Differences between revisions 89 and 91 (spanning 2 versions)
 ⇤ ← Revision 89 as of 2008-04-18 13:21:57 → Size: 46892 Editor: was Comment: ← Revision 91 as of 2008-04-24 21:58:46 → ⇥ Size: 48827 Editor: schilly Comment: taylor series animated Deletions are marked like this. Additions are marked like this. Line 386: Line 386: === Taylor Series ===by Harald Schilly{{{var('x')x0 = 0f = sin(x)*e^(-x)p = plot(f,-1,5, thickness=2)dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))@interactdef _(order=(1..12)):    ft = f.taylor(x,x0,order)    pt = plot(ft,-1, 5, color='green', thickness=2)    html('$f(x)\;=\;%s$'%latex(f))    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))    show(dot + p + pt, ymin = -.5, ymax = 1)}}}attachment:taylor_series_animated.gif Line 661: Line 679: === Factor Trees ===by William Stein{{{import randomdef 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@interactdef 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

# 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

### Profile a snippet of code

html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'),
do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
if do_preparse: cmd = preparse(cmd)
print "<html>"  # trick to avoid word wrap
if cprof:
cProfile.run(cmd)
else:
profile.run(cmd)
print "</html>"

attachment:profile.png

### 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

### A Random Walk

by William Stein

html('<h1>A Random Walk</h1>')
vv = []; nn = 0
@interact
def foo(pts = checkbox(True, "Show points"),
refresh = checkbox(False, "New random walk every time"),
steps = (50,(10..500))):
# We cache the walk in the global variable vv, so that
# checking or unchecking the points checkbox doesn't change
# the random walk.
html("<h2>%s steps</h2>"%steps)
global vv
if refresh or len(vv) == 0:
s = 0; v = [(0,0)]
for i in range(steps):
s += random() - 0.5
v.append((i, s))
vv = v
elif len(vv) != steps:
# Add or subtract some points
s = vv[-1][1]; j = len(vv)
for i in range(steps - len(vv)):
s += random() - 0.5
vv.append((i+j,s))
v = vv[:steps]
else:
v = vv
L = line(v, rgbcolor='#4a8de2')
if pts: L += points(v, pointsize=10, rgbcolor='red')
show(L, xmin=0, figsize=[8,3])

attachment:randomwalk.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

### Taylor Series

by Harald Schilly

var('x')
x0  = 0
f   = sin(x)*e^(-x)
p   = plot(f,-1,5, thickness=2)
dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):
ft = f.taylor(x,x0,order)
pt = plot(ft,-1, 5, color='green', thickness=2)
html('$f(x)\;=\;%s$'%latex(f))
html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
show(dot + p + pt, ymin = -.5, ymax = 1)

attachment:taylor_series_animated.gif

## 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>'
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

### Vector Fields and Euler's Method

by Mike Hansen (tested and updated by William Stein)

x,y = var('x,y')
@interact
def _(f = input_box(default=y), g=input_box(default=-x*y+x^3-x),
xmin=input_box(default=-1), xmax=input_box(default=1),
ymin=input_box(default=-1), ymax=input_box(default=1),
start_x=input_box(default=0.5), start_y=input_box(default=0.5),
step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
old_f = f
f = f.function(x,y)
old_g = g
g = g.function(x,y)
steps = int(steps)

points = [ (start_x, start_y) ]
for i in range(steps):
xx, yy = points[-1]
try:
points.append( (xx+step_size*f(xx,yy), yy+step_size*g(xx,yy)) )
except (ValueError, ArithmeticError, TypeError):
break

starting_point = point(points[0], pointsize=50)
solution = line(points)
vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )

result = vector_field + starting_point + solution

html(r"<h2>$\frac{dx}{dt} = %s$  $\frac{dy}{dt} = %s$</h2>"%(latex(old_f),latex(old_g)))
print "Step size: %s"%step_size
print "Steps: %s"%steps
result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)

attachment:euler.png

### Vector Field with Runga-Kutta-Fehlberg

by Harald Schilly

# Solve ODEs using sophisticated Methods like Runga-Kutta-Fehlberg
# by Harald Schilly, April 2008
var('x y')
@interact
def _(fin = input_box(default=y+exp(x/10)-1/3*((x-1/2)^2+y^3)*x-x*y^3), gin=input_box(default=x^3-x+1/100*exp(y*x^2+x*y^2)-0.7*x),
xmin=input_box(default=-1), xmax=input_box(default=1.8),
ymin=input_box(default=-1.3), ymax=input_box(default=1.5),
x_start=(-1,(-2,2)), y_start=(0,(-2,2)), error=(0.5,(0,1)),
t_length=(23,(0, 100)) , num_of_points = (1500,(5,2000)),
algorithm = selector([
("rkf45" , "runga-kutta-felhberg (4,5)"),
("rk2" , "embedded runga-kutta (2,3)"),
("rk4" , "4th order classical runga-kutta"),
("rk8pd" , 'runga-kutta prince-dormand (8,9)'),
("rk2imp" , "implicit 2nd order runga-kutta at gaussian points"),
("rk4imp" , "implicit 4th order runga-kutta at gaussian points"),
("bsimp" , "implicit burlisch-stoer (requires jacobian)"),
("gear1" , "M=1 implicit gear"),
("gear2" , "M=2 implicit gear")
])):
f(x,y)=fin
g(x,y)=gin

ff = f._fast_float_(*f.args())
gg = g._fast_float_(*g.args())

#solve
path = []
err = error
xerr = 0
for yerr in [-err, 0, +err]:
T=ode_solver()
T.algorithm=algorithm
T.function = lambda t, yp: [ff(yp[0],yp[1]), gg(yp[0],yp[1])]
T.jacobian = lambda t, yp: [[diff(fun,dval)(yp[0],yp[1]) for dval in [x,y]] for fun in [f,g]]
T.ode_solve(y_0=[x_start + xerr, y_start + yerr],t_span=[0,t_length],num_points=num_of_points)
path.append(line([p[1] for p in T.solution]))

#plot
vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )
starting_point = point([x_start, y_start], pointsize=50)
show(vector_field + starting_point + sum(path), aspect_ratio=1, figsize=[8,9])

attachment:ode_runga_kutta.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

### 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')
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):
verbose=False)
R = get_data('NASDAQ')
R = get_data("NYSE")
R = R.splitlines()
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:
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:
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
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 += '&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)