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| These animations were drawn by [[http://www.sagemath.org|Sage]]. <<TableOfContents>> == The witch of Maria Agnesi == by Marshall Hampton {{{#!python numbers=none xtreme = 4.1 myaxes = line([[-xtreme,0],[xtreme,0]],rgbcolor = (0,0,0)) myaxes = myaxes + line([[0,-1],[0,2.1]],rgbcolor = (0,0,0)) a = 1.0 t = var('t') npi = RDF(pi) def agnesi(theta): mac = circle((0,a),a,rgbcolor = (0,0,0)) maL = line([[-xtreme,2*a],[xtreme,2*a]]) maL2 = line([[0,0],[2*a*cot(theta),2*a]]) p1 = [2*a*cot(theta),2*a*sin(theta)^2] p2 = [2*a*cot(theta)-cot(theta)*(2*a-2*a*sin(theta)^2),2*a*sin(theta)^2] maL3 = line([p2,p1,[2*a*cot(theta),2*a]], rgbcolor = (1,0,0)) map1 = point(p1) map2 = point(p2) am = line([[-.05,a],[.05,a]], rgbcolor = (0,0,0)) at = text('a',[-.1,a], rgbcolor = (0,0,0)) yt = text('y',[0,2.2], rgbcolor = (0,0,0)) xt = text('x',[xtreme + .1,-.1], rgbcolor = (0,0,0)) matext = at+yt+xt ma = mac+myaxes+maL+am+matext+maL2+map1+maL3+map2 return ma def witchy(theta): ma = agnesi(theta) agplot = parametric_plot([2*a*cot(t),2*a*sin(t)^2],[t,.001,theta], rgbcolor = (1,0,1)) return ma+agplot a2 = animate([witchy(i) for i in srange(.1,npi-.1,npi/60)]+[witchy(i) for i in srange(npi-.1,.1,-npi/60)], xmin = -3, xmax = 3, ymin = 0, ymax = 2.3, figsize = [6,2.3], axes = False) a2.show() }}} {{attachment:witch.gif}} == A simpler hypotrochoid == The following animates a hypotrochoid much to the same effect as the previous script, but much more concisely. {{{#!python numbers=off import operator # The colors for various elements of the plot: class color: stylus = (1, 0, 0) outer = (.8, .8, .8) inner = (0, 0, 1) plot = (0, 0, 0) center = (0, 0, 0) tip = (1, 0, 0) # and the corresponding line weights: class weight: stylus = 1 outer = 1 inner = 1 plot = 1 center = 5 tip = 5 scale = 1 # The scale of the image animation_delay = .1 # The delay between frames, in seconds # Starting and ending t values t_i = 0 t_f = 2*pi # The t values of the animation frames tvals = srange(t_i, t_f, (t_f-t_i)/60) r_o = 8 # Outer circle radius r_i = 2 # Inner circle radius r_s = 3 # Stylus radius # Coordinates of the center of the inner circle x_c = lambda t: (r_o - r_i)*cos(t) y_c = lambda t: (r_o - r_i)*sin(t) # Parametric coordinates for the plot x = lambda t: x_c(t) + r_s*cos(t*(r_o/r_i)) y = lambda t: y_c(t) - r_s*sin(t*(r_o/r_i)) # Maximum x and y values of the plot x_max = r_o - r_i + r_s y_max = find_maximum_on_interval(y, t_i, t_f)[0] # The plots of the individual elements. Order is important; plots # are stacked from bottom to top as they appear. elements = ( # The outer circle lambda t_f: circle((0, 0), r_o, rgbcolor=color.outer, thickness=weight.outer), # The plot itself lambda t_f: parametric_plot((x, y), t_i, t_f, rgbcolor=color.plot, thickness=weight.plot), # The inner circle lambda t_f: circle((x_c(t_f), y_c(t_f)), r_i, rgbcolor=color.inner, thickness=weight.inner), # The inner circle's center lambda t_f: point((x_c(t_f), y_c(t_f)), rgbcolor=color.center,pointsize=weight.center), # The stylus lambda t_f: line([(x_c(t_f), y_c(t_f)), (x(t_f), y(t_f))], rgbcolor=color.stylus, thickness=weight.stylus), # The stylus' tip lambda t_f: point((x_c(t_f), y_c(t_f)), rgbcolor=color.tip, pointsize=weight.tip), ) # Create the plots and animate them. The animate function renders an # animated gif from the frames provided as its first argument. # Though avid python programmers will find the syntax clear, an # explanation is provided for novices. animation = animate([sum(f(t) for f in elements) for t in tvals], xmin=-x_max, xmax=x_max, ymin=-y_max, ymax=y_max, figsize=(x_max*scale, y_max*scale * y_max/x_max)) animation.show(delay=animation_delay) # The previous could be expressed more pedagogically as follows: # # Evaluate each function in the elements array for the provided t # value: # # plots = lambda t: f(t) for f in elements # # Join a group of plots together to form a single plot: # # def join_plots(plots): # result = plots[0] # for plot in plots[1:]: # result += plot # return result # # or # # join_plots = sum # # Create an array of plots, one for each provided t value: # # frames = [join_plots(plots(t)) for t in tvals] # # Finally, animate the frames: # # animation = animate(frames) }}} == The Tamer and the Lion by Provencal and Labbe == A tamer wants to escape within a circle without being eaten by a lion who lives on the circle. The speed of the lion is 4 times that of the tamer. How can the tamer escape? There is a nice and clever solution in 2d, but does the naive solution where the tamer always moves oppositely to the lion works? In November 2009, Sage and a small script written by Xavier Provençal and Sébastien Labbé in Montpellier answers the question. {{attachment:tamer.gif}} To create the above animation, download [[attachment:tamer.sage]] and type {{{#!python numbers=none sage: load tamer.sage sage: l = range(0,1200,10) sage: a = anime(l) sage: a Animation with 120 frames sage: show(a) }}} |
