Sage Interactions - Graphics
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Curves of Pursuit
by Marshall Hampton.
xxxxxxxxxxnpi = RDF(pi)def rot(t): from math import cos, sin return matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])def pursuit(n,x0,y0,lamb,steps = 100, threshold = .01): paths = [[[x0,y0]]] for i in range(1,n): rx,ry = list(rot(2*npi*i/n)*vector([x0,y0])) paths.append([[rx,ry]]) oldpath = [x[-1] for x in paths] for q in range(steps): diffs = [[oldpath[(j+1)%n][0]-oldpath[j][0],oldpath[(j+1)%n][1]-oldpath[j][1]] for j in range(n)] npath = [[oldpath[j][0]+lamb*diffs[j][0],oldpath[j][1]+lamb*diffs[j][1]] for j in range(n)] for j in range(n): paths[j].append(npath[j]) oldpath = npath return pathshtml('<h3>Curves of Pursuit</h3>')def curves_of_pursuit(n = slider([2..20],default = 5, label="# of points"),steps = slider([floor(1.4^i) for i in range(2,18)],default = 10, label="# of steps"), stepsize = slider(srange(.01,1,.01),default = .2, label="stepsize"), colorize = selector(['BW','Line color', 'Filled'],default = 'BW')): outpaths = pursuit(n,0,1,stepsize, steps = steps) mcolor = (0,0,0) outer = line([q[0] for q in outpaths]+[outpaths[0][0]], rgbcolor = mcolor) polys = Graphics() if colorize=='Line color': colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))] elif colorize == 'BW': colors = [(0,0,0) for j in range(len(outpaths[0]))] else: colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))] polys = sum([polygon([outpaths[(i+1)%n][j+1],outpaths[(i+1)%n][j], outpaths[i][j+1]], rgbcolor = colors[j]) for i in range(n) for j in range(len(outpaths[0])-1)]) #polys = polys[0] colors = [(0,0,0) for j in range(len(outpaths[0]))] nested = sum([line([q[j] for q in outpaths]+[outpaths[0][j]], rgbcolor = colors[j]) for j in range(len(outpaths[0]))]) lpaths = [line(x, rgbcolor = mcolor) for x in outpaths] show(sum(lpaths)+nested+polys, axes = False, figsize = [5,5], xmin = -1, xmax = 1, ymin = -1, ymax =1)
Catalog of 3D Parametric Plots
xxxxxxxxxxvar('u,v')plots = ['Two Interlinked Tori', 'Star of David', 'Double Heart', 'Heart', 'Green bowtie', "Boy's Surface", "Maeder's Owl", 'Cross cap']plots.sort()def _(example=selector(plots, buttons=True, nrows=2), tachyon=("Raytrace", False), frame = ('Frame', False), opacity=(1,(0.1,1))): url = '' if example == 'Two Interlinked Tori': f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v)) f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u)) p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), color="red", opacity=opacity) p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), color="blue",opacity=opacity) P = p1 + p2 elif example == 'Star of David': f_x = cos(u)*cos(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200) f_y = cos(u)*sin(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200) f_z = sin(u)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200) P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi),opacity=opacity) elif example == 'Double Heart': f_x = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) + abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*sin(v) f_y = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) - abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*cos(v) f_z = sin(u)*(abs(cos(4*u/4))^1 + abs(sin(4*u/4))^1)^(-1/1) P = parametric_plot3d([f_x, f_y, f_z], (u, 0, pi), (v, -pi, pi),opacity=opacity) elif example == 'Heart': f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u)) f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u)) f_z = v P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red",opacity=opacity) elif example == 'Green bowtie': f_x = sin(u) / (sqrt(2) + sin(v)) f_y = sin(u) / (sqrt(2) + cos(v)) f_z = cos(u) / (1 + sqrt(2)) P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -pi, pi), frame=False, color="green",opacity=opacity) elif example == "Boy's Surface": url = "http://en.wikipedia.org/wiki/Boy's_surface" fx = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) fy = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) fz = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, pi), plot_points = [90,90], frame=False, color="orange",opacity=opacity) elif example == "Maeder's Owl": fx = v *cos(u) - 0.5* v^2 * cos(2* u) fy = -v *sin(u) - 0.5* v^2 * sin(2* u) fz = 4 *v^1.5 * cos(3 *u / 2) / 3 P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, 1),plot_points = [90,90], frame=False, color="purple",opacity=opacity) elif example =='Cross cap': url = 'http://en.wikipedia.org/wiki/Cross-cap' fx = (1+cos(v))*cos(u) fy = (1+cos(v))*sin(u) fz = -tanh((2/3)*(u-pi))*sin(v) P = parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red",opacity=opacity) else: print("Bug selecting plot?") return pretty_print(html('<h2>%s</h2>'%example)) if url: pretty_print(html('<h3><a target="_new" href="%s">%s</a></h3>'%(url,url))) show(P, viewer='tachyon' if tachyon else 'threejs', frame=frame)
Interactive rotatable raytracing with Tachyon3d
xxxxxxxxxxC = cube(color=['red', 'green', 'blue'], aspect_ratio=[1,1,1], viewer='tachyon') + sphere((1,0,0),0.2)def example(theta=(0,2*pi), phi=(0,2*pi), zoom=(1,(1,4))): show(C.rotate((0,0,1), theta).rotate((0,1,0),phi), zoom=zoom)
Interactive 3d plotting
xxxxxxxxxxvar('x,y')def example(clr=Color('orange'), f=4*x*exp(-x^2-y^2), xrange='(-2, 2)', yrange='(-2,2)', zrot=(0,pi), xrot=(0,pi), zoom=(1,(1/2,3)), square_aspect=('Square Frame', False), tachyon=('Ray Tracer', True)): xmin, xmax = sage_eval(xrange); ymin, ymax = sage_eval(yrange) P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax), color=clr) html('<h1>Plot of $f(x,y) = %s$</h1>'%latex(f)) aspect_ratio = [1,1,1] if square_aspect else [1,1,1/2] show(P.rotate((0,0,1), -zrot).rotate((1,0,0),xrot), viewer='tachyon' if tachyon else 'threejs', figsize=6, zoom=zoom, frame=False, frame_aspect_ratio=aspect_ratio)
Somewhat Silly Egg Painter
by Marshall Hampton (refereed by William Stein)
xxxxxxxxxxvar('s,t')g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s))def P(color, rng): return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1)colorlist = ['red','blue','red','blue']def _(band_number = selector(range(1,5)), current_color = Color('red'), auto_update=False): html('<h1 align=center>Egg Painter</h1>') colorlist[band_number-1] = current_color egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)]) show(egg)
Plot Coloring
by Timothy Clemans
xxxxxxxxxxdef color_experimenter(expression=input_box('x^2', 'Expression', str), color=Color('red')): if expression: try: plot(SR(expression), rgbcolor=color).show() except TypeError: print("There's a problem with your expression.") else: print("Be sure to enter a plottable expression")
Interactive 2D Plotting
by Timothy Clemans
xxxxxxxxxxdef error_msg(msg): pretty_print(html('<p style="font-family:Arial, sans-serif;color:#000"><span style="color:red;font-weight:bold">Error</span>: %s</p>' % msg))def interactive_2d_plotter(expression=input_box('sin(x)', 'Expression', str), x_range=range_slider(-10,10,1,(0,10), label='X Range'), square=checkbox(True, 'Square'), axes=checkbox(False, 'Show Axes')): if expression: try: expression = SR(expression) # turn string into a Sage expression except TypeError: print(error_msg('This is not an expression.')) return try: xmin, xmax = x_range if square or not axes: print("var('%s')\nplot(%s).show(%s%s%s)" % (expression.variables()[0], repr(expression), 'aspect_ratio=1' if square else '', ', ' if square and not axes else '', 'axes=False' if not axes else '')) if square: plot(expression, xmin, xmax).show(aspect_ratio=1, axes=axes) else: plot(expression, xmin, xmax).show(axes=axes) else: print("var('%s')\nplot(%s)" % (expression.variables()[0], repr(expression))) plot(expression, xmin, xmax).show(axes=axes) except ValueError: print(error_msg('This expression has more than one variable.')) return except TypeError: print(error_msg("This expression contains an unknown function.")) return
