Differences between revisions 1 and 76 (spanning 75 versions)
 ⇤ ← Revision 1 as of 2008-03-11 18:14:23 → Size: 505 Editor: was Comment: ← Revision 76 as of 2008-04-10 04:34:29 → ⇥ Size: 36643 Editor: jason Comment: Deletions are marked like this. Additions are marked like this. Line 3: Line 3: Post code (and screen shots) of the use of interact in Sage here. We'll likely restructure and reorganize this, or move it out of the wiki (?) once we have some nontrivial content and get a sense of how it is laid out. == Graphics == 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):{{{@interactdef _(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):{{{@interactdef _(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 Line 8: Line 56: === A contour map and 3d plot of two inverse distance functions ===by William Stein{{{@interactdef _(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('

Tangent line grapher

')@interactdef 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 appropriatebutton 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')@interactdef _(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('
$f = %s$
'%latex(f))    html('
$g = %s$
'%latex(g))    html('
$h = %s = %s$
'%(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 HoltzThis 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/interactState = Data = None # globals to allow incremental additions to graphics@interactdef 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( """$%s$""" % (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( """%s""" % (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=== Coordinate Transformations ===by Jason Grout{{{var('u v')from sage.ext.fast_eval import fast_float@interactdef trans(x=input_box(u^2-v^2, label="x=",type=SR), \    y=input_box(u*v, label="y=",type=SR), \    t_val=slider(0,10,0.2,6, label="Length of curves"), \    u_percent=slider(0,1,0.05,label="u", default=.7),    v_percent=slider(0,1,0.05,label="v", default=.7),    u_range=input_box(range(-5,5,1), label="u lines"),    v_range=input_box(range(-5,5,1), label="v lines")):    thickness=4    u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent    v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent    t_min = -t_val    t_max = t_val    g1=sum([parametric_plot((SR(u.subs(u=i))._fast_float_('v'),v.subs(u=i)._fast_float_('v')), t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])    g2=sum([parametric_plot((u.subs(v=i)._fast_float_('u'),SR(v.subs(v=i))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])    vline_straight=parametric_plot((SR(u.subs(v=v_val))._fast_float_('u'),SR(v.subs(v=v_val))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)    uline_straight=parametric_plot((SR(u.subs(u=u_val))._fast_float_('v'),SR(v.subs(u=u_val))._fast_float_('v')), t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)    (g1+g2+vline_straight+uline_straight).save("uv_coord.png",aspect_ratio=1, figsize=[5,5], axes_labels=['$u$','$v$'])    g3=sum([parametric_plot((x.subs(u=i)._fast_float_('v'),y.subs(u=i)._fast_float_('v')), t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])    g4=sum([parametric_plot((x.subs(v=i)._fast_float_('u'),y.subs(v=i)._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])    vline=parametric_plot((SR(x.subs(v=v_val))._fast_float_('u'),SR(y.subs(v=v_val))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)    uline=parametric_plot((SR(x.subs(u=u_val))._fast_float_('v'),SR(y.subs(u=u_val))._fast_float_('v')), t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)    (g3+g4+vline+uline).save("xy_coord.png", aspect_ratio=1, figsize=[5,5], axes_labels=['$x$','$y$'])    print jsmath("x=%s, \: y=%s"%(latex(x), latex(y)))    print "
"}}}attachment:coordinate-transform-1.pngattachment:coordinate-transform-2.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 = ''    if headers:        for q in headers:            s = s + '
' + str(q) + '
' + str(q) + '
'    for x in y:        s = s + ''        for q in x:            s = s + ''        s = s + ''    s = s + ''    return svar('x y')@interactdef 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== 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('

Numerical instability of the classical Gram-Schmidt algorithm

')@interactdef 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) + '
')    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 GroutA 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.{{{@interactdef 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 linvar('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]]@interactdef 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('

Singular value decomposition: image of the unit circle and the singular vectors

')    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@interactdef 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("

Function plot and its discrete Fourier transform

")    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){{{@interactdef 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. = 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 Line 11: Line 522: === Continued Fraction Plotter ===by William Stein{{{@interactdef _(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{{{@interactdef _(N=(100,(2..2000))):    html("$\pi(x)$ and $x/(\log(x)-1)$ 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){{{@interactdef _(m=selector([1..15],nrows=2), n=(7,(3..10))):    G = DirichletGroup(m)    s = "

First n=%s Bernoulli numbers attached to characters with modulus m=%s

"%(n,m)    s += ''    s += '
$\\chi$Conductor$B_{%s,\chi}$
$%s$
%s%s
' + \           ''.join(''%k for k in [1..n]) + ''    for eps in G.list():        v = ''.join([''%latex(eps.bernoulli(k)) for k in [1..n]])        s += '%s\n'%(             eps, eps.conductor(), v)    s += ''    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+Rfundamental_domain = [[x-1,y] for x,y in fundamental_domain]@interactdef _(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@interactdef _(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 Line 18: Line 624: 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@interactdef 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("

