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This is called forward differencing"Z Perspective   Perspective projection is by far the most popular projection method in games It looks the most like reality to us Perspective rendering does involve some interesting math and surprising visual results  Affine Mapping of Texture UVs(Over the next few slides, we ll apply a texture to a pair of triangles We ll use screen-space affine mappings to compute the texture coordinates for each pixel First, let s look at a special case: a pair of triangles parallel to the view plane.Polygon Parallel to View: Affine Interpolation//(*       (Looking good so far&   That worked correctly  maybe we can get away with using affine mappings (and thus forward differencing) for all of our texturing& Not so fast  next, we ll tilt the top of the square away from the camera& .Polygon Tilted in Depth: Affine Interpolation//(*       What Happened?  \We were interpolating using an affine mapping in (2D) screen space. Perspective doesn t preserve affine maps Let s look at the inverse mapping: namely, we ll derive the mapping from world space to screen space If this mapping isn t affine, then the inverse (screen to triangle) can t be affine, either //6Example  Projecting a LineWe ll project a 2D line segment (y,z) into 1D using perspective:Projecting a Line Segment >Special Case  Parallel to View (  General Case Correct InterpolationThe perspective projection breaks our nice, simple affine mapping Affine in world space becomes projective in screen space Per-vertex depth matters The correct mapping from screen space to color or UVs is projective:6BSEBSEPerspective-correct SteppingThe per-pixel method for stepping our projective mapping in screen space is: Affine forward diff to step numerator Affine forward diff to step denominator Division to compute final value This requires an expensive per-pixel division, or even several divisions `MnI" MnI -Polygon Tilted in Depth: Correct Perspective..(.   Colors versus Texture UVs*       Textures need to be perspective-correct because the detailed features in most textures make errors obvious This is why the example images that we have shown involve texturing Gouraud-shaded colors are more gradual and can often get by without perspective correction6kD[kD[T:Cheating  Faster Perspective  2SW and HW systems have optimized perspective correction by approximation Subdivide tris or scanlines into smaller bits and use affine stepping (very popular) Use a quadratic function to approximate the projective interpolation Reorder pixel rendering to render pixels of constant depth together (very rare)&II,S Cheating - ArtifactsSome tricks are better than others, and some break down in extreme cases Look at some early PS1 games, and you ll see plenty of interesting and different perspective-correction artifacts As with most approximations, you tend to trade speed for accuracy References   Eberly, David H., 3D Game Engine Design, Morgan Kaufmann Publishers, San Francisco, 2001 Hecker, Chris, Behind the Screen: Perspective Texture Mapping (Series), Game Developer Magazine, Miller Freeman, 1995-19966A75T  *    !    u   References   Van Verth, James, Bishop, Lars, Essential Mathematics for Games and Interactive Applications: A Programmer s Guide, Morgan Kaufmann Publishers, San Francisco, 2004 Woo, Mason, et al, OpenGL Programming Guide: The Official Guide to Learning OpenGL, Addison-Wesley, 1999^ RE*;*{        ` f33` 3f3` ___>?" dd@,?" dd@  " @ `"  n?" dd@   @@``@n?" dd@  @@``PR    @ ` ` p>>D$ 0Essential Math for Games r(    N?-gֳgֳ ?P - X Click to edit Master title style!!  2  HA-gֳgֳ ? - RClick to edit Master text styles Second level Third level Fourth level Fifth level!    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Bishop  #   .@"Arialw@ r}ww0- . 2 ( . . 2 lars . . 2 *@&. .2 P essentialmath   . .2 7.com)  .--"System 0-&TNPP & ՜.+,D՜.+,\    On-screen ShowRed Storm Entertainment96 <Times New RomanArial WingdingsLecture TemplateMicrosoft Equation 3.0Adobe Photoshop Elements ImageRendering and RasterizationRendering Overview Rendering: The Last StepDestination: The FramebufferTop-level StepsRepresenting Surfaces: Points%Representing Surfaces: Line Segments!Representing Surfaces: TrianglesA Simple CubeThe Power of TrianglesVertices Heavy PointsCommon Per-vertex Values%Efficient Rendering of Triangle SetsIndexed Geometry ExampleEven More EfficientTristrip ExampleTriangle Set Implementation Geometry Representation SummaryShading - Assigning ColorsTriangle Shading Methods Flat ShadingGouraud (Smooth) Shading Gouraud ShadingGenerating Source ColorsDynamic LightingImaged-based TexturingImaged-based TexturingTexture Applied to Surface#Rasterization: Coloring the PixelsComputing Per-pixel ColorsConceptual Rasterization OrderStepping in Screen SpaceAffine Mappings on TrianglesAffine in Screen SpaceForward DifferencesForward Differencing PerspectiveAffine Mapping of Texture UVs/Polygon Parallel to View: Affine InterpolationLooking good so far/Polygon Tilted in Depth: Affine InterpolationWhat Happened?Example Projecting a LineProjecting a Line Segment Special Case Parallel to View General CaseCorrect InterpolationPerspective-correct Stepping.Polygon Tilted in Depth: Correct PerspectiveColors versus Texture UVsCheating Faster PerspectiveCheating - Artifacts References References  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles68_AdHocReviewCycleID_EmailSubject _AuthorEmail_AuthorEmailDisplayNameL5SlidesE lmb@ndl.comLars M. 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