University:
California State University, Northridge
Course:
MATH 496A-Z | Experimental Topics in Modern Mathematics
Academic year:

2024
Views:

390

Pages:

10
Author:

barretina7q0

Math for 3D Graphics math for 3D graphics main idea: describe a scene in 3D space and render it to a 2D surface still looking 3D graphics hardware computes pixel data from vertex data ... of course doing this is not trivial why math? geometry: project 3D object onto a plane but there is more: how can the scene be described efficiently? how is the information about the color, the light, and the texture of an object stored? and how does the hardware manipulate and display it? the rendering pipeline - overview a 3D scene consists of separate objects, each defined by a set of vertices and a graphics primitive information about the object and its properties is stored at each vertex how that information is manipulated and displayed requires several transformations and a number of mathematical operations the sequence of transformations from vertices (input) to display (output) is known as the rendering pipeline in chapter 0, the book provides a detailed description of this process, which we'll summarize here the rendering pipeline - hardware on modern computers graphics are handled by Graphical Processing Unit (GPU), separate from CPU CPU communicates with graphics application application sends rendering commands to graphics library (e.g., OpenGL) OpenGL sends commands to GPU (through a graphics driver) GPU processes vertex data to produce pixel data pixel data is displayed in image buffers the rendering pipeline - hardware CPU Main Memory Application OpenGL pixel data vertex data Graphics Driver GPU Image Buffers Depth Buffer Texture Maps the rendering pipeline - transforms several 3D coordinate systems some local, some global, vertex data is stored in 3D object space - local position of each object is given in world space - global also camera/eye space: x and y align with display, and z in viewing direction model-view transformation object space camera space world space concatenate transforms: + the rendering pipeline - transforms several 3D coordinate systems some local, some global, vertex data is stored in 3D object space - local position of each object is given in world space - global also camera/eye space: x and y align with display, and z in viewing direction model-view transformation object space camera space the rendering pipeline - transforms more transforms: projection: apply perspective performed in four-dimensional homogeneous clip space graphics primitives clipped to visible area in clip space x, y, and z are in [-1,1], coordinates represent position vector of vertices viewport transformation of vertices from normalized coordinates to pixel coordinates - vertices now in window space the rendering pipeline - transforms now some calculations: how much light reaches each vertex? how much is reflected?... per-vertex lighting - pixel light is interpolated from vertices vertices may also carry texture coordinates all these information is interpolated to determine the final color for each pixel in the viewport... rasterization and fragment operations once in viewport coordinates: rasterization - what pixels are covered by what primitive? fragment - depth, color, texture, and location of pixel fragment/pixel shading - fragment data is used to determine the final color and depth for each pixel several test are performed so as to determine what pixels will be visible before calculating fragment shading so as to avoid unnecessary operations if a pixel/fragment passes all tests, its color is blended to the buffer

Math for 3D Graphics

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