Differences between revisions 16 and 17
Revision 16 as of 2010-12-03 17:44:03
Size: 10955
Editor: Eviatar
Comment:
Revision 17 as of 2011-02-10 22:53:42
Size: 15503
Editor: pang
Comment: Added Crofton's formula
Deletions are marked like this. Additions are marked like this.
Line 243: Line 243:

== Crofton's formula ==
by Pablo Angulo. Illustrates [[http://en.wikipedia.org/wiki/Crofton%27s_formula| Crofton's formula]] by throwing some random lines and computing the intersection number with a given curve. May use either solve for exact computation of the intersections, or may also approximate the curve by straight segments (this is the default).

{{{
from collections import defaultdict

var('t x y')
pin = pi.n()

def longitud(curva, t0, t1):
    dxdt = derivative(curva[0], t)
    dydt = derivative(curva[1], t)
    integrando(t) = sqrt(dxdt^2 + dydt^2)
    i,_ = numerical_integral(integrando, t0, t1)
    return i

def random_line(cota):
    theta = random()*pin
    k = 2*cota*random() - cota
    return sin(theta)*x + cos(theta)*y + k

def crofton_exact(curva, t0, t1, L, M):
    forget()
    assume(t>t0)
    assume(t<t1)
    pp = parametric_plot(curva, (t, t0, t1), color='red')
    cortesd = defaultdict(int)
    for k in range(L):
        rl = random_line(M)
        ss = solve(rl(x=curva[0], y=curva[1]), t)
        cortes = 0
        for s in ss:
            tt = s.rhs()
            x0,y0 = curva[0](t=tt), curva[1](t=tt)
            if x0 in RR and y0 in RR:
                pp += point2d((x0,y0), pointsize = 30)
                cortes += 1
        if cortes:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='green')
        else:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='blue')
        cortesd[cortes] += 1
    return cortesd, pp
    
def random_line_n(cota):
    theta = random()*pin
    k = 2*cota*random() - cota
    return sin(theta), cos(theta), k

def interseccion_sr(punto1, punto2, recta):
    'Devuelve el punto de interseccion de una recta y un segmento, o None si no se cortan'
    x1, y1 = punto1
    x2, y2 = punto2
    a, b, c = recta
    num = (-c - a*x1 - b*y1)
    den = (a*(x2 - x1) + b*(y2 - y1))
    if (0 < num < den) or (den < num < 0):
        t_i = num/den
        return ((1-t_i)*x1 + t_i*x2, (1-t_i)*y1 + t_i*y2)
    else:
        return None

def interseccion_cr(curva, t0, t1, recta, partes=50):
    '''Devuelve el numero de puntos de interseccion de una curva y una recta'''
    x,y = curva
    paso = (t1 - t0)/partes
    puntos = [(x(t=tr), y(t=tr)) for tr in srange(t0, t1 + paso, paso)]
    intersecciones = (interseccion_sr(puntos[j], puntos[j+1], recta)
                      for j in xrange(partes-1))
    return [p for p in intersecciones if p ]
    
def crofton_aprox(curva, t0, t1, L, M):
    cortesd = defaultdict(int)
    pp = parametric_plot(curva, (t, t0, t1), color='red')
    for k in range(L):
        a,b,c = random_line_n(M)
        rl = a*x + b*y + c
        cortes = interseccion_cr(curva, t0, t1, (a,b,c))
        if cortes:
            pp += sum(point2d(p, pointsize = 30) for p in cortes)
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='green')
        else:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='blue')
        cortesd[len(cortes)] += 1
    return cortesd, pp

def print_stats(d):
    print 'Number of lines with k intersection points:'
    print ', '.join('%d:%d'%(k,v) for k,v in d.iteritems())
    
}}}
{{{
@interact
def crofton_interact(u1 = text_control('x and y coordinates of curve'),
                     curvax = input_box(t^2, label='x(t)' ),
                     curvay = input_box(2*t-1, label='y(t)' ),
                     u2 = text_control('Interval of definition'),
                     t0 = 0, t1 = 1,
                     u3 = text_control('Draw L lines randomly cos(t)x + sin(t)y + K, |K|&lt;M, 0 <= t < 2pi'),
                     M = 2,
                     L = 5,
                     u4 = text_control('Use function "solve" from maxima for exact computations?'),
                     exact = checkbox(False),
                     u5 = text_control('Otherwise, a curve is approximated by how many segments?'),
                     steps = slider(4, 40, 4, 8)):

    if exact:
        cortesd, p = crofton_exact((curvax, curvay), t0, t1, L, M)
    else:
        cortesd, p = crofton_aprox((curvax, curvay), t0, t1, L, M)
    p.show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2,ymax=2)
    print 'A curve of lenght %f'%longitud((curvax, curvay), t0, t1)
    print_stats(cortesd)
    cortes_tot = sum(k*v for k,v in cortesd.iteritems())
    print 'Aprox lenght using Crofton\'s formula: %f'%((cortes_tot/L)*(pi*M))
}}}
{{attachment:crofton4.png}}

