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| Post code that demonstrates the use of the interact command in Sage here. It should be easy for people to just scroll through and paste examples out of here into their own sage notebooks. We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out. If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected]. [[TableOfContents]] == Miscellaneous == === Evaluate a bit of code in a given system === by William Stein (there is no way yet to make the text box big): {{{ @interact def _(system=selector([('sage0', 'Sage'), ('gp', 'PARI'), ('magma', 'Magma')]), code='2+2'): print globals()[system].eval(code) }}} attachment:evalsys.png == Graph Theory == === Automorphism Groups of some Graphs === by William Stein (I spent less than five minutes on this): {{{ @interact def _(graph=['CycleGraph', 'CubeGraph', 'RandomGNP'], n=selector([1..10],nrows=1), p=selector([10,20,..,100],nrows=1)): print graph if graph == 'CycleGraph': print "n (=%s): number of vertices"%n G = graphs.CycleGraph(n) elif graph == 'CubeGraph': if n > 8: print "n reduced to 8" n = 8 print "n (=%s): dimension"%n G = graphs.CubeGraph(n) elif graph == 'RandomGNP': print "n (=%s) vertices"%n print "p (=%s%%) probability"%p G = graphs.RandomGNP(n, p/100.0) print G.automorphism_group() show(plot(G)) }}} attachment:autograph.png == Calculus == === A contour map and 3d plot of two inverse distance functions === by William Stein {{{ @interact def _(q1=(-1,(-3,3)), q2=(-2,(-3,3)), cmap=['autumn', 'bone', 'cool', 'copper', 'gray', 'hot', 'hsv', 'jet', 'pink', 'prism', 'spring', 'summer', 'winter']): x,y = var('x,y') f = q1/sqrt((x+1)^2 + y^2) + q2/sqrt((x-1)^2+(y+0.5)^2) C = contour_plot(f, (-2,2), (-2,2), plot_points=30, contours=15, cmap=cmap) show(C, figsize=3, aspect_ratio=1) show(plot3d(f, (x,-2,2), (y,-2,2)), figsize=5, viewer='tachyon') }}} attachment:mountains.png === A simple tangent line grapher === by Marshall Hampton {{{ html('<h2>Tangent line grapher</h2>') @interact def tangent_line(f = input_box(default=sin(x)), xbegin = slider(0,10,1/10,0), xend = slider(0,10,1/10,10), x0 = slider(0, 1, 1/100, 1/2)): prange = [xbegin, xend] x0i = xbegin + x0*(xend-xbegin) var('x') df = diff(f) tanf = f(x0i) + df(x0i)*(x-x0i) fplot = plot(f, prange[0], prange[1]) print 'Tangent line is y = ' + tanf._repr_() tanplot = plot(tanf, prange[0], prange[1], rgbcolor = (1,0,0)) fmax = f.find_maximum_on_interval(prange[0], prange[1])[0] fmin = f.find_minimum_on_interval(prange[0], prange[1])[0] show(fplot + tanplot, xmin = prange[0], xmax = prange[1], ymax = fmax, ymin = fmin) }}} attachment:tangents.png === Function tool === Enter symbolic functions $f$, $g$, and $a$, a range, then click the appropriate button to compute and plot some combination of $f$, $g$, and $a$ along with $f$ and $g$. This is inspired by the Matlab funtool GUI. {{{ x = var('x') @interact def _(f=sin(x), g=cos(x), xrange=input_box((0,1)), yrange='auto', a=1, action=selector(['f', 'df/dx', 'int f', 'num f', 'den f', '1/f', 'finv', 'f+a', 'f-a', 'f*a', 'f/a', 'f^a', 'f(x+a)', 'f(x*a)', 'f+g', 'f-g', 'f*g', 'f/g', 'f(g)'], width=15, nrows=5, label="h = "), do_plot = ("Draw Plots", True)): try: f = SR(f); g = SR(g); a = SR(a) except TypeError, msg: print msg[-200:] print "Unable to make sense of f,g, or a as symbolic expressions." return if not (isinstance(xrange, tuple) and len(xrange) == 2): xrange = (0,1) h = 0; lbl = '' if action == 'f': h = f lbl = 'f' elif action == 'df/dx': h = f.derivative(x) lbl = '\\frac{df}{dx}' elif action == 'int f': h = f.integrate(x) lbl = '\\int f dx' elif action == 'num f': h = f.numerator() lbl = '\\text{numer(f)}' elif action == 'den f': h = f.denominator() lbl = '\\text{denom(f)}' elif action == '1/f': h = 1/f lbl = '\\frac{1}{f}' elif action == 'finv': h = solve(f == var('y'), x)[0].rhs() lbl = 'f^{-1}(y)' elif action == 'f+a': h = f+a lbl = 'f + a' elif action == 'f-a': h = f-a lbl = 'f - a' elif action == 'f*a': h = f*a lbl = 'f \\times a' elif action == 'f/a': h = f/a lbl = '\\frac{f}{a}' elif action == 'f^a': h = f^a lbl = 'f^a' elif action == 'f^a': h = f^a lbl = 'f^a' elif action == 'f(x+a)': h = f(x+a) lbl = 'f(x+a)' elif action == 'f(x*a)': h = f(x*a) lbl = 'f(xa)' elif action == 'f+g': h = f+g lbl = 'f + g' elif action == 'f-g': h = f-g lbl = 'f - g' elif action == 'f*g': h = f*g lbl = 'f \\times g' elif action == 'f/g': h = f/g lbl = '\\frac{f}{g}' elif action == 'f(g)': h = f(g) lbl = 'f(g)' html('<center><font color="red">$f = %s$</font></center>'%latex(f)) html('<center><font color="green">$g = %s$</font></center>'%latex(g)) html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h))) if do_plot: P = plot(f, xrange, color='red', thickness=2) + \ plot(g, xrange, color='green', thickness=2) + \ plot(h, xrange, color='blue', thickness=2) if yrange == 'auto': show(P, xmin=xrange[0], xmax=xrange[1]) else: yrange = sage_eval(yrange) show(P, xmin=xrange[0], xmax=xrange[1], ymin=yrange[0], ymax=yrange[1]) }}} attachment:funtool.png === Newton-Raphson Root Finding === by Neal Holtz This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued. {{{ # ideas from 'A simple tangent line grapher' by Marshall Hampton # http://wiki.sagemath.org/interact State = Data = None # globals to allow incremental additions to graphics @interact def newtraph(f = input_box(default=8*sin(x)*exp(-x)-1, label='f(x)'), xmin = input_box(default=0), xmax = input_box(default=4*pi), x0 = input_box(default=3, label='x0'), show_calcs = ("Show Calcs",True), step = ['Next','Reset'] ): global State, Data prange = [xmin,xmax] state = [f,xmin,xmax,x0,show_calcs] if (state != State) or (step == 'Reset'): # when any of the controls change X = [RR(x0)] # restart the plot df = diff(f) Fplot = plot(f, prange[0], prange[1]) Data = [X, df, Fplot] State = state X, df, Fplot = Data i = len(X) - 1 # compute and append the next x value xi = X[i] fi = RR(f(xi)) fpi = RR(df(xi)) xip1 = xi - fi/fpi X.append(xip1) msg = xip1s = None # now check x value for reasonableness is_inf = False if abs(xip1) > 10E6*(xmax-xmin): is_inf = True show_calcs = True msg = 'Derivative is 0!' xip1s = latex(xip1.sign()*infinity) X.pop() elif not ((xmin - 0.5*(xmax-xmin)) <= xip1 <= (xmax + 0.5*(xmax-xmin))): show_calcs = True msg = 'x value out of range; probable divergence!' if xip1s is None: xip1s = '%.4g' % (xip1,) def Disp( s, color="blue" ): if show_calcs: html( """<font color="%s">$ %s $</font>""" % (color,s,) ) Disp( """f(x) = %s""" % (latex(f),) + """~~~~f'(x) = %s""" % (latex(df),) ) Disp( """i = %d""" % (i,) + """~~~~x_{%d} = %.4g""" % (i,xi) + """~~~~f(x_{%d}) = %.4g""" % (i,fi) + """~~~~f'(x_{%d}) = %.4g""" % (i,fpi) ) if msg: html( """<font color="red"><b>%s</b></font>""" % (msg,) ) c = "red" else: c = "blue" Disp( r"""x_{%d} = %.4g - ({%.4g})/({%.4g}) = %s""" % (i+1,xi,fi,fpi,xip1s), color=c ) Fplot += line( [(xi,0),(xi,fi)], linestyle=':', rgbcolor=(1,0,0) ) # vert dotted line Fplot += points( [(xi,0),(xi,fi)], rgbcolor=(1,0,0) ) labi = text( '\nx%d\n' % (i,), (xi,0), rgbcolor=(1,0,0), vertical_alignment="bottom" if fi < 0 else "top" ) if is_inf: xl = xi - 0.05*(xmax-xmin) xr = xi + 0.05*(xmax-xmin) yl = yr = fi else: xl = min(xi,xip1) - 0.02*(xmax-xmin) xr = max(xi,xip1) + 0.02*(xmax-xmin) yl = -(xip1-xl)*fpi yr = (xr-xip1)*fpi Fplot += points( [(xip1,0)], rgbcolor=(0,0,1) ) # new x value labi += text( '\nx%d\n' % (i+1,), (xip1,0), rgbcolor=(1,0,0), vertical_alignment="bottom" if fi < 0 else "top" ) Fplot += line( [(xl,yl),(xr,yr)], rgbcolor=(1,0,0) ) # tangent show( Fplot+labi, xmin = prange[0], xmax = prange[1] ) Data = [X, df, Fplot] }}} attachment:newtraph.png == Differential Equations == === Euler's Method in one variable === by Marshall Hampton. This needs some polishing but its usable as is. {{{ def tab_list(y, headers = None): ''' Converts a list into an html table with borders. ''' s = '<table border = 1>' if headers: for q in headers: s = s + '<th>' + str(q) + '</th>' for x in y: s = s + '<tr>' for q in x: s = s + '<td>' + str(q) + '</td>' s = s + '</tr>' s = s + '</table>' return s var('x y') @interact def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Exact solution = '), y_prime_in = input_box('sin(x)', type = str, label = "y' = "), start = input_box(0.0, label = 'x starting value: '), stop = input_box(6.0, label = 'x stopping value: '), startval = input_box(0.0, label = 'y starting value: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Number of steps: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Number of steps shown in table: ')): y_exact = lambda x: eval(y_exact_in) y_prime = lambda x,y: eval(y_prime_in) stepsize = float((stop-start)/nsteps) steps_shown = max(nsteps,show_steps) sol = [startval] xvals = [start] for step in range(nsteps): sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1])) xvals.append(xvals[-1] + stepsize) sol_max = max(sol + [find_maximum_on_interval(y_exact,start,stop)[0]]) sol_min = min(sol + [find_minimum_on_interval(y_exact,start,stop)[0]]) show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min) if nsteps < steps_shown: table_range = range(len(sol)) else: table_range = range(0,floor(steps_shown/2)) + range(len(sol)-floor(steps_shown/2),len(sol)) html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y'])) }}} attachment:eulermethod.png |
Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected]. * [:interact/graph_theory:] - Graph Theory * [:interact/calculus:] - Calculus * [:interact/diffeq:] - Differential Equations |
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| === Factor Trees === by William Stein {{{ import random def ftree(rows, v, i, F): if len(v) > 0: # add a row to g at the ith level. rows.append(v) w = [] for i in range(len(v)): k, _, _ = v[i] if k is None or is_prime(k): w.append((None,None,None)) else: d = random.choice(divisors(k)[1:-1]) w.append((d,k,i)) e = k//d if e == 1: w.append((None,None)) else: w.append((e,k,i)) if len(w) > len(v): ftree(rows, w, i+1, F) def draw_ftree(rows,font): g = Graphics() for i in range(len(rows)): cur = rows[i] for j in range(len(cur)): e, f, k = cur[j] if not e is None: if is_prime(e): c = (1,0,0) else: c = (0,0,.4) g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) if not k is None and not f is None: g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], alpha=0.5) return g @interact def factor_tree(n=100, font=(10, (8..20)), redraw=['Redraw']): n = Integer(n) rows = [] v = [(n,None,0)] ftree(rows, v, 0, factor(n)) show(draw_ftree(rows, font), axes=False) }}} attachment:factortree.png |
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| === Stock Market data, fetched from Yahoo and Google === by William Stein {{{ import urllib class Day: def __init__(self, date, open, high, low, close, volume): self.date = date self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close) self.volume=int(volume) def __repr__(self): return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high, self.low, self.close, self.volume) class Stock: def __init__(self, symbol): self.symbol = symbol.upper() def __repr__(self): return "%s (%s)"%(self.symbol, self.yahoo()['price']) def yahoo(self): url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7') values = urllib.urlopen(url).read().strip().strip('"').split(',') data = {} data['price'] = values[0] data['change'] = values[1] data['volume'] = values[2] data['avg_daily_volume'] = values[3] data['stock_exchange'] = values[4] data['market_cap'] = values[5] data['book_value'] = values[6] data['ebitda'] = values[7] data['dividend_per_share'] = values[8] data['dividend_yield'] = values[9] data['earnings_per_share'] = values[10] data['52_week_high'] = values[11] data['52_week_low'] = values[12] data['50day_moving_avg'] = values[13] data['200day_moving_avg'] = values[14] data['price_earnings_ratio'] = values[15] data['price_earnings_growth_ratio'] = values[16] data['price_sales_ratio'] = values[17] data['price_book_ratio'] = values[18] data['short_ratio'] = values[19] return data def historical(self): try: return self.__historical except AttributeError: pass symbol = self.symbol def get_data(exchange): name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()), verbose=False) return open(name).read() R = get_data('NASDAQ') if "Bad Request" in R: R = get_data("NYSE") R = R.splitlines() headings = R[0].split(',') self.__historical = [] try: for x in reversed(R[1:]): date, opn, high, low, close, volume = x.split(',') self.__historical.append(Day(date, opn,high,low,close,volume)) except ValueError: pass self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x) return self.