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Rendering and Rasterization Lars M. Bishop (lars@essentialmath.com)

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Essential Math for Games Rendering Overview Owing to time pressures, we will only cover the basic concepts of rendering Designed to be a high-level overview to help you understand other references Details of each stage may be found in the references at the end of the section No OpenGL/D3D specifics will be given

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Essential Math for Games Rendering: The Last Step The final goal is drawing to the screen Drawing involves creating a digital image of the projected scene Involves assigning colors to every pixel on the screen This may be done by: Dedicated hardware (console, modern PC) Hand-optimized software (cell phone, old PC)

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Essential Math for Games Destination: The Framebuffer We store the rendered image in the framebuffer 2D digital image (grid of pixels) Generally RGB(A) The graphics hardware reads these color values out to the screen and displays them to the user

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Essential Math for Games Top-level Steps We have a few steps remaining Tessellation: How do we represent the surfaces of objects? Shading: How do we assign colors to points on these surfaces? Rasterization: How do we color pixels based on the shaded geometry?

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Essential Math for Games Representing Surfaces: Points The geometry pipeline leaves us with: Points in 2D screen (pixel) space (x s, y s ) Per-point depth value (z ndc ) We could render these points directly by coloring the pixel containing each point If we draw enough of these points, maybe we could represent objects with them But that could take millions of points (or more)

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Essential Math for Games Representing Surfaces: Line Segments Could connect pairs of points with line segments Draw lines in the framebuffer, just like a 2D drafting program This is called wireframe rendering Useful in engineering and app debugging Not very realistic But the idea is good – use sets of vertices to define higher-level primitives

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Essential Math for Games Representing Surfaces: Triangles We saw earlier that three points define a triangle Triangles are planar surfaces, bounded by three edges In real-time 3D, we use triangles to join together discrete points into surfaces

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Essential Math for Games A Simple Cube Cube represented by 8 points (trust us) Cube represented by triangles (defined by 8 points)

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Essential Math for Games The Power of Triangles Flexible We can approximate a wide range of surfaces using sets of triangles Tunable A smooth surface can be approximated by more or fewer triangles Allows us to trade off speed and accuracy Simple but Effective We already know how to transform and project the three vertices that define a triangle

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Essential Math for Games Vertices – “Heavy” Points As we will see, we often need to store additional data at each distinct point that define our triangles We call these data structures vertices Vertices provide a way of storing the added values we need to compute the color of the surface while rendering

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Essential Math for Games Common Per-vertex Values Position The “points” we’ve been transforming Color The color of the surface at/near the point Normal vector A vector perpendicular to the surface at/near the point Texture coordinates A value used to apply a digital image (akin to a decal on a plastic model) to the surface

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Essential Math for Games Efficient Rendering of Triangle Sets Indexed geometry allows vertices to be shared by several triangles Uses an array of indices into the array of vertices Each set of three indices determine a triangle Uses much fewer verts than (3 x Tris)

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Essential Math for Games Indexed Geometry Example 1 2 3 4 5 6 0 Index list for shared vertices (0,1,2),(0,2,3),(0,3,4),(0,4,5),(0,5,6),(0,6,1) 18 Individual vertices (exploded view) 7 shared verticesConfiguration

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Essential Math for Games Even More Efficient Triangle Strips (tristrips) Every vertex after the first two defines a triangle: (i-2, i-1, i) Tristrips use much shorter index lists (2 + Tris) versus (3 x Tris) You may not quite get this stated efficiency You may have to include degenerate triangles to fit some topologies For example, try to build our earlier 6-tri fan…

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Essential Math for Games Tristrip Example Index list for shared vertices (0, 1, 2, 3, 4, 5, 6, 7) 0246 1357

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Essential Math for Games Triangle Set Implementation Most 3D hardware is optimized for tristrips However, transformed vertices are often “cached” in a limited-size cache Indexed geometry is not “perfect” – using a vertex in two triangles may not gain much performance if the vertex is kicked out of the cache between them Lots of papers available from the hardware vendors on how to optimize for their caches

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Essential Math for Games Geometry Representation Summary Generally, we draw triangles Triangles are defined by three screen- space vertices Vertices include additional information required to store or compute colors Indexed primitives allow us to represent triangles more efficiently

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Essential Math for Games Shading - Assigning Colors The next step is known as shading, and involves assigning colors to any and all points on the surface of each triangle There are several common shading techniques We’ll present them from least complex (and expensive) to most complex

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Essential Math for Games Triangle Shading Methods Per-triangle: Called “Flat” shading Looks faceted Per-vertex: Called “Smooth” or “Gouraud” shading Looks smooth, but still low-detail Per-pixel: Image-based texturing is one example Programmable shaders are more general

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Essential Math for Games Flat Shading Uses colors defined per-triangle directly as the color for the entire triangle

