World and Screen Spaces

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2/2/2005 Lecture 2 3 2/2/2005 Lecture 2 4 World and Screen Spaces Lecture 2 Comp 236 Spring 2005 From last time Our world view seemed rather limited: - - A box ranging from [-,] in x, and [-,] in y

1 2/2/2005 Lecture 2 3 2/2/2005 Lecture 2 4 World and Screen Spaces Lecture 2 Comp 236 Spring 2005 From last time Our world view seemed rather limited: - - A box ranging from [-,] in x, and [-,] in y Today s issues: Reality pixels on the screen Abstraction the world as we wish to see it (on the screen) 2/2/2005 Lecture 2 2 More than you might suspect An aside a visit to Fractal Land A lot of stuff is going in our little world Fractal Julia Sets A visualization of a (seemingly) simple iterative function defined over the imaginary plane Chaotic behavior Small changes have dramatic effects Recipe for a Julia Set: ) A complex number, c 2) The set of all points, P, in the imaginary plane: Julia Set of c: (p in P) such that p i+ = p i2 + c converges to a fixed point in the limit as i Complex Numbers 2-tuples, with parts called real and imaginary Definitions c = a + b : cre = are + bre cim = aim + bim c = a*b : cre = are*bre aim*bim cim = are*bim + aim*bre c = a 2 : cre = are 2 aim 2 cim = 2 * are * aim a = sqrt(are 2 + aim 2 )

Convergence example 2/2/2005 Lecture 2 7 Convergence example 2/2/2005 Lecture 2 8 Real numbers are a subset of complex numbers: consider c = [0, 0], and p = [x, 0] for what values of x is x i+ = x i2

2 Convergence example 2/2/2005 Lecture 2 7 Convergence example 2/2/2005 Lecture 2 8 Real numbers are a subset of complex numbers: consider c = [0, 0], and p = [x, 0] for what values of x is x i+ = x i2 convergent? x 0 = 05 x = 025 x 2 = x 3 = Real numbers are a subset of complex numbers: consider c = [0, 0], and p = [x, 0] for what values of x is x i+ = x i2 convergent? x 0 = x = 2 x 2 = 464 x 3 = /2/2005 Lecture 2 5 2/2/2005 Lecture 2 6 Convergence Properties Suppose c = [0,0], for what complex values of p does the series converge? A peek at the Fractal Code class Imaginary: def init (self, r = 0, i = 0): selfre = r selfim = I For real numbers: If x i > then the series diverges For complex numbers If p i > 2 then the series diverges BTW, The black points are the ones in c s Julia set def Julia(p, c): maxiterations = 256 n = Imaginary() for i in range(maxiterations): nre = pre*pre - pim*pim + cre nim = 2*pre*pim + cim rsqr = nre*nre + nim*nim if rsqr > 40: break pre, pim = nre, nim return (i, rsqr) The return values i and rsqr are used to assign a color

2/2/2005 Lecture 2 2/2/2005 Lecture 2 2 Now, Some Graphics Code def display(): delta = ( - -)/float(400) p = Imaginary() for j in range(height): y = - + j*delta for i in range(width): x = - + i*delta

3 2/2/2005 Lecture 2 2/2/2005 Lecture 2 2 Now, Some Graphics Code def display(): delta = ( - -)/float(400) p = Imaginary() for j in range(height): y = - + j*delta for i in range(width): x = - + i*delta pre, pim = x, y n, rsqr = Julia(p, c) if n == 255: glcolor3d(0,0,0) else: r = mathsqrt(rsqr)/float(3) g = n/float(28) b = mathsqrt(rsqr)/float(n+) glcolor3d(r,g,b) glbegin(gl_polygon) glvertex2d(x, y) glvertex2d(x, y+delta) glvertex2d(x+delta, y+delta) glvertex2d(x+delta, y) glend() How can we see more? Our world view only allows us to see so much We need to define a mapping from our desired world view to our screen 2/2/2005 Lecture 2 9 2/2/2005 Lecture 2 0 Screen Space Pixel Independence In today s world, graphics are generally presented by establishing colors for a set of discrete samples called pixels Pixels are displayed on screen in windows Logically, pixels are addressed as two-dimensional arrays, whose indices are called Sceen-space coordinates (0,0) (width-,0) (0,height-) (width-,height-) Often, we desire to structure our graphics independent of screen or window sizes We do so by defining a world-space where we specify our graphics Often, with physical justification 25 Meters 2 Meters 500 cubits 800 cubits