__historical def plot_average(self, spline_samples=10): d = self.historical() if len(d) == 0: return text('no historical data at Google Finance about %s'%self.symbol, (0,3)) avg = list(enumerate([(z.high+z.low)/2 for z in d])) P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \ text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black') if spline_samples > 0: k = 250//spline_samples spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]]) P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7)) P.xmax(260) return P def plot_diff(self): d = self.historical() if len(d) == 0: return text('no historical data at Google Finance about %s'%self.symbol, (0,3)) diff = [] for i in range(1, len(d)): z1 = d[i]; z0 = d[i-1] diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2)) P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \ text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black') P.xmax(260) return P symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort() stocks = dict([(s,Stock(s)) for s in symbols]) @interact def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])): if other_symbol != '': symbol = other_symbol S = Stock(symbol) html('<h1 align=center><font color="darkred">%s</font></h1>'%S) S.plot_average(spline_samples).save('avg.png', figsize=[10,2]) S.plot_diff().save('diff.png', figsize=[10,2]) Y = S.yahoo() k = Y.keys(); k.sort() html('Price during last 52 weeks:<br>Grey line is a spline through %s points (do not take seriously!):<br> <img src="cell://avg.png">'%spline_samples) html('Difference from previous day:<br> <img src="cell://diff.png">') html('<table align=center>' + '\n'.join('<tr><td>%s</td><td>%s</td></tr>'%(k[i], Y[k[i]]) for i in range(len(k))) + '</table>') }}} attachment:stocks.png |
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While support for R is rapidly improving, scipy.stats has a lot of useful stuff too. This only scratches the surface. |
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| @interact def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009), Update = selector(['Update'], buttons=True, label = '')): co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines() datalines = [] for a_line in co2data: if a_line.find('Creation:') != -1: cdate = a_line if a_line[0] != '#': temp = a_line.replace('\n','').split(' ') temp = [float(q) for q in temp if q != ''] datalines.append(temp) html('<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>') html('<h4>'+cdate+'</h4>') |
import scipy.stats as Stat co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines() datalines = [] for a_line in co2data: if a_line.find('Creation:') != -1: cdate = a_line if a_line[0] != '#': temp = a_line.replace('\n','').split(' ') temp = [float(q) for q in temp if q != ''] datalines.append(temp) trdf = RealField(16) @interact def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)): htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>' htmls2 = '<h4>'+cdate+'</h4>' |
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| show(list_plot([[q[2],q[4]] for q in datalines], plotjoined=True, rgbcolor=(1,0,0)), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False) }}} attachment:mauna_loa_co2.png |
slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data) html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>') var('x,y') show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False) }}} attachment:co2c.png |
Sage Interactions
Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].
- [:interact/graph_theory:] - Graph Theory
- [:interact/calculus:] - Calculus
- [:interact/diffeq:] - Differential Equations
Linear Algebra
Numerical instability of the classical Gram-Schmidt algorithm
by Marshall Hampton (tested by William Stein, who thinks this is really nice!)
def GS_classic(a_list):
'''
Given a list of vectors or a matrix, returns the QR factorization using the classical (and numerically unstable) Gram-Schmidt algorithm.
'''
if type(a_list) != list:
cols = a_list.cols()
a_list = [x for x in cols]
indices = range(len(a_list))
q = []
r = [[0 for i in indices] for j in indices]
v = [a_list[i].copy() for i in indices]
for i in indices:
for j in range(0,i):
r[j][i] = q[j].inner_product(a_list[i])
v[i] = v[i] - r[j][i]*q[j]
r[i][i] = (v[i]*v[i])^(1/2)
q.append(v[i]/r[i][i])
q = matrix([q[i] for i in indices]).transpose()
return q, matrix(r)
def GS_modern(a_list):
'''
Given a list of vectors or a matrix, returns the QR factorization using the 'modern' Gram-Schmidt algorithm.