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Essential Math for Games Gouraud (Smooth) Shading Gouraud shading defines a smooth interpolation of the colors at the three vertices of a triangle The colors at the vertices (C 0,C 1,C 2 ) define an affine mapping from barycentric coordinates (s,t) on a triangle to a color at that point:

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Essential Math for Games Gouraud Shading

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Essential Math for Games Generating Source Colors Both Flat and Gouraud shading require source colors to interpolate There are several common sources: Artist-supplied colors (modeling package) Dynamic lighting

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Essential Math for Games Dynamic Lighting Dynamic lighting assigns colors on a per-frame basis by computing an approximation of the light incident upon the point to be colored Uses the vertex position, normal, and some per-object material color information Dynamic lighting is detailed in most basic rendering texts, including ours

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Essential Math for Games Imaged-based Texturing Extremely powerful shading method Unlike Flat and Gouraud shading, texturing allows for sub-triangle, sub- vertex detail Per-vertex values specify how to map an image onto the surface Visual result is as if a digital image were “pasted” onto the surface

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Essential Math for Games Imaged-based Texturing Per-vertex texture coordinates (or UVs) define an affine mapping from barycentric coords to a point in R 2. Resulting point (u,v) is a coordinate in a texture image: (0,0)(1,0) (0,1)(1,1) U Axis V Axis

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Essential Math for Games Texture Applied to Surface The vertices of this cylinder include UVs that wrap the texture around it: V=1.0 V=0.0

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Essential Math for Games Rasterization: Coloring the Pixels The final step in rendering is called rasterization, and involves 1)Determining which parts of a triangle are visible (Visible Surface Determination) 2)Determining which screen pixels are covered by each triangle 3)Computing the color at each pixel 4)Writing the pixel color to the framebuffer

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Essential Math for Games Computing Per-pixel Colors Even though rasterization is almost universally done in hardware today, it involves some interesting and instructive mathematical concepts As a result, we’ll cover this topic in greater detail Specifically, we’ll look at perspective projections and texturing

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Essential Math for Games Conceptual Rasterization Order Conceptually, we draw triangles to the framebuffer by rendering adjacent pixels in screen space one after the other Rasterization draws pixel (x,y), then (x+1,y), then (x+2,y) etc. across each horizontal span covered by a triangle Then, it draws the next horizontal span

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Essential Math for Games Stepping in Screen Space If we have a fast way to compute the color of a triangle at pixel (x+1,y) from the color of pixel (x,y), we can draw a triangle quite quickly The same is true for computing texture coordinates Affine mappings are perfect for this

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Essential Math for Games Affine Mappings on Triangles Gouraud shading defines an affine mapping from world-space points in a triangle to colors Texturing defines an affine mapping from world-space points on a triangle to texture coordinates (UVs) Note, however, these are mappings from world space to colors and UVs

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Essential Math for Games Affine in Screen Space For the moment, assume that we have an affine mapping from screen space (x s, y s ) to color: Note that the depth value is ignored

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Essential Math for Games Forward Differences Note the difference in colors between a pixel and the pixel directly to the right: Wow – that’s nice!

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Essential Math for Games Forward Differencing This simple difference leads to a useful trick: Given the color for a base pixel in a triangle, the color of the other pixels are: To step from pixel to adjacent pixel (i=1 and/or j=1) is just an addition! This is called forward differencing

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Essential Math for Games 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

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Essential Math for Games 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

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Essential Math for Games Polygon Parallel to View: Affine Interpolation Textured view CORRECT Wire-frame view

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Essential Math for Games 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…

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Essential Math for Games Polygon Tilted in Depth: Affine Interpolation Textured view WRONG! Wire-frame view

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Essential Math for Games 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

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Essential Math for Games Example – Projecting a Line We’ll project a 2D line segment (y,z) into 1D using perspective: Line in world space: Projection of any point (not necessarily on line) to 1D:

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Essential Math for Games Projecting a Line Segment Y-axis Z-axis P D View Plane

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Essential Math for Games Special Case – Parallel to View Y-axis Z-axis P D=(D Y,0) Affine when projected - That’s why this case worked View Plane

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Essential Math for Games General Case Y-axis Z-axis P D Not affine when projected – That’s why this case was wrong

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Essential Math for Games Correct Interpolation The 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:

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Essential Math for Games Perspective-correct Stepping The per-pixel method for stepping our projective mapping in screen space is: 1)Affine forward diff to step numerator 2)Affine forward diff to step denominator 3)Division to compute final value This requires an expensive per-pixel division, or even several divisions

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Essential Math for Games Polygon Tilted in Depth: Correct Perspective Textured view CORRECT! Wire-frame view

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Essential Math for Games 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 correction

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Essential Math for Games Cheating – Faster Perspective SW 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)

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Essential Math for Games Cheating - Artifacts Some 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

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Essential Math for Games 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, 1995-1996

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Essential Math for Games 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

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