Interact with matplotlib
xxxxxxxxxx# Simple example demonstrating how to interact with matplotlib directly.# Comment plt.clf() to get the plots overlay in each update.# Gokhan Sever & Harald Schilly (2010-01-24)from scipy import statsimport numpy as npimport matplotlib.pyplot as pltdef plot_norm(loc=(0,(0,10)), scale=(1,(1,10))): rv = stats.norm(loc, scale) x = np.linspace(-10,10,1000) plt.plot(x,rv.pdf(x)) plt.grid(True) plt.savefig('plt.png') plt.clf()
Spirograph
xxxxxxxxxx#---------------------------## Javier Pérez Lázaro ## Logroño (Spain) ## javier.perezl@unirioja.es ##---------------------------##introductionpretty_print(html('<h1><center>Spirograph</center></h1>'))text1='Spirograph is a tool for drawing hypotrochoids and epitrochoids.'text2='Assume that a A is a point attached to a circle. A can be attached to the boundary of the circle or to any exterior or interior place. If the circle rolls around the outside of a fixed circle, the curve traced by the point A is called an epitrochoid. In case the circle rolls around the inside of a fixed circle, the curve is an hypotrochoid.'text3='If the quotient between the radii of the circles is a rational number, then the curves are periodic.'#the codedef fun(tex1=text_control(text1), tex2=text_control(text2), tex3=text_control(text3),h=('Select:',list(['epitrochoid','hypotrochoid'])),tex4=text_control('Radius of the circle. Should be a rational number with shape p/q.'),b=input_box(default=7/30,label='radius'),tex5=text_control("Rate between the distance of the point to the circle's center and the radius."),rate=input_box(default=1),u=selector(['Plot the curve. Slider of % below enabled.','Build an animation of the plot with the number of frames specified below.'],label='Choose:'),per=slider(0,100,1,default=100,label='graph %'),frames=100,cir_bool=checkbox(True, "Show circles?"),auto_update=false): draw=True if h == 'hypotrochoid' and (b >= 1 or b <= 0): print("In a hypotrochoid, radius must be between 0 and 1.") draw = False if h == 'epitrochoid' and b <= 0: print("In a epitrochoid, radius must be positive") draw=False if draw==True: if h=='hypotrochoid': b=-b var('t') cx=(1+b)*cos(t*b/(1+b)) cy=(1+b)*sin(t*b/(1+b)) px=cx-b*rate*cos(t) py=cy-b*rate*sin(t) axeM=1+max([0,b+abs(b)*rate]) if u=='Plot the curve. Slider of % below enabled.': tMax=pi*denominator(b/(b+1))*per/50 L=parametric_plot((px,py),(t,0,max([0.001,tMax])),plot_points=10*rate*tMax) if cir_bool: p=point((px(t=tMax),py(t=tMax)),pointsize=30,color='blue') c=point((cx(t=tMax),cy(t=tMax)),pointsize=30,color='red') cir=circle((cx(t=tMax),cy(t=tMax)),b,color='red') lin=line([(cx(t=tMax),cy(t=tMax)),(px(t=tMax),py(t=tMax))]) L+=circle((0,0),1)+cir+lin+p+c show(L,aspect_ratio=1,xmin=-axeM,xmax=axeM,ymin=-axeM,ymax=axeM) if u=='Build an animation of the plot with the number of frames specified below.': tMax=2*pi*denominator(b/(b+1)) step=tMax/(frames-1) curva=Graphics() v=[] for a in srange(step,tMax,step): curva+=parametric_plot((px,py),(t,a-step,a)) L=curva if cir_bool: cx_a=cx(t=a) cy_a=cy(t=a) px_a=cx_a-b*rate*cos(a) py_a=cy_a-b*rate*sin(a) p=point((px_a,py_a),pointsize=30,color='blue') c=point((cx_a,cy_a),pointsize=30,color='red') cir=circle((cx_a,cy_a),b,color='red') lin=line([(cx_a,cy_a),(px_a,py_a)]) L+=circle((0,0),1)+cir+lin+c+p v.append(L) animate(v,xmin=-axeM,xmax=axeM,ymin=-axeM,ymax=axeM,aspect_ratio=1).show()