Review of Selected Surface Graphics Topics (1)Visible Light3-Component ColorRGBRaster DisplaysPolygon MeshRepresenting Polygon MeshViewing PipelinePowerPoint PresentationSlide 10Slide 11Homogeneous CoordinatesTranslation in Homogenous CoordinatesWhy these properties are importantRotation in Homogeneous SpaceAffine TransformationAffine TransformationsAffine Transformation in 3DViewingSlide 20Slide 21Slide 22Slide 23View VolumeWhy do clippingDifficultyCanonical View VolumePerspective TransformAbout Perspective TransformPerspective + Projection MatrixCamera Control and ViewingComplete PerspectiveSlide 33Slide 34One Plane At a Time Clipping (a.k.a. Sutherland-Hodgeman Clipping)Slide 36Slide 37Slide 38Slide 394D ClippingWhy Homogeneous ClippingSlide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Spanning Scan Line AlgorithmSlide 55Slide 56Warnock’s AlgorithmAt Single Pixel LevelSlide 59Weiler -Atherton AlgorithmSlide 61List Priority AlgorithmsSlide 63Slide 64Slide 65Slide 66Slide 67Slide 68Review of Selected Surface Graphics Topics (1)Jian Huang, CS 594, Spring 2002Visible Light3-Component Color•The de facto representation of color on screen display is RGB.•Some printers use CMY(K)•Why?–The color spectrum can not be represented by 3 basis functions?–But the cone receptors in human eye are of 3 types, roughly sensitive to 430nm, 560nm, and 610nmRGB•The de facto standardRaster Displays•Display synchronized with CRT sweep•Special memory for screen update•Pixels are the discrete elements displayed•Generally, updates are visiblePolygon 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 edgesViewing 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.Homogeneous 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.Affine Transformation•Property: preserving parallel lines•The coordinates of three corresponding points uniquely determine any Affine Transform!!Affine Transformations•Translation•Rotation•Scaling•ShearingTAffine Transformation in 3D•Translation•Rotate•Scale•ShearViewing•Object space to World space: affine transformation•World space to Eye space: how?•Eye space to Clipping space involves projection and viewing frustumPinhole 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. PPFPerspective 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)FImageWorldIWSimple 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 historical reasonSimilar TrianglesYZ[0, d][0, 0][Y, Z] [(d/Z)Y, d]•Similar situation with x-coordinate•Similar Triangles: point [x,y,z] projects to [(d/z)x, (d/z)y, d]View Volume•Defines visible region of space, pyramid edges are clipping planes•Frustum :truncated pyramid with near and far clipping planes–Near (Hither) plane ? Don’t care about behind the camera –Far (Yon) plane, define field of interest, allows z to be scaled to a limited fixed-point value for z-buffering.Why do clipping•Clipping is a visibility preprocess. In real-world scenes clipping can remove a substantial percentage of the environment from consideration.•Clipping offers an important optimizationDifficulty•It is difficult to do clipping directly in the viewing frustumCanonical View Volume•Normalize the viewing frustum to a cube, canonical view volume•Converts perspective frustum to orthographic frustum – perspective transformationPerspective Transform•The equationsalpha = yon/(yon-hither)beta = yon*hither/(hither - yon)s: size of window on the image planezz’1alphayonhitherAbout Perspective Transform•Some propertiesPerspective + Projection Matrix•AR: aspect ratio correction, ResX/ResY•s= ResX,•Theta: half view angle, tan(theta) = s/dCamera Control and ViewingFocal length (d), image size/shape and clipping planes included in perspective transformation Angle or Field of view (FOV) AR Aspect Ratio of view-portHither, Yon Nearest and farthest vision limits (WS).eyecoihitheryonLookat - coiLookfrom - eyeView angle - FOVComplete Perspective•Specify near and far clipping planes - transform z between znear and zfar on to a fixed range•Specify field-of-view (fov) angle•OpenGL’s glFrustum and gluPerspective do theseMore Viewing Parameters Camera, Eye or Observer:lookfrom:location of focal point or cameralookat: point to be centered in imageCamera orientation about the look at -lookfrom axisvup: a vector that is pointing straight up in the image. This is like an orientation.Point Clipping(x, y) is inside iff XminXmaxYminYmaxXminx Xmax Yminy Ymax ANDOne Plane At a Time Clipping(a.k.a. Sutherland-Hodgeman Clipping)•The Sutherland-Hodgeman triangle clipping algorithm uses a
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