'''
if type(a_list) != list:
cols = a_list.cols()
a_list = [x for x in cols]
indices = range(len(a_list))
q = []
r = [[0 for i in indices] for j in indices]
v = [a_list[i].copy() for i in indices]
for i in indices:
r[i][i] = v[i].norm(2)
q.append(v[i]/r[i][i])
for j in range(i+1, len(indices)):
r[i][j] = q[i].inner_product(v[j])
v[j] = v[j] - r[i][j]*q[i]
q = matrix([q[i] for i in indices]).transpose()
return q, matrix(r)
html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>')
@interact
def gstest(precision = slider(range(3,53), default = 10), a1 = input_box([1,1/1000,1/1000]), a2 = input_box([1,1/1000,0]), a3 = input_box([1,0,1/1000])):
myR = RealField(precision)
displayR = RealField(5)
html('precision in bits: ' + str(precision) + '<br>')
A = matrix([a1,a2,a3])
A = [vector(myR,x) for x in A]
qn, rn = GS_classic(A)
qb, rb = GS_modern(A)
html('Classical Gram-Schmidt:')
show(matrix(displayR,qn))
html('Stable Gram-Schmidt:')
show(matrix(displayR,qb))attachment:GramSchmidt.png
Linear transformations
by Jason Grout
A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.
@interact
def linear_transformation(theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)):
A=matrix([[1,-1],[-1,1/2]])
v=vector([r*cos(theta), r*sin(theta)])
w = A*v
circles = sum([circle((0,0), radius=i, rgbcolor=(0,0,0)) for i in [1..2]])
print jsmath("v = %s,\; %s v=%s"%(v.n(4),latex(A),w.n(4)))
show(v.plot(rgbcolor=(1,0,0))+w.plot(rgbcolor=(0,0,1))+circles,aspect_ratio=1)attachment:Linear-Transformations.png
Singular value decomposition
by Marshall Hampton
import scipy.linalg as lin
var('t')
def rotell(sig,umat,t,offset=0):
temp = matrix(umat)*matrix(2,1,[sig[0]*cos(t),sig[1]*sin(t)])
return [offset+temp[0][0],temp[1][0]]
@interact
def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= selector(['Off','On'],label='offset image from domain')):
rf_low = RealField(12)
my_mat = matrix(rf_low,2,2,[a11,a12,a21,a22])
u,s,vh = lin.svd(my_mat.numpy())
if ofs == 'On':
offset = 3
fsize = 6
colors = [(1,0,0),(0,0,1),(1,0,0),(0,0,1)]
else:
offset = 0
fsize = 5
colors = [(1,0,0),(0,0,1),(.7,.2,0),(0,.3,.7)]
vvects = sum([arrow([0,0],matrix(vh).row(i),rgbcolor = colors[i]) for i in (0,1)])
uvects = Graphics()
for i in (0,1):
if s[i] != 0: uvects += arrow([offset,0],vector([offset,0])+matrix(s*u).column(i),rgbcolor = colors[i+2])
html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>')
print jsmath("A = %s = %s %s %s"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist()))))
image_ell = parametric_plot(rotell(s,u,t, offset),0,2*pi)
graph_stuff=circle((0,0),1)+image_ell+vvects+uvects
graph_stuff.set_aspect_ratio(1)
show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])attachment:svd1.png
Discrete Fourier Transform
by Marshall Hampton
import scipy.fftpack as Fourier
@interact
def discrete_fourier(f = input_box(default=sum([sin(k*x) for k in range(1,5,2)])), scale = slider(.1,20,.1,5)):
var('x')
pbegin = -float(pi)*scale
pend = float(pi)*scale
html("<h3>Function plot and its discrete Fourier transform</h3>")
show(plot(f, pbegin, pend, plot_points = 512), figsize = [4,3])
f_vals = [f(ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)]
my_fft = Fourier.fft(f_vals)
show(list_plot([abs(x) for x in my_fft], plotjoined=True), figsize = [4,3])attachment:dfft1.png
Algebra
Groebner fan of an ideal
by Marshall Hampton; (needs sage-2.11 or higher, with gfan-0.3 interface)
@interact
def gfan_browse(p1 = input_box('x^3+y^2',type = str, label='polynomial 1: '), p2 = input_box('y^3+z^2',type = str, label='polynomial 2: '), p3 = input_box('z^3+x^2',type = str, label='polynomial 3: ')):
R.<x,y,z> = PolynomialRing(QQ,3)
i1 = ideal(R(p1),R(p2),R(p3))
gf1 = i1.groebner_fan()
testr = gf1.render()
html('Groebner fan of the ideal generated by: ' + str(p1) + ', ' + str(p2) + ', ' + str(p3))
show(testr, axes = False, figsize=[8,8*(3^(.5))/2])attachment:gfan_interact.png
Number Theory
Factor Trees
by William Stein
import random
def ftree(rows, v, i, F):
if len(v) > 0: # add a row to g at the ith level.
