponding points uniquely determine any Affine Transform!!Affine Transformations•Translation•Rotation•Scaling•ShearingTHow to determine an Affine 2D Transformation?•We set up 6 linear equations in terms of our 6 unknowns. In this case, we know the 2D coordinates before and after the mapping, and we wish to solve for the 6 entries in the affine transform matrixAffine Transformation in 3D•Translation•Rotate•Scale•ShearMore Rotation•Which axis of rotation is this?Viewing•Object space to World space: affine transformation•World space to Eye space: how?•Eye space to Clipping space involves projection and viewing frustumPerspective Projectionand Pin Hole Camera•Projection point sees anything on ray through pinhole F•Point W projects along the ray through F to appear at I (intersection of WF with image plane)FImageWorldIWImage FormationFImageWorldProjecting shapes•project points onto image plane•lines are projected by projecting its end points onlyOrthographic Projection•focal point at infinity •rays are parallel and orthogonal to the image planeImageWorldFFImageWorldIWComparisonSimple Perspective Camera•camera looks along z-axis•focal point is the origin•image plane is parallel to xy-plane at distance d•d is call focal length for
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