Differences between revisions 11 and 32 (spanning 21 versions)
 ⇤ ← Revision 11 as of 2009-05-23 11:22:36 → Size: 3572 Editor: pang Comment: slight improvements over previous version of Evolutes ← Revision 32 as of 2013-11-25 10:30:11 → ⇥ Size: 18327 Editor: pang Comment: fixing geodesics interact