Sage Interactions - Geometry

goto interact main page

Intersecting tetrahedral reflections

by Marshall Hampton. Inspired by a question from Hans Schepker of Glass Geometry.

#Pairs of tetrahedra, one the reflection of the other in the internal face, are joined by union operations:
p1 = Polyhedron(vertices = [[1,1,1],[1,1,0],[0,1,1],[1,0,1]])
p2 = Polyhedron(vertices = [[1/3,1/3,1/3],[1,1,0],[0,1,1],[1,0,1]])
p12 = p1.union(p2)
p3 = Polyhedron(vertices = [[0,0,1],[0,0,0],[0,1,1],[1,0,1]])
p4 = Polyhedron(vertices = [[2/3,2/3,1/3],[0,0,0],[0,1,1],[1,0,1]])
p34 = p3.union(p4)
p5 = Polyhedron(vertices = [[1,0,0],[1,0,1],[0,0,0],[1,1,0]])
p6 = Polyhedron(vertices = [[1/3,2/3,2/3],[1,0,1],[0,0,0],[1,1,0]])
p56 = p5.union(p6)
p7 = Polyhedron(vertices = [[0,1,0],[0,0,0],[1,1,0],[0,1,1]])
p8 = Polyhedron(vertices = [[2/3,1/3,2/3],[0,0,0],[1,1,0],[0,1,1]])
p78 = p7.union(p8)
pti = p12.intersection(p34).intersection(p56).intersection(p78)
@interact
def tetra_plot(opac = slider(srange(0,1.0,.25), default = .25)):
    p12r = p12.render_wireframe()+p12.render_solid(opacity = opac)
    p34r = p34.render_wireframe()+p34.render_solid(rgbcolor = (0,0,1),opacity = opac)
    p56r = p56.render_wireframe()+p56.render_solid(rgbcolor = (0,1,0),opacity = opac)
    p78r = p78.render_wireframe()+p78.render_solid(rgbcolor = (0,1,1),opacity = opac)
    ptir = pti.render_wireframe()+pti.render_solid(rgbcolor = (1,0,1),opacity = .9)
    show(p12r+p34r+p56r+p78r+ptir, frame = False)

tetrareflect.png

Evolutes

by Pablo Angulo. Computes the evolute of a plane curve given in parametric coordinates. The curve must be parametrized from the interval [0,2pi].

var('t');
def norma(v):
    return sqrt(sum(x^2 for x in v))    
paso_angulo=5

@interact
def _( gamma1=input_box(default=sin(t)), gamma2=input_box(default=1.3*cos(t)),
    draw_normal_lines=True, 
    rango_angulos=range_slider(0,360,paso_angulo,(0,90),label='Draw lines for these angles'), 
    draw_osculating_circle=True, 
    t0=input_box(default=pi/3,label='parameter value for the osculating circle'), 
    auto_update=False ):
        
    gamma=(gamma1,gamma2)
    gammap=(gamma[0].derivative(),gamma[1].derivative())
    normal=(gammap[1]/norma(gammap), -gammap[0]/norma(gammap))
    gammapp=(gammap[0].derivative(),gammap[1].derivative())
    
    np=norma(gammap)
    npp=norma(gammapp)
    pe=gammap[0]*gammapp[0]+gammap[1]*gammapp[1]
    curvatura=(gammap[1]*gammapp[0]-gammap[0]*gammapp[1])/norma(gammap)^3
    radio=1/curvatura
        
    centros=(gamma[0]+radio*normal[0],gamma[1]+radio*normal[1])
            
    curva=parametric_plot(gamma,(t,0,2*pi))
    evoluta=parametric_plot(centros,(t,0,2*pi), color='red')
    grafica=curva+evoluta