rows.append(v)
w = []
for i in range(len(v)):
k, _, _ = v[i]
if k is None or is_prime(k):
w.append((None,None,None))
else:
d = random.choice(divisors(k)[1:-1])
w.append((d,k,i))
e = k//d
if e == 1:
w.append((None,None))
else:
w.append((e,k,i))
if len(w) > len(v):
ftree(rows, w, i+1, F)
def draw_ftree(rows,font):
g = Graphics()
for i in range(len(rows)):
cur = rows[i]
for j in range(len(cur)):
e, f, k = cur[j]
if not e is None:
if is_prime(e):
c = (1,0,0)
else:
c = (0,0,.4)
g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c)
if not k is None and not f is None:
g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)],
alpha=0.5)
return g
@interact
def factor_tree(n=100, font=(10, (8..20)), redraw=['Redraw']):
n = Integer(n)
rows = []
v = [(n,None,0)]
ftree(rows, v, 0, factor(n))
show(draw_ftree(rows, font), axes=False)attachment:factortree.png
Continued Fraction Plotter
by William Stein
@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
c = list(continued_fraction(RealField(prec)(number))); print c
show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])attachment:contfracplot.png
Illustrating the prime number thoerem
by William Stein
@interact
def _(N=(100,(2..2000))):
html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)
show(plot(prime_pi, 0, N, rgbcolor='red') + plot(x/(log(x)-1), 5, N, rgbcolor='blue'))attachment:primes.png
Computing Generalized Bernoulli Numbers
by William Stein (Sage-2.10.3)
@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
G = DirichletGroup(m)
s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
s += '<table border=1>'
s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
for eps in G.list():
v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
eps, eps.conductor(), v)
s += '</table>'
html(s)attachment:bernoulli.png
Fundamental Domains of SL_2(ZZ)
by Robert Miller
L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
global fundamental_domain
if gen == 't+1':
fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
elif gen == 't-1':
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
elif gen == '-1/t':
new_dom = []
for x,y in fundamental_domain:
sq_mod = x^2 + y^2
new_dom.append([(-1)*x/sq_mod, y/sq_mod])
fundamental_domain = new_dom
P = polygon(fundamental_domain)
P.ymax(1.2); P.ymin(-0.1)
P.show()attachment:fund_domain.png
Computing modular forms
by William Stein
j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),
group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
M = CuspForms(group(N),k)
print j; global j; j += 1
print M; print '\n'*3
print "Computing basis...\n\n"
if M.dimension() == 0:
print "Space has dimension 0"
else:
prec = max(prec, M.dimension()+1)
for f in M.basis():
view(f.q_expansion(prec))
print "\n\n\nDone computing basis."attachment:modformbasis.png
Computing the cuspidal subgroup
by William Stein
html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
A = J0(N)
print A.cuspidal_subgroup()attachment:cuspgroup.png
A Charpoly and Hecke Operator Graph
by William Stein
# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
p = selector(prime_range(2,12),nrows=1),
three_d = ("Three Dimensional", False)):
S = SupersingularModule(N)
T = S.hecke_matrix(p)
G = Graph(T, multiedges=True, loops=not three_d)
html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
show(T.charpoly().factor())
if three_d:
show(G.plot3d(), aspect_ratio=[1,1,1])
else:
show(G.plot(),figsize=7)attachment:heckegraph.png
Demonstrating the Diffie-Hellman Key Exchange Protocol
by Timothy Clemans (refereed by William Stein)
@interact
def diffie_hellman(button=selector(["New example"],label='',buttons=True),
bits=("Number of bits of prime", (8,12,..512))):
maxp = 2^bits
p = random_prime(maxp)
k = GF(p)
if bits>100:
g = k(2)
else:
g = k.multiplicative_generator()
a = ZZ.random_element(10, maxp)
b = ZZ.random_element(10, maxp)
print """
<html>
<style>
.gamodp {
background:yellow
}
.gbmodp {
background:orange
}
.dhsame {
color:green;
font-weight:bold
}
</style>
<h2>%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font:12px Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li>
<li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
""" % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,
(g^ b)^a, g^a, b, p, (g^a)^b)attachment:dh.png
Plotting an elliptic curve over a finite field
E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
show(E)
print "p = %s"%p
show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)attachment:ellffplot.png
Web applications
Stock Market data, fetched from Yahoo and Google
by William Stein
import urllib
class Day:
def __init__(self, date, open, high, low, close, volume):
self.date = date
self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close)
self.volume=int(volume)
def __repr__(self):
return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high,
self.low, self.close, self.volume)
class Stock:
def __init__(self, symbol):
self.symbol = symbol.upper()
def __repr__(self):
return "%s (%s)"%(self.symbol, self.yahoo()['price'])
def yahoo(self):
url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7')
values = urllib.urlopen(url).read().strip().strip('"').split(',')
data = {}
data['price'] = values[0]
data['change'] = values[1]
data['volume'] = values[2]
data['avg_daily_volume'] = values[3]
data['stock_exchange'] = values[4]
data['market_cap'] = values[5]
data['book_value'] = values[6]
data['ebitda'] = values[7]
data['dividend_per_share'] = values[8]
data['dividend_yield'] = values[9]
data['earnings_per_share'] = values[10]
data['52_week_high'] = values[11]
data['52_week_low'] = values[12]
data['50day_moving_avg'] = values[13]
data['200day_moving_avg'] = values[14]
data['price_earnings_ratio'] = values[15]
data['price_earnings_growth_ratio'] = values[16]
data['price_sales_ratio'] = values[17]
data['price_book_ratio'] = values[18]
data['short_ratio'] = values[19]
return data
def historical(self):
try:
return self.__historical
except AttributeError:
pass
symbol = self.symbol
def get_data(exchange):
name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()),
verbose=False)
return open(name).read()
R = get_data('NASDAQ')
if "Bad Request" in R:
R = get_data("NYSE")
R = R.splitlines()
headings = R[0].split(',')
self.__historical = []
try:
for x in reversed(R[1:]):
date, opn, high, low, close, volume = x.split(',')
self.__historical.append(Day(date, opn,high,low,close,volume))
except ValueError:
pass
self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x)
return self.__historical
def plot_average(self, spline_samples=10):
d = self.historical()
if len(d) == 0:
return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
avg = list(enumerate([(z.high+z.low)/2 for z in d]))
P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \
text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black')
if spline_samples > 0:
k = 250//spline_samples
spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]])
P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7))
P.xmax(260)
return P
def plot_diff(self):
d = self.historical()
if len(d) == 0:
return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
diff = []
for i in range(1, len(d)):
z1 = d[i]; z0 = d[i-1]
diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2))
P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \
text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black')
P.xmax(260)
return P
symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort()
stocks = dict([(s,Stock(s)) for s in symbols])
@interact
def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])):
if other_symbol != '':
symbol = other_symbol
S = Stock(symbol)
html('<h1 align=center><font color="darkred">%s</font></h1>'%S)
S.plot_average(spline_samples).save('avg.png', figsize=[10,2])
S.plot_diff().save('diff.png', figsize=[10,2])
Y = S.yahoo()
k = Y.keys(); k.sort()
html('Price during last 52 weeks:<br>Grey line is a spline through %s points (do not take seriously!):<br> <img src="cell://avg.png">'%spline_samples)
html('Difference from previous day:<br> <img src="cell://diff.png">')
html('<table align=center>' + '\n'.join('<tr><td>%s</td><td>%s</td></tr>'%(k[i], Y[k[i]]) for i in range(len(k))) + '</table>')attachment:stocks.png
CO2 data plot, fetched from NOAA
by Marshall Hampton
While support for R is rapidly improving, scipy.stats has a lot of useful stuff too. This only scratches the surface.