Charpoly and Hecke Graph: Level %s, T_%s

"%(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){{{@interactdef 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 """

%s-Bit Diffie-Hellman Key Exchange

1. Alice and Bob agree to use the prime number p=%s and base g=%s.
2. Alice chooses the secret integer a=%s, then sends Bob (ga mod p):
%s%s mod %s = %s.
3. Bob chooses the secret integer b=%s, then sends Alice (gb mod p):
%s%s mod %s = %s.
4. Alice computes (gb mod p)a mod p:
%s%s mod %s = %s.
5. Bob computes (ga mod p)b mod p:
%s%s mod %s = %s.
""" % (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')@interactdef _(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 ===== CO2 data plot, fetched from NOAA ===by Marshall Hampton{{{import urllib2 as U@interactdef 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('

CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data

')    html('

'+cdate+'

')    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])    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=== Pie Chart from the Google Chart API ===by Harald Schilly{{{# Google Chart API: http://code.google.com/apis/chartimport urllib2 as inetfrom pylab import imshow@interactdef 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('
'%url)}}}attachment:interact_with_google_chart_api.png== Bioinformatics ===== Web app: protein browser ===by Marshall Hampton (tested by William Stein){{{import urllib2 as U@interactdef 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_framed_field = RealField(10)@interactdef 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()@interactdef _(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('

%s

'%example)    if url:        html('

%s

'%(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)@interactdef 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')@interactdef 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('

Plot of $f(x,y) = %s$

'%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[[Anchor(eggpaint)]]=== 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']@interactdef _(band_number = selector(range(1,5)), current_color = Color('red')):    html('

Egg Painter

')    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

# Sage Interactions

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].

## 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

### Coordinate Transformations

by Jason Grout

var('u v')
from sage.ext.fast_eval import fast_float
@interact
def trans(x=input_box(u^2-v^2, label="x=",type=SR), \
y=input_box(u*v, label="y=",type=SR), \
t_val=slider(0,10,0.2,6, label="Length of curves"), \
u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7),
v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7),
u_range=input_box(range(-5,5,1), label="u lines"),
v_range=input_box(range(-5,5,1), label="v lines")):
thickness=4
u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
t_min = -t_val
t_max = t_val

g1=sum([parametric_plot((SR(u.subs(u=i))._fast_float_('v'),v.subs(u=i)._fast_float_('v')), t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])
g2=sum([parametric_plot((u.subs(v=i)._fast_float_('u'),SR(v.subs(v=i))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])
vline_straight=parametric_plot((SR(u.subs(v=v_val))._fast_float_('u'),SR(v.subs(v=v_val))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
uline_straight=parametric_plot((SR(u.subs(u=u_val))._fast_float_('v'),SR(v.subs(u=u_val))._fast_float_('v')), t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)

(g1+g2+vline_straight+uline_straight).save("uv_coord.png",aspect_ratio=1, figsize=[5,5], axes_labels=['$u$','$v$'])

g3=sum([parametric_plot((x.subs(u=i)._fast_float_('v'),y.subs(u=i)._fast_float_('v')),  t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])
g4=sum([parametric_plot((x.subs(v=i)._fast_float_('u'),y.subs(v=i)._fast_float_('u')),  t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])
vline=parametric_plot((SR(x.subs(v=v_val))._fast_float_('u'),SR(y.subs(v=v_val))._fast_float_('u')),  t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
uline=parametric_plot((SR(x.subs(u=u_val))._fast_float_('v'),SR(y.subs(u=u_val))._fast_float_('v')),  t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
(g3+g4+vline+uline).save("xy_coord.png", aspect_ratio=1, figsize=[5,5], axes_labels=['$x$','$y$'])

print jsmath("x=%s, \: y=%s"%(latex(x), latex(y)))
print "<html><table><tr><td><img src='cell://uv_coord.png'/></td><td><img src='cell://xy_coord.png'/></td></tr></table></html>"

attachment:coordinate-transform-1.png attachment:coordinate-transform-2.png

## Differential Equations

### Euler's Method in one variable

by Marshall Hampton. This needs some polishing but its usable as is.

'''
Converts a list into an html table with borders.
'''
s = '<table border = 1>'
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

## 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

### 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

### CO2 data plot, fetched from NOAA

by Marshall Hampton

import urllib2 as U
@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 = '')):
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>')
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])
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

### Pie Chart from the Google Chart API

by Harald Schilly

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 += '&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)

## 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)
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

interact (last edited 2021-08-23 15:58:42 by anewton)