    if draw_normal_lines:
        f=2*pi/360
        lineas=sum(line2d( [ (gamma[0](t=i*f), gamma[1](t=i*f)), 
                             (centros[0](t=i*f), centros[1](t=i*f)) ], 
                           thickness=1,rgbcolor=(1,0.8,0.8)) 
                   for i in range(rango_angulos[0], rango_angulos[1]+paso_angulo, paso_angulo))
        grafica+=lineas
    
    if draw_osculating_circle and 0<t0<2*pi:
        punto=point((gamma[0](t=t0), gamma[1](t=t0)), rgbcolor=hue(0),pointsize=30)
        circulo=circle( (centros[0](t=t0), centros[1](t=t0)), radio(t=t0) )  
        grafica+=punto+circulo

    show(grafica,aspect_ratio=1,xmin=-2,xmax=2,ymin=-2,ymax=2)

evoluta3.png

Geodesics on a parametric surface

by Antonio Valdés and Pablo Angulo. A first interact allows the user to introduce a parametric surface, and draws it. Then a second interact draws a geodesic within the surface. The separation is so that after the first interact, the geodesic equations are "compiled", and then the second interact is faster.

u, v, t = var('u v t')
@interact
def _(x = input_box(3*sin(u)*cos(v), 'x'),
      y = input_box(sin(u)*sin(v), 'y'),
      z = input_box(2*cos(u), 'z'),
      _int_u = input_grid(1, 2, default = [[0,pi]], label = 'u -interval'), 
      _int_v = input_grid(1, 2, default = [[-pi,pi]], label = 'v -interval')):
    
    global F, Fu, Fv, func, S_plot, int_u, int_v
    int_u = _int_u[0]
    int_v = _int_v[0] 
    
    F = vector([x, y, z])

    S_plot = parametric_plot3d( F, 
                                (u, int_u[0], int_u[1]), 
                                (v, int_v[0], int_v[1]))
    S_plot.show(aspect_ratio = [1, 1, 1])
    
    dFu = F.diff(u)
    dFv = F.diff(v)
    
    Fu = fast_float(dFu, u, v)
    Fv = fast_float(dFv, u, v)
    
    ufunc = function('ufunc', t)
    vfunc = function('vfunc', t)
    
    dFtt = F(u=ufunc, v=vfunc).diff(t, t)
    
    ec1 = dFtt.dot_product(dFu(u=ufunc, v=vfunc))
    ec2 = dFtt.dot_product(dFv(u=ufunc, v=vfunc))
    
    dv, ddv, du, ddu = var('dv, ddv, du, ddu')
    
    diffec1 = ec1.subs_expr(diff(ufunc, t) == du, 
                            diff(ufunc, t, t) == ddu,     
                            diff(vfunc, t) == dv, 
                            diff(vfunc, t, t) == ddv, 
                            ufunc == u, vfunc == v)
    diffec2 = ec2.subs_expr(diff(ufunc, t) == du, 
                            diff(ufunc, t, t) == ddu,     
                            diff(vfunc, t) == dv, 
                            diff(vfunc, t, t) == ddv, 
                            ufunc == u, vfunc == v)
    sols = solve([diffec1 == 0 , diffec2 == 0], ddu, ddv)
    
    ddu_rhs = (sols[0][0]).rhs().full_simplify()
    ddv_rhs = (sols[0][1]).rhs().full_simplify()
        
    ddu_ff = fast_float(ddu_rhs, du, dv, u, v)
    ddv_ff = fast_float(ddv_rhs, du, dv, u, v)
    
    def func(y,t):
        v = list(y)
        return [ddu_ff(*v), ddv_ff(*v), v[0], v[1]]

geodesics1.png

from scipy.integrate import odeint

def fading_line3d(points, rgbcolor1, rgbcolor2, *args, **kwds):
    L = len(points)
    vcolor1 = vector(RDF, rgbcolor1)
    vcolor2 = vector(RDF, rgbcolor2)
    return sum(line3d(points[j:j+2], 
                      rgbcolor = tuple( ((L-j)/L)*vcolor1 + (j/L)*vcolor2 ), 
                      *args, **kwds) 
               for j in srange(L-1))

steps = 100

@interact
def _(u_0 = slider(int_u[0], int_u[1], (int_u[1] - int_u[0])/100, 
                   default = (int_u[0] + int_u[1])/2, label = 'u_0'),
      v_0 = slider(int_v[0], int_v[1], (int_v[1] - int_v[0])/100, 
                   default = (int_v[0] + int_v[1])/2, label = 'v_0'),
      V_u = slider(-10, 10, 1/10, default = 1, label = 'V_u'),
      V_v = slider(-10, 10, 1/10, default = 0, label = 'V_v'), 
      int_s = slider(0, 10, 1/10, 
                           default = (int_u[1] - int_u[0])/2, 
                           label = 'geodesic interval'),
      sliding_color = checkbox(True,'change color along the geodesic')):
        
