CSE 167: Introduction to Computer Graphics Lecture #4: Vertex Transformation

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1 CSE 167: Introduction to Computer Graphics Lecture #4: Vertex Transformation Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013

2 Announcements Project 2 due Friday, October 11 Compare on-line homework scores to your notes and let us know if they do not match To get graded early, please see course assistants during office hours New: Friday grading with two sign-up boards in labs 260 and 270 Changes to office hours in lab: see course web page 2

3 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 3

4 View Volumes View volume = 3D volume seen by camera Orthographic view volume Camera coordinates Perspective view volume Camera coordinates World coordinates World coordinates 4

5 Projection Matrix Camera coordinates Projection matrix Canonical view volume Viewport transformation 5 Image space (pixel coordinates)

6 Orthographic View Volume Specified by 6 parameters: Right, left, top, bottom, near, far Or, if symmetrical: Width, height, near, far 6

7 Orthographic Projection Matrix In OpenGL: glortho(left, right, bottom, top, near, far) 7 2 right + left 0 0 right left right left 2 top + bottom 0 0 P ortho (right,left,top,bottom,near, far) = top bottom top bottom 2 far + near 0 0 far near far near width P ortho (width,height,near, far) = height No equivalent in OpenGL 2 far + near 0 0 far near far near

8 Perspective View Volume General view volume Camera coordinates Defined by 6 parameters, in camera coordinates Left, right, top, bottom boundaries Near, far clipping planes Clipping planes to avoid numerical problems Divide by zero Low precision for distant objects Usually symmetric, i.e., left=-right, top=-bottom 8

9 Perspective View Volume Symmetrical view volume y y=top FOV -z z=-near Only 4 parameters Vertical field of view (FOV) Image aspect ratio (width/height) Near, far clipping planes z=-far aspect ratio= tan(fov / 2) = top near right left top bottom = right top 9

10 Perspective Projection Matrix General view frustum with 6 parameters Camera coordinates In OpenGL: glfrustum(left, right, bottom, top, near, far) 10

11 Perspective Projection Matrix Symmetrical view frustum with field of view, aspect ratio, near and far clip planes Camera coordinates y y=top FOV z=-near -z z=-far aspect tan(fov / 2) P persp (FOV,aspect,near, far) = tan(fov / 2) near + far 0 0 near far In OpenGL: gluperspective(fov, aspect, near, far) 11 2 near far near far

12 Canonical View Volume Goal: create projection matrix so that User defined view volume is transformed into canonical view volume: cube [-1,1]x[-1,1]x[-1,1] Multiplying corner vertices of view volume by projection matrix and performing homogeneous divide yields corners of canonical view volume Perspective and orthographic projection are treated the same way Canonical view volume is last stage in which coordinates are in 3D Next step is projection to 2D frame buffer 12

13 Viewport Transformation After applying projection matrix, scene points are in normalized viewing coordinates Per definition within range [-1..1] x [-1..1] x [-1..1] Next is projection from 3D to 2D (not reversible) Normalized viewing coordinates can be mapped to image (=pixel=frame buffer) coordinates Range depends on window (view port) size: [x0 x1] x [y0 y1] Scale and translation required: 13 D( x 0, x 1, y 0, y 1 )= ( x 1 x 0 ) x 0 + x 1 0 ( y 1 y 0 ) 2 0 y 0 + y ( ) 2 ( ) 2

14 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 14

15 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Object space M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix 15

16 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Object space World space M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix 16

17 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space 17

18 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space Canonical view volume 18

19 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Image space M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space Canonical view volume 19

20 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Pixel coordinates: M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix 20

21 The Complete Vertex Transformation Object Coordinates Model Matrix World Coordinates Camera Matrix Camera Coordinates Canonical View Volume Coordinates Window Coordinates Projection Matrix Viewport Matrix 21

22 Complete Vertex Transformation in OpenGL Mapping a 3D point in object coordinates to pixel coordinates: OpenGL GL_MODELVIEW matrix M: Object-to-world matrix C: camera matrix P: projection matrix D: viewport matrix OpenGL GL_PROJECTION matrix 22