4 Normalized Device Coordinates Rather than construct a single mapping from world-toscreen, we will instead introduce an intermediate rendering-space and compose these two mappings This rendering-space is what we ve been using a our default to this point This space is called Normalized Device Coordinates (NDC) Sometimes called - canonical screen space - 2/2/2005 Lecture 2 3 Why introduce NDC? Easy to convert NDC to a fixed-point representation Why? That s just a standard two s complement representation, with a binary point It represents number in the range [-,) Simplifies many rendering operations and makes them more amenable to a H/W implementation (Clipping, Computing coefficients used in interpolation) Separates the bulk of geometric processing from the specifics of rasterization (sampling) 2/2/2005 Lecture 2 4 For a World to NDC mapping: We know x (-) Wx Wl = (-) Wr Wl Wx Wl x = 2 Wr Wl x = A Wx + B Where: How does it work? 2 A = Wr Wl Worldtop - Worldbottom x? - Worldx Worldleft That s just a Linear Mapping We know of a compact notation for such mappings Wr + Wl B = Wr Wl Worldright 2/2/2005 Lecture 2 5 Mapping as a Matrix Multiplication Since our world space to rendering space mappings are linear, they can be accomplished via a matrix multiplication x y = Wr 2 Wl Wt 2 Wb 0 Wx Wy 2/2/2005 Lecture 2 6 Wr+ Wl Wr Wl Wt+ Wb Wt Wb

From NDC to screen coordinates 2/2/2005 Lecture 2 9 Mapping from world to screen 2/2/2005 Lecture 2 20 Same approach Sx SOx x ( ) = width ( ) Solve for Sx Sx = x 2 + width + SOx Sx = Ax + B where

5 From NDC to screen coordinates 2/2/2005 Lecture 2 9 Mapping from world to screen 2/2/2005 Lecture 2 20 Same approach Sx SOx x ( ) = width ( ) Solve for Sx Sx = x 2 + width + SOx Sx = Ax + B where ScreenOy - ScreenOx - width width A = B = + SOx 2 2 2/2/2005 Lecture 2 7 height width The NDC-to-screen mapping could be implemented as a matrix transform, but, generally, it is not Window System: (0,0) (uses dedicated Hardware) World-to-NDC floating point NDC-to-screen fixed-point Graphics System: (0,0) Inserts an opportunity to compensate for differences in coordinate frames OpenGL provides a lot of helpful tools, but not enough We often need to construct a screen-to-world mapping 2/2/2005 Lecture 2 8 OpenGL Tools Available Typical OpenGL code to establish a window: glutinitwindowsize(400,400) glutinitwindowposition(00,00) Code to set up a viewport: glviewport(0, 0, w, h) To establish a world space coordinate system: glortho2d(worldl, worldr, worldb, worldt) Screen-to-World in Practice def : global world glmatrixmode(gl_projection) gluortho2d(worldl, worldr, worldb, worldt) glmatrixmode(gl_modelview) glcolor3d(0,0,) glbegin(gl_polygon) glvertex2d(worldl, worldb) glvertex2d(worldr, worldb) glvertex2d(worldr, worldt) glvertex2d(worldl, worldt) glend()

6 The GLUTs of the program What is GLUT, Tony? def main(): glutinit(sysargv) glutinitdisplaymode(glut_single GLUT_RGB) glutinitwindowsize(width, height) glutinitwindowposition(00, 00) glutcreatewindow("julia Set (aka Whoville)") glutdisplayfunc(display) glutmousefunc(onmousebutton) glutkeyboardfunc(onkeypress) glutreshapefunc(onresize) glutmainloop() A O/S agnostic OpenGL environment Advantages: Portable: Windows, Cygwin, Linux, Mac-OS minimal-overhead (Hides away details of opening windows, etc) Appeals to C-hackers (console for printf() s, etc) Disadvantages Ugly (lacks look-and-feel of real app, outdated call-backbased event-handling model) Limited Interaction Global variables galore 2/2/2005 Lecture 2 2 2/2/2005 Lecture 2 22 Getting GLUT Web site: Windows: wwwxmissioncom/~nate/gluthtml Others: wwwopenglorg/developers/documentation/gluthtml wwwsourceforgenet/projects/uncpythontools (under uncopengl link) Overview: Appendix D of OpenGL Programming Guide Back to the Code import sys import math from GL import * from GLU import * from GLUT import * Lots of global variables (and an old Friend) class Imaginary: def init (self, r = 0, i = 0): selfre = r selfim = i class Extent: def init (self, l = 0, r = 0, b = 0, t = 0): selfl = l selfr = r selfb = b selft = t world = Extent(-,, -, ) c = Imaginary(009, 0603) width = 200 height = 200 def Julia(p, c): maxiterations = 256 n = Imaginary() for i in range(maxiterations): nre = pre*pre - pim*pim + cre nim = 2*pre*pim + cim rsqr = nre*nre + nim*nim if rsqr > 40: break pre, pim = nre, nim return (i, rsqr) 2/2/2005 Lecture /2/2005 Lecture 2 24

7 Initialize Speaking of Resize Executed before any other OpenGL calls, called when major changes occur: def : global world glmatrixmode(gl_projection) gluortho2d(worldl, worldr, worldb, worldt) glmatrixmode(gl_modelview) Note: by default GLUT set s up a viewport, we only need to mess with it if the window size changes Resize gets called when the window size changes Half of the new width is added and subtracted from the world s center def onresize(w, h): width = w height = h glviewport(0, 0, w, h) glmatrixmode(gl_projection) l = 05*(worldr + worldl) - (05*(worldt - worldb)*w)/h r = 05*(worldr + worldl) + (05*(worldt - worldb)*w)/h worldl = l worldr = r gluortho2d(worldl, worldr, worldb, worldt) glmatrixmode(gl_modelview) 2/2/2005 Lecture /2/2005 Lecture 2 26 An Updated Old Friend Recap all the GLUT stuff def display(): delta = (worldr - worldl)/float(width) p = Imaginary() for j in range(height): y = worldb + j*delta for i in range(width): x = worldl + i*delta pre, pim = x, y n, rsqr = Julia(p, c) if n == 255: glcolor3d(0,0,0) else: r = mathsqrt(rsqr)/float(3) g = n/float(28) b = mathsqrt(rsqr)/float(n+) The graphics code is the same glcolor3d(r,g,b) glbegin(gl_polygon) glvertex2d(x, y) glvertex2d(x, y+delta) glvertex2d(x+delta, y+delta) glvertex2d(x+delta, y) glend() glflush() 2/2/2005 Lecture 2 27 def main(): glutinit(sysargv) glutinitdisplaymode(glut_single GLUT_RGB) glutinitwindowsize(width, height) glutinitwindowposition(00, 00) glutcreatewindow("julia Set (aka Whoville)") glutdisplayfunc(display) glutmousefunc(onmousebutton) glutkeyboardfunc(onkeypress) glutreshapefunc(onresize) glutmainloop() if name == " main ": main() 2/2/2005 Lecture 2 28

8 Now the GUI stuff Now the GUI stuff def onmousebutton(button, state, x, y): if (button == GLUT_LEFT_BUTTON) & (state == GLUT_DOWN): mousex = xscreentoworld(x) mousey = yscreentoworld(y) dx = (worldr - worldl)/float(4) dy = (worldt - worldb)/float(4) worldl = mousex - dx worldr = mousex + dx worldb = mousey - dy worldt = mousey + dy display() elif elif (button == GLUT_RIGHT_BUTTON) & (state == GLUT_DOWN): mousex = xscreentoworld(x) mousey = yscreentoworld(y) dx = worldr - worldl dy = worldt - worldb worldl = mousex - dx worldr = mousex + dx worldb = mousey - dy worldt = mousey + dy display() 2/2/2005 Lecture /2/2005 Lecture 2 30 Screen-to-World Mapping The Key Code def xscreentoworld(x): return ((worldr - worldl) * x / float(width)) + worldl That sure was a lot of lecture for so little code! 2/2/2005 Lecture 2 3 def onkeypress(key, x, y): global c print key if (key == 'r') (key == 'R'): cre, cim = 009, 0603 worldl = - worldr = worldb = - worldt = display() elif (key == 'c') (key == 'C'): print "got here" cre, cim = 0, 0 worldl = - worldr = worldb = - worldt = display() 2/2/2005 Lecture 2 32

9 A Horse Race Next Time Python - Interpreted C# - Compiled 2D Graphics and Imaging in OpenGL Preview of st Programming Assignment 2/2/2005 Lecture /2/2005 Lecture 2 40

From last time. Julia set. Screen space & world space. More than you might suspect

From last time. Julia set. Screen space & world space. More than you might suspect From last time Screen space & world space Computer Graphics COMP 770 (236) Spring 2007 - Limited world view A box on the domain [-,-] x [,] Today s issues: Reality: pixels on the screen Abstraction: the