        du, dv, u, v = var('du dv u v')
        Point = [u_0, v_0]
        velocity = [V_u, V_v]
        Point = map(float, Point)
        velocity = map(float, velocity)
        
        geo2D_aux = odeint(func,
                           y0 = [velocity[0], velocity[1], Point[0], Point[1]],
                           t = srange(0, int_s, 0.01))
    
        geo3D = [F(u=l,v=r) for [j, k, l, r] in geo2D_aux]
        
        if sliding_color:
            g_plot = fading_line3d(geo3D, rgbcolor1 = (1, 0, 0), rgbcolor2 = (0, 1, 0), thickness=4)
        else:
            g_plot = line3d(geo3D, rgbcolor=(0, 1, 0), thickness=4)
        
        P = F(u=Point[0], v=Point[1])
        P_plot = point3d((P[0], P[1], P[2]), rgbcolor = (0, 0, 0), pointsize = 30)
        V = velocity[0] * Fu(u = Point[0], v = Point[1]) + \
            velocity[1] * Fv(u= Point[0], v = Point[1])
        V_plot = arrow3d(P, P + V, color = 'black')
        
        show(g_plot + S_plot + V_plot + P_plot,aspect_ratio = [1, 1, 1])

geodesics2.png

Dimensional Explorer

By Eviatar Bach

Renders 2D images (perspective or spring-layout) and 3D models of 0-10 dimensional hypercubes. It also displays number of edges and vertices.

@interact
def render(Display=selector(['2D Perspective', '2D Spring-layout', '3D']), Dimension=slider(0,10,default=4, step_size=1), Size=slider(0,10,default=5,step_size=1), Vertices=False, Calculations=False):
    
    if Display=='2D Perspective':
        
        if Dimension==0:
            g=graphs.GridGraph([1])
            print 'Vertices:', len(g.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(g.edges()), ('(%s*(%s/2))' %(len(g.vertices()), Dimension) if Calculations else '')
            g.show(figsize=[Size,Size], vertex_size=30, vertex_labels=False, transparent=True, vertex_colors='black')
            
        else:
            g=graphs.CubeGraph(Dimension)
            print 'Vertices:', len(g.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(g.edges()), ('(%s*(%s/2))' %(len(g.vertices()), Dimension) if Calculations else '')
            g.show(figsize=[Size,Size], vertex_size=(20 if Vertices else 0), vertex_labels=False, transparent=True, vertex_colors='black')
            
    if Display=='2D Spring-layout':
        
        if Dimension==0:
            s=graphs.GridGraph([1])
            print 'Vertices:', len(s.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(s.edges()), ('(%s*(%s/2))' %(len(s.vertices()), Dimension) if Calculations else '')
            s.show(figsize=[Size,Size], vertex_size=30, vertex_labels=False, transparent=True, vertex_colors='black')
            
        else:
            s=graphs.GridGraph([2]*Dimension)
            print 'Vertices:', len(s.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(s.edges()), ('(%s*(%s/2))' %(len(s.vertices()), Dimension) if Calculations else '')
            s.show(figsize=[Size,Size], vertex_size=(20 if Vertices else 0), vertex_labels=False, transparent=True, vertex_colors='black')
            
    if Display=='3D':
        if Dimension==0:
            d=graphs.GridGraph([1])
            print 'Vertices:', len(d.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(d.edges()), ('(%s*(%s/2))' %(len(d.vertices()), Dimension) if Calculations else '')
            d.show3d(figsize=[Size/2,Size/2], vertex_size=0.001)

        else:
            d=graphs.CubeGraph(Dimension)
            print 'Vertices:', len(d.vertices()), ('(2^%s)'%Dimension if Calculations else ''), '\nEdges:', len(d.edges()), ('(%s*(%s/2))' %(len(d.vertices()), Dimension) if Calculations else '')
            d.show3d(figsize=[Size,Size], vertex_size=(0.03 if Vertices else 0.001))

dimensions.png

Crofton's formula

by Pablo Angulo. Illustrates Crofton's formula by throwing some random lines and computing the intersection number with a given curve. May use either solve for exact computation of the intersections, or may also approximate the curve by straight segments (this is the default).