23 Complete Vertex Transformation in OpenGL GL_MODELVIEW, C -1 M Defined by the programmer. Think of the ModelView matrix as where you stand with the camera and the direction you point it. GL_PROJECTION, P Utility routines to set it by specifying view volume: glfrustum(), glperspective(), glortho() Think of the projection matrix as describing the attributes of your camera, such as field of view, focal length, etc. Viewport, D Specify implicitly via glviewport() No direct access with equivalent to GL_MODELVIEW or GL_PROJECTION 23

24 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 24

25 Rendering Pipeline Scene data Rendering pipeline Hardware and software which draws 3D scenes on the screen Consists of several stages Simplified version here Most operations performed by specialized hardware (GPU) Access to hardware through low-level 3D API (OpenGL, DirectX) All scene data flows through the pipeline at least once for each frame 25 Image

26 Rendering Pipeline Scene data Modeling and viewing transformation Shading Textures, lights, etc. Geometry Vertices and how they are connected Triangles, lines, points, triangle strips Attributes such as color Projection 26 Rasterization, visibility Image Specified in object coordinates Processed by the rendering pipeline one-by-one

27 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Transform object to camera coordinates Specified by GL_MODELVIEW matrix in OpenGL User computes GL_MODELVIEW matrix as discussed Rasterization, visibility MODELVIEW matrix 27 Image

28 Rendering Pipeline Scene data Modeling and viewing transformation Shading Look up light sources Compute color for each vertex Projection Rasterization, visibility 28 Image

29 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Rasterization, visibility Project 3D vertices to 2D image positions GL_PROJECTION matrix 29 Image

30 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Draw primitives (triangles, lines, etc.) Determine what is visible Rasterization, visibility 30 Image

31 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Rasterization, visibility 31 Image Pixel colors

32 Rendering Engine Scene data Rendering pipeline Rendering Engine: Additional software layer encapsulating low-level API Higher level functionality than OpenGL Platform independent Layered software architecture common in industry Game engines Graphics middleware 32 Image

33 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 33

34 Culling Goal: Discard geometry that does not need to be drawn to speed up rendering Types of culling: View frustum culling Occlusion culling Small object culling Backface culling Degenerate culling 34

35 View Frustum Culling Triangles outside of view frustum are off-screen Done on canonical view volume Images: SGI OpenGL Optimizer Programmer's Guide 35

36 Videos Rendering Optimisations - Frustum Culling View Frustum Culling Demo 36

37 Bounding Box How to cull objects consisting of may polygons? Cull bounding box Rectangular box, parallel to object space coordinate planes Box is smallest box containing the entire object 37 Image: SGI OpenGL Optimizer Programmer's Guide

38 Occlusion Culling Geometry hidden behind occluder cannot be seen Many complex algorithms exist to identify occluded geometry Images: SGI OpenGL Optimizer Programmer's Guide 38

39 Video Umbra 3 Occlusion Culling explained 39

40 Small Object Culling Object projects to less than a specified size Cull objects whose screen-space bounding box is less than a threshold number of pixels 40

41 Backface Culling Consider triangles as one-sided, i.e., only visible from the front Closed objects If the back of the triangle is facing the camera, it is not visible Gain efficiency by not drawing it (culling) Roughly 50% of triangles in a scene are back facing 41

42 Backface Culling Convention: Triangle is front facing if vertices are ordered counterclockwise p2 p1 p0 p0 Front-facing OpenGL allows one- or two-sided triangles One-sided triangles: glenable(gl_cull_face); glcullface(gl_back) Two-sided triangles (no backface culling): gldisable(gl_cull_face) p1 Back-facing p2 42

43 Backface Culling Compute triangle normal after projection (homogeneous division) Third component of n negative: front-facing, otherwise back-facing Remember: projection matrix is such that homogeneous division flips sign of third component 43

44 Degenerate Culling Degenerate triangle has no area Vertices lie in a straight line Vertices at the exact same place Normal n=0 Source: Computer Methods in Applied Mechanics and Engineering, Volume 194, Issues

45 Rendering Pipeline Primitives Modeling and Viewing Transformation Shading Projection Scan conversion, visibility Culling, Clipping Discard geometry that will not be visible 45 Image

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