import urllib2 as U
import scipy.stats as Stat
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
if a_line.find('Creation:') != -1:
cdate = a_line
if a_line[0] != '#':
temp = a_line.replace('\n','').split(' ')
temp = [float(q) for q in temp if q != '']
datalines.append(temp)
trdf = RealField(16)
@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
htmls2 = '<h4>'+cdate+'</h4>'
sel_data = [[q[2],q[4]] for q in datalines if start_date < q[2] < end_date]
c_max = max([q[1] for q in sel_data])
c_min = min([q[1] for q in sel_data])
slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>')
var('x,y')
show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)attachment:co2c.png
Pie Chart from the Google Chart API
by Harald Schilly
# Google Chart API: http://code.google.com/apis/chart
import urllib2 as inet
from pylab import imshow
@interact
def gChart(title="Google Chart API plots Pie Charts!", color1=Color('purple'), color2=Color('black'), color3=Color('yellow'), val1=slider(0,1,.05,.5), val2=slider(0,1,.05,.3), val3=slider(0,1,.05,0.1), label=("Maths Physics Chemistry")):
url = "http://chart.apis.google.com/chart?cht=p3&chs=600x300"
url += '&chtt=%s&chts=000000,25'%title.replace(" ","+")
url += '&chco=%s'%(','.join([color1.html_color()[1:],color2.html_color()[1:],color3.html_color()[1:]]))
url += '&chl=%s'%label.replace(" ","|")
url += '&chd=t:%s'%(','.join(map(str,[val1,val2,val3])))
print url
html('<div style="border:3px dashed;text-align:center;padding:50px 0 50px 0"><img src="%s"></div>'%url)attachment:interact_with_google_chart_api.png
Bioinformatics
Web app: protein browser
by Marshall Hampton (tested by William Stein)
import urllib2 as U
@interact
def protein_browser(GenBank_ID = input_box('165940577', type = str), file_type = selector([(1,'fasta'),(2,'GenPept')])):
if file_type == 2:
gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&id='
else:
gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&dopt=fasta&id='
f = U.urlopen(gen_str + GenBank_ID)
g = f.read()
f.close()
html(g)attachment:biobrowse.png
Coalescent simulator
by Marshall Hampton
def next_gen(x, selection=1.0):
'''Creates the next generation from the previous; also returns parent-child indexing list'''
next_x = []
for ind in range(len(x)):
if random() < (1 + selection)/len(x):
rind = 0
else:
rind = int(round(random()*(len(x)-1)+1/2))
next_x.append((x[rind],rind))
next_x.sort()
return [[x[0] for x in next_x],[x[1] for x in next_x]]
def coal_plot(some_data):
'''Creates a graphics object from coalescent data'''
gens = some_data[0]
inds = some_data[1]
gen_lines = line([[0,0]])
pts = Graphics()
ngens = len(gens)
gen_size = len(gens[0])
for x in range(gen_size):
pts += point((x,ngens-1), hue = gens[0][x]/float(gen_size*1.1))
p_frame = line([[-.5,-.5],[-.5,ngens-.5], [gen_size-.5,ngens-.5], [gen_size-.5,-.5], [-.5,-.5]])
for g in range(1,ngens):
for x in range(gen_size):
old_x = inds[g-1][x]
gen_lines += line([[x,ngens-g-1],[old_x,ngens-g]], hue = gens[g-1][old_x]/float(gen_size*1.1))
pts += point((x,ngens-g-1), hue = gens[g][x]/float(gen_size*1.1))
return pts+gen_lines+p_frame
d_field = RealField(10)
@interact
def coalescents(pop_size = slider(2,100,1,15,'Population size'), selection = slider(-1,1,.1,0, 'Selection for first taxon'), s = selector(['Again!'], label='Refresh', buttons=True)):
print 'Population size: ' + str(pop_size)
print 'Selection coefficient for first taxon: ' + str(d_field(selection))
start = [i for i in range(pop_size)]
gens = [start]
inds = []
while gens[-1][0] != gens[-1][-1]:
g_index = len(gens) - 1
n_gen = next_gen(gens[g_index], selection = selection)
gens.append(n_gen[0])
inds.append(n_gen[1])
coal_data1 = [gens,inds]
print 'Generations until coalescence: ' + str(len(gens))
show(coal_plot(coal_data1), axes = False, figsize = [8,4.0*len(gens)/pop_size], ymax = len(gens)-1)attachment:coalescent.png
Miscellaneous Graphics
Catalog of 3D Parametric Plots
var('u,v')
plots = ['Two Interlinked Tori', 'Star of David', 'Double Heart',
'Heart', 'Green bowtie', "Boy's Surface", "Maeder's Owl",
'Cross cap']
plots.sort()
@interact
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
html('<h2>%s</h2>'%example)
if url:
html('<h3><a target="_new" href="%s">%s</a></h3>'%(url,url))
show(P, viewer='tachyon' if tachyon else 'jmol', frame=frame)attachment:parametricplot3d.png
Interactive rotatable raytracing with Tachyon3d
C = cube(color=['red', 'green', 'blue'], aspect_ratio=[1,1,1],
viewer='tachyon') + sphere((1,0,0),0.2)
@interact
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)attachment:tachyonrotate.png
Interactive 3d plotting
var('x,y')
@interact
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 'jmol',
figsize=6, zoom=zoom, frame=False,
frame_aspect_ratio=aspect_ratio)attachment:tachyonplot3d.png
Somewhat Silly Egg Painter
by Marshall Hampton (refereed by William Stein)
var('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']
@interact
def _(band_number = selector(range(1,5)), current_color = Color('red')):
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)attachment:eggpaint.png