from collections import defaultdict

var('t x y')
pin = pi.n()

def longitud(curva, t0, t1):
    dxdt = derivative(curva[0], t)
    dydt = derivative(curva[1], t)
    integrando(t) = sqrt(dxdt^2 + dydt^2)
    i,_ = numerical_integral(integrando, t0, t1)
    return  i

def random_line(cota):
    theta = random()*pin
    k = 2*cota*random() - cota
    return sin(theta)*x + cos(theta)*y + k

def crofton_exact(curva, t0, t1, L, M):
    forget()
    assume(t>t0)
    assume(t<t1)
    pp = parametric_plot(curva, (t, t0, t1), color='red')
    cortesd = defaultdict(int)
    for k in range(L):
        rl = random_line(M)
        ss = solve(rl(x=curva[0], y=curva[1]), t)
        cortes = 0
        for s in ss:
            tt = s.rhs()
            x0,y0 = curva[0](t=tt), curva[1](t=tt)
            if x0 in RR and y0 in RR:
                pp += point2d((x0,y0), pointsize = 30)
                cortes += 1
        if cortes:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='green')
        else:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='blue')
        cortesd[cortes] += 1
    return cortesd, pp
    
def random_line_n(cota):
    theta = random()*pin
    k = 2*cota*random() - cota
    return sin(theta), cos(theta), k

def interseccion_sr(punto1, punto2, recta):
    'Devuelve el punto de interseccion de una recta y un segmento, o None si no se cortan'
    x1, y1 = punto1
    x2, y2 = punto2
    a, b, c   = recta
    num = (-c - a*x1 - b*y1)
    den = (a*(x2 - x1) + b*(y2 - y1))
    if (0 < num < den) or (den < num < 0):
        t_i = num/den
        return ((1-t_i)*x1 + t_i*x2, (1-t_i)*y1 + t_i*y2)
    else:
        return None

def interseccion_cr(curva, t0, t1, recta, partes=50):
    '''Devuelve el numero de puntos de interseccion de una curva y una recta'''
    x,y = curva
    paso = (t1 - t0)/partes
    puntos = [(x(t=tr), y(t=tr)) for tr in srange(t0, t1 + paso, paso)]
    intersecciones = (interseccion_sr(puntos[j], puntos[j+1], recta)
                      for j in xrange(partes-1))
    return [p for p in intersecciones if p ]
    
def crofton_aprox(curva, t0, t1, L, M):
    cortesd = defaultdict(int)
    pp = parametric_plot(curva, (t, t0, t1), color='red')
    for k in range(L):
        a,b,c = random_line_n(M)
        rl = a*x + b*y + c
        cortes = interseccion_cr(curva, t0, t1, (a,b,c))
        if cortes:
            pp += sum(point2d(p, pointsize = 30) for p in cortes)
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='green')
        else:
            pp += implicit_plot(rl, (x,-M,M), (y,-M,M), color='blue')
        cortesd[len(cortes)] += 1
    return cortesd, pp

def print_stats(d):
    print 'Number of lines with k intersection points:'
    print ', '.join('%d:%d'%(k,v) for k,v in d.iteritems())

@interact
def crofton_interact(u1 = text_control('x and y coordinates of curve'),
                     curvax = input_box(t^2, label='x(t)' ),
                     curvay = input_box(2*t-1, label='y(t)' ),
                     u2 = text_control('Interval of definition'),
                     t0 = 0, t1 = 1,
                     u3 = text_control('Draw L lines randomly cos(t)x + sin(t)y + K, |K|&lt;M, 0 <= t < 2pi'),
                     M  = 2,
                     L  = 5,
                     u4    = text_control('Use function "solve" from maxima for exact computations?'),
                     exact = checkbox(False),
                     u5    = text_control('Otherwise, a curve is approximated by how many segments?'),
                     steps = slider(4, 40, 4, 8)):

    if exact:
        cortesd, p = crofton_exact((curvax, curvay), t0, t1, L, M)
    else:
        cortesd, p = crofton_aprox((curvax, curvay), t0, t1, L, M)
    p.show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2,ymax=2)
    print 'A curve of lenght %f'%longitud((curvax, curvay), t0, t1)
    print_stats(cortesd)
    cortes_tot = sum(k*v for k,v in cortesd.iteritems())
    print 'Aprox lenght using Crofton\'s formula: %f'%((cortes_tot/L)*(pi*M))

crofton4.png

interact/geometry (last edited 2019-11-15 08:20:36 by chapoton)