2 Transformations and Homogeneous Coordinates

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Everett Dickerson
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1 Brief solutions to Exam in Computer Graphics Time and place: 08:00 3:00 Tuesday March 7, 2009, Gimogatan 4, sal Grades TD388: 3: 20pts; 4: 26pts; 5: 34pts. Glossary API Application Programmer s Interface. A source code interface that a computer system or program library provides in order to support requests for services to be made of it by a computer program. A layer between software and hardware. GPU Graphics Processing Unit. The GPU is a dedicated graphics rering device for a computer. Modern GPUs are very efficient at manipulating and displaying computer graphics. Their highly parallel structure makes them more effective than general-purpose CPUs. OpenGL Open Graphics Library. OpenGL is a standard specification defining a cross-language cross-platform API for writing applications that produce 2D and 3D computer graphics (developed by Silicon Graphics Inc. (SGI) in 992). It competes with Direct3D on Microsoft Windows platforms. Frame buffer A framebuffer loads a memory buffer containing a complete frame of data. The information in the buffer typically consists of colour values for every pixel on the display screen. RGBA Red Green Blue Alpha. A use of the RGB colour model, with extra information A enabling alpha bling and alpha compositing. The alpha channel is normally used as an opacity channel. If a pixel has a value of 0 in its alpha channel, then it is fully transparent (invisible), whereas a value of gives an opaque pixel. Values between 0 and make it possible for pixels to show through a background like a glass (translucency), an effect not possible with binary (transparent or opaque) transparency. 2 Transformations and Homogeneous Coordinates (a) Assume that v has unit length. Choose a vector x that is not parallell with v (This can be checked using the dot product). The vectors u and w can then be computed in the following way: (b) Let u = v x v x w = u v B = v x u y w x x v y u y w y y v z u z w z z

2 From the given conditions, we find that AB = , i.e. the identity matrix, and hence A = B. A represents a change of coordinate system. 3 The Phong Reflection Model (a) The Phong reflection model uses four vectors to calculate a colour for an arbitrary surface point p: (b) the vector n is the normal at p the vector v is in the direction from p to the viewer (or the centre of projection, COP) the vector l is in the direction from p to the (point) light source the vector r is in the direction that a perfectly reflected ray from l would take; r is determined by vectors n and l I = ( k a L a + See Chapter 6.3 for description of variables. ) a + bd + cd 2 (k dl d (n l) + k s L s (v r) α ) (c) We can modify the Phong illumination model to include anistropic highlights by varying the specular exponent alpha deping on the direction to the lightsource and the viewer. We assume that we have a vector anisotropy-direction, perpicular to the shading normal N, that represents the main orientation of the surface roughness. The roughness of the surface along, and perpicular to, this direction is given by two values alphax and alphay, respectively. (p) 4 Hidden Surface Removal (a) Object space approach Works on object level, comparing objects with each other. Examples: Painter s algorithm, Depth sort. Image/pixel space approach Works on pixel level, comparing pixel values. Examples: z-buffer algorithm, Ray-casting/ray-tracing. (b) The z-buffer algorithm has an image space approach, even though it is often built object by object. However, the comparisons are performed on pixel level. fill z-buffer with infinite (or maximal) distance for all polygons for each pixel calculate z-value if z(x,y) is closer than z-buffer(x,y) draw pixel z-buffer(x,y)=z(x,y) 2

3 5 Clipping (a) The idea is to discard polygons and clip out parts of polygons that lie outside the view frustum. See handouts for illustrations in 2D and 3D. (b) The normalized device coordinates can be obtained by using perspective division, but the coordinate is not projected to the front plane. See handouts for illustration. Anders 6 OpenGL and Interaction #include <GL/glut.h> float pos_x=-; float pos_y=-; int win_x =300; int win_y=300; void initglut( int &argc, char **argv) { glutinit( &argc,argv ); glutinitdisplaymode(glut_double GLUT_RGB); glutinitwindowsize( win_x, win_y ); glutinitwindowposition( 00, 00 ); glutcreatewindow( "OpenGL Program" ); void initgl() { glclearcolor( 0.0, 0.0, 0.0, 0.0 ); glmatrixmode( GL_PROJECTION ); glloadidentity(); glortho(0.0,.0, 0.0,.0, -.0,.0 ); glmatrixmode ( GL_MODELVIEW ); // mouse motion callback void motion(int x, int y) { pos_x=x/double(win_x); pos_y=-(y/double(win_y)); glutpostredisplay(); //display callback void draw() { glclear( GL_COLOR_BUFFER_BIT ); //draw a red point glcolor3f( 0.8, 0.2, 0.2 ); glpointsize(20.0); glbegin( GL_POINTS ); glvertex2f( pos_x, pos_y ); glend(); 3

4 glutswapbuffers(); //reshape callback void reshape(int width, int height){ glviewport(0,0,width,height); win_x=width; win_y=height; int main(int argc, char **argv) { initglut(argc, argv); initgl(); //register display callback glutdisplayfunc( draw ); //register mouse motion callback glutpassivemotionfunc(motion); //register reshape callback glutreshapefunc(reshape); //Start the GLUT main event loop. glutmainloop(); return 0; 7 Shader programming varying vec3 normal, lightdir, pointdir, texcoord0; 2 uniform sampler2d diffusemap; 3 4 void main() 5 { 6 if (texture2d(diffusemap, texcoord0.xy).x < 0.5) 7 discard; 8 9 vec3 h = normalize(pointdir + lightdir); 0 vec3 n = normalize(normal); vec3 l = normalize(lightdir); 2 3 float ln = clamp(dot(l, n), 0.0,.0); 4 float hn = clamp(dot(h, n), 0.0,.0); 5 float s = pow(hn, 256.0); 6 float bf = dot(vec3(0.0, 0.0,.0), n) >= 0.0?.0 : 0.0; 7 float rim = (.0 - pow(ln, 5.0)) * bf; 8 9 float f = (ln > 0.7?.0 : (ln > 0.5? 0.8 : (ln > 0.2? 0.7 : 0.5))); 20 float g = s > 0.6?.0 : 0.0; Gustav 4

5 2 22 vec3 diffuse = vec3(.0, 0.0, 0.0); 23 vec3 specular = vec3(0.0,.0, 0.0); vec3 col = (g > 0.0 rim > 0.9)? specular : diffuse * f; gl_fragcolor = vec4(col, 0); 28 The actual numbers used to define the discrete colour levels does not matter, as long as the logic is correct and that darker areas receive darker colours. 8 Curves (a) If we distribute the control points uniformly in parameter space, we get p(0) = c 0 p( 2 ) = c c + ( 2 )2 c 2 p() = c 0 + c + c 2 Or, in matrix notation 0 0 A = 2 ( 2 )2. (b) Using Horner s rule, we get p(u) = c 0 + u(c + c 2 u). 5

OpenGL/GLUT Intro. Week 1, Fri Jan 12

OpenGL/GLUT Intro. Week 1, Fri Jan 12 University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2007 Tamara Munzner OpenGL/GLUT Intro Week 1, Fri Jan 12 http://www.ugrad.cs.ubc.ca/~cs314/vjan2007 News Labs start next week Reminder: