ࡱ> 5@ Ebjbj22 ,XX=|GGGGdH |jR|IIIIIIIInjpjpjpjpjpjpj$AlRnJj0JII0J0JjIIjhhh0JIInjh0Jnjh"h>h>hIpI pZGY2 >hhj0j>hn en>hn>hIIhI IpIIIjj||dDGh ||G  PyRoll: A Python Gravity Well Simulation Documentation Mark TenPas Math 198 May, 2005 Introduction Abstract PyRoll, a program written in Python with GLUT (PyGLUT), is designed to model a gravity well. The program has a framework that makes it easy to modify it to allow for the modeling of other surfaces. Running the Program & User Input To execute PyRoll, go to the directory where PyRoll.py is held and run the following command: python pyroll.py This should call up a GLUT window with a funnel and magenta ball inside of it. To pause the program, press space bar. (Press space bar again to unpause) To restart the ball with the same initial velocity, press r. To exit the program, press q. The Program In-Depth Lines 21-38 Program Details The framework for the program was borrowed from tr0.py, a program that models a tauras (a donut-shaped 3D object.) It was written by George Francis in April, 2005, and is available as an illiLesson. Global Variables in the Program: The first eight global variables were present in tr0. My motto is, If it aint broken, dont fix it, and since I didnt need to modify these to implement my program, I left them as they were: Lines 21-28: global wd global ht global MouseX global MouseY global lux global lu global pwr global aff The next nine variables were needed to keep track of different aspects of the ball. xdis, ydis, and zdis all correspond to the displacements of x, y, and z, respectively; xvel, yvel, and zvel all correspond to their velocities, and xnorm, ynorm, and znorm all correspond to the adjustment in x, y, and z, that must be made to put the edge of our ball- rather than its center- on our surface. I had originally hoped to have a ball object store this data, but since Im not that familiar with Python, I was a bit concerned about the overhead involved in making things object-oriented, and how that would be optimized. In the end, I decided to put my energy into other aspects of the program. Lines 30-38: global xdis global ydis global zdis global xvel global yvel global zvel global xnorm global ynorm global znorm The Program In-Depth Lines 40-59 The next two variables that I include are references for the starting points. These are defined in main() Lines 40-41: global startvelx global startvely Finally, the last two variables I use have to do with keeping track of user preferences. The pause variable keeps track of whether the program is paused. The friction variable is designed to allow the user to turn friction on by pressing f. When the user presses f, this variable is set to 1; when the user presses f again, it is set back to 0. Lines 43-44: global pause global friction Lines 47-54 make several constant declarations that were present in the old program: Lines 47-54: lux =(0.3, 0.5, 0.8) lu = [1,0,0] # this should not be necessary pwr = 10. aff = [1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0] DrawVert(th, ta) The first thing to note about DrawVert is that it takes th (theta) and ta (tau) as parameters: Line 56: def drawvert(th,ta): In fact, this function gets called from a nested set of for loops (first on th, then on ta) on line 89. Folks familiar with polar coordinates and/or parametric functions may be noticing that were setting up a parametric function over an angle theta and a radius, tau. Lines 58 and 59 modify our theta and tau so that they work on a scale thats OK for us: rad=10.0/ta z=ta/10.0 The Program In-Depth Lines 60-79 Lines 60-62 define our normal vectors; we need them to handle lighting. These vectors came from the partial derivative of our function (n0, n1, and n2 correspond to x, y, and z, respectively). Normally, n0 and n1 should both be divided by r^2, but in order to reduce the number of divisions we make, we multiply everything by r^2, including n2: Lines 60-62 n0=C(th) n1=S(th) n2=-z*z We calculate the unit vector, nn, in lines 63-64: Lines 63-64: d=sqrt(n0*n0+n1*n1+n2*n2) nn=(n0/d,n1/d,n2/d) Lines 65-78, used to calculate lighting, were already present in tr0: lmb = nn[0]*lu[0]+nn[1]*lu[1]+nn[2]*lu[2] lmb = min(max(lmb, 0.0), 1.0) #clamp spec=min(1., 1. - pwr + pwr*lmb) dog = (th- 15.)/(360.-15.) cat = (ta- 1.)/(100.-1.) rr = dog gg = (.25 + abs(cat-.5)) bb = (1-cat) amb = .25 rrr=max( rr*amb, max(rr*lmb, spec)) ggg=max( gg*amb, max(gg*lmb, spec)) bbb=max( bb*amb, max(bb*lmb, spec)) glColor3f(rrr,ggg,bbb) Line 79 draws out our parametric function: X= rad*C(th) Y= rad*S(th) Z=z glVertex3f(rad*C(th), rad*S(th), z) The Program In-Depth Lines 82-105 DrawFun() Lines 82-90: def drawfun(): global lmb global vv global nn for th in range(0,360,15): glBegin(GL_TRIANGLE_STRIP) for ta in range (2,30, 1): drawvert(th,ta); drawvert(th+12,ta) glEnd() #end drawfun DrawFun() iterates through ta and th and draws a series of triangle slices, with positions defined by drawvert (see previous page). DrawBall(rr, gg, bb) Lines 94-96: def drawball(rr,gg,bb): glColor3f(rr,gg,bb) glutWireSphere(.20,8,8) Given parameters for red (rr), green (gg), and blue (bb), this function draws a glut wire sphere of radius .2 with 8 lattitudinal and 8 longitudinal wires, with color dependant on input. Calculite(aff) Lines 99-105: def calculite(aff): global lu for ii in range(3): lu[ii]= 0 for jj in range(3): lu[ii]+=aff[ii][jj]*lux[jj] #end calculate Calculite is a program that was already present in tr0. I think its used for calculating luminosity, but the luminosity that most programmers will have to worry about will be in drawVert. The Program In-Depth Lines 107-127 Lines 107-113 set up the global variables mentioned on page 3: Lines 107-113: #assign initial window and mouse settings wd = 800 ht = 800 MouseX = wd/2 MouseY = ht/2 brake = 512. Calc(): The first few lines call the global variables we declared earlier (see page 3). Ive set up fric as a way of implementing friction, and hh is our dt. HH will need to be set smaller on faster computers and higher on slower computers. Finally, right before we get into the full swing of calculating the balls new position, we make sure our program isnt paused. If it is pause (pause=1), then we do nothing. Lines 115-123: def calc(): global xdis, ydis, zdis, xvel, yvel, zvel, xnorm, ynorm, znorm global pause, friction fric = 0*friction #dt: hh = .05 # updates xdis ydis,zdis if pause == 0: In the next four lines, we first calculate the slope of our surface at our current displacement (dzdx and dzdy). We use the limit definition of a partial derivative to approximate this. We then calculate the acceleration, based off of that derivative: Lines 124-127: dzdx=(-1.0/sqrt(xdis*xdis+ydis*ydis)+1/sqrt((xdis-.001)*(xdis- .001)+ydis*ydis))/.001 dzdy=(-1.0/sqrt(xdis*xdis+ydis*ydis)+1/sqrt(xdis*xdis+(ydis- .001)*(ydis-.001)))/.001 xacc=-1*dzdx/(dzdx*dzdx+1) yacc=-1*dzdy/(dzdy*dzdy+1) Please note that since our parameterized function and explicit function are equivalent, we can handle everything in terms of x, y and z. Conceptually, its a lot easier. The Program In-Depth Lines 128-142 Based on: V(t+1)~=V(t)+A(t) and D(t+1)~=D(t)+V(t), we can approximate our velocity and displacement: Lines 128-131: xvel=xvel+xacc*hh yvel=yvel+yacc*hh xdis=xdis+xvel*hh ydis=ydis+yvel*hh To make sure our ball stays on the surface, we calculate zdis based on our function for the gravity well and our x and y displacements: Line 132: zdis=1.0/sqrt(xdis*xdis+ydis*ydis) We then calculate the normal vector, and adjust it so that it becomes a parallel vector of length .2 (the radius of our sphere/ball.) We want to make sure that our balls edge falls on the surface, so we need to adjust the position of its center in the direction of the surfaces normal vector a distance of .2. However, we cant adjust the actual position of the ball, since we need to use that to calculate the slope in our next iteration. Lines 133-142 #calculate the normal vector to adjust the position of the ball. xnorm=xdis ynorm=ydis zcalc=sqrt(xdis*xdis+ydis*ydis) znorm=zcalc*zcalc*zcalc normallength=sqrt(xnorm*xnorm+ynorm*ynorm+znorm*znorm) inverse=-1.0/normallength xnorm=xnorm*0.2*inverse ynorm=ynorm*0.2*inverse znorm=znorm*0.2*inverse The Program In-Depth Lines 145-174 Display() Display is a function that GLUT calls to know the specifics of what its supposed to display. We have to worry about our surface and our ball. The surface, and the rotations affected by the mouse, were already in tr0.py. The important things that happen in this function are that we scale our matrix (line 151) to make everything about 1/3 the size, and we then translate our new matrix such that we can place the ball where we want it to be: Lines 145-157: def display(): global aff, xdis, ydis, zdis glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT) glMatrixMode(GL_MODELVIEW) glLoadIdentity() glMultMatrixf(aff) glScalef(.3,.3,.3) drawfun() calc() glTranslatef(xdis+xnorm,ydis+ynorm,zdis+znorm) drawball(1,0,1) glutSwapBuffers() calculite(aff) Keyboard(key, x, y) Keyboard is a function that gets called whenever theres an input. The variable key holds the character that just got typed. In this program, we need to worry about q, r, and space for quit, restart, and pause, respectively (See page 2 for user inputs.) Ive also accounted for f, in case future programmers want to include friction in the program: Lines 163-174: def keyboard(key, x, y): global pause, xdis, ydis, zdis, xvel, friction, startvelx, startvely if key == chr(27) or key == 'q': sys.exit(0) if key == ' ' and pause == 1: pause=2 if key == ' ' and pause == 0: pause=1 print "paused" if key == ' ' and pause == 2: pause=0 print "unpaused" The Program In-Depth Lines 175-208 Lines 175-185: if key == 'r': print "restart" xdis=2.0 ydis=2.0 zdis=0.0 xvel=startvelx yvel=startvely if key == 'f' and friction == 1: friction=0 if key == 'f' and friction == 0: friction=1 glutPostRedisplay() #end keyboard() Reshape(width, height) I pretty much left Reshape() alone when implementing my program. The one thing that may be helpful to future developers, however, is that glOrtho controls the range in which our program gets displayed. Lines 191-208 def reshape(width, height): global wd global ht glClearColor(0.0, 0.0, 0.0, 0.0) if height == 0: height = 1 wd = width ht = height glViewport(0,0,wd,ht) glMatrixMode(GL_PROJECTION) glLoadIdentity() if wd<=ht: glOrtho(-2.0,2.0,-2.0*ht/wd,2.0*ht/wd,1.0,200.0) else: glOrtho(-2.0*wd/ht,2.0*wd/ht,-2.0,2.0,1.0,200.0) glMatrixMode(GL_MODELVIEW) glLoadIdentity() #end reshape() The Program In-Depth Lines 211-233 Chaptrack() Chaptrack() rotates our matrix based on mouse inputs. I left it unchanged from tr0.py. The important conceptual thing to note is that Chaptrack currently rotates everything thats been put in our matrix. Put another way, assuming you want your objects to stay in place relative to everything else, (rather than stay in place relative to the monitor) you dont have to touch Chaptrack(). Lines 211-227 def chaptrack(): global MouseX global MouseY global wd global ht global aff dx = (MouseX-wd/2)/brake dy = (MouseY-ht/2)/brake glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity() glRotatef(dx,0,1.0,0.0) glRotatef(dy,1.0,0.0,0.0) glMultMatrixf(aff) aff = glGetFloatv(GL_MODELVIEW_MATRIX) glPopMatrix() #end chaptrack() Idle() This is the function OpenGL calls when its idling. This kinda allows us to have multiple threads going at the same time in the function, but it probably wont be something that future programmers will have to worry about: Lines 229-233 def idle(): chaptrack() glutPostRedisplay() #end idle() The Program In-Depth Lines 237-273 Mousemotion() Mousemotion keeps track of the mouses position: Lines 237-242 def mousemotion(x,y): global MouseX global MouseY MouseX = x MouseY = y #end mousemotion() Main(argv) The Main function handles a lot. Its used to initialize most of our global variables, turn our matrix over 180 degrees so we can see the top, and otherwise set everything up in GLUT: Lines 245-273: def main(argv) : global wd global ht global xdis, ydis, zdis, xvel, yvel, pause, friction pause=0 friction=0 xdis=2.0 ydis=2.0 zdis=0.0 startvelx=-0.4 startvely=0.3 xvel=startvelx yvel=startvely glutInitDisplayMode(GLUT_RGB | GLUT_DEPTH | GLUT_DOUBLE) glutInitWindowPosition(50, 50) glutInitWindowSize(wd, ht) glutInit(argv) glutCreateWindow("PyRoll") glutKeyboardFunc(keyboard) glutDisplayFunc(display) glutIdleFunc(idle) glutPassiveMotionFunc(mousemotion) glEnable(GL_DEPTH_TEST) glShadeModel(GL_SMOOTH) glMatrixMode(GL_PROJECTION) glLoadIdentity() glOrtho(-2.0,2.0,-2.0,2.0,-2.0,2.0) glRotatef(180,0,1.0,0.0) glutMainLoop() Possible Improvements The Cave/Cube: Porting this program over to the cave or the cube, assuming Professor doesnt get around to it first, might be an exciting feat. Currently, the easiest way to do this would be to convert my global variables into an object. Adding On New Surfaces and Force Regimes: My project was pretty simple; construct a gravity well that operates under the force of gravity. What happens when forces become dependant on position or velocity? A few ideas for other potential wells (operating under the force of gravity) might include electrons between positive and negative charges and a buoy in a series of waves. An idea for another force regime might involve a charged particle moving through a magnetic field. The infrastructure in this program should be in place to handle all of that. Four Dimensions Would it be possible to display a 3D cross-section of the equivalent of a 4-Dimensional surface? Would such a thing be interesting? Its something Ive thought about, but something I havent had the time for. As of 2005, it seems like a desktop Linux computer might be able to just barely pull off constantly redrawing a relatively small surface. Im sure that given a year or two, a desktop might be in a much stronger position to handle it. Conclusion & Acknowledgements Well, thats my project. I enjoyed figuring out all of the math involved, as well as working with all of my classmates & professor. Id especially like to thank Prof. Francis and Jacob Lee (another student) for their help on the project. Jacob helped me figure out Python, and Prof. Francis helped me with a lot in OpenGL. If anyone has any questions about my project, Id love to talk about it; and if anyone decides to build on my project, Id love to hear about and celebrate their work. 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