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== Banchoff-Pohl area == by Pablo Angulo. Computes the Banchoff-Pohl "area enclosed by a spatial curve", by throwing some random lines and computing the linking number with the given curve. Lines not linked to the given curve are displayed in red, linked lines are displayed in green. {{{#!sagecell from collections import defaultdict var('t') a = 0; b= 2*pi def random_line3d(bound): '''Random line in R^3: first choose the guiding vector of the line, then choose a point in the plane p perpendicular to that vector. ''' p = vector([(2*random() - 1) for _ in range(3)]) v = p/norm(p) v2, v3 = matrix(v).right_kernel().basis() if det(matrix([v, v2, v3]))<0: v2, v3 = v3, v2 v2 = v2/norm(v2) v3 = v3 - (v3*v2)*v2 v3 = v3/norm(v3) return v, (2*random()*bound - bound, v2), (2*random()*bound - bound, v3) def winding_number(x, y, x0, y0): x -= x0 y -= y0 r2 = x^2 + y^2 xp = x.derivative(t) yp = y.derivative(t) integrando(t) = (x*yp -y*xp)/r2 i,e = numerical_integral(integrando, a, b) return round(i/(2*pi)) def linking_number(curve, line): v, (a2, v2), (a3, v3) = line M = matrix([v, v2, v3]) # curve2d = (curve*M.inverse())[1:] #This is VERY slow, for some reason! curve2d = sum(c*v for c,v in zip(curve,M.inverse()))[1:] x, y = curve2d return winding_number(x, y, a2,a3) def color_ln(number): if number: return (0,1-1/(1+number),0) else: return (1,0,0) def banchoff_pohl(curve, L, M): ln_d = defaultdict(int) pp = parametric_plot3d(curve, (t,0,2*pi), thickness=2) for k in range(L): a_line = random_line3d(M) ln = abs(linking_number(curve, a_line)) v, (l1, v1), (l2, v2) = a_line pp += parametric_plot3d(l1*v1+l2*v2+t*v,(t,-2,2), color=color_ln(ln)) ln_d[ln] += 1 return ln_d, pp def print_stats(d): print 'Number of lines with linking number k:' print ', '.join('%d:%d'%(k,v) for k,v in d.iteritems()) @interact def bp_interact( u1 = text_control('x, y, z coordinates of a closed space curve in [0,2*pi]'), curvax = input_box(cos(t), label='x(t)' ), curvay = input_box(sin(t), label='y(t)' ), curvaz = input_box(0, label='y(t)' ), u2 = text_control('The curve should be contained in a ball of radius M'), M = 1, u3 = text_control('Use L lines chosen randomly'), L = 10): ln_d, p = banchoff_pohl(vector((curvax, curvay, curvaz)), L, M) p.show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2,ymax=2) bp_area_aprox = (sum(k^2*v for k,v in ln_d.iteritems())/L)*2*pi*M^2 print 'Bahnchoff-Pohl area of the curve(aprox): %f'%bp_area_aprox print_stats(ln_d) }}} {{attachment:banchoff-pohl.png}} |
Sage Interactions - Geometry
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Contents
Intersecting tetrahedral reflections
by Marshall Hampton. Inspired by a question from Hans Schepker of Glass Geometry.
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].
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.
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.
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).
Banchoff-Pohl area
by Pablo Angulo. Computes the Banchoff-Pohl "area enclosed by a spatial curve", by throwing some random lines and computing the linking number with the given curve. Lines not linked to the given curve are displayed in red, linked lines are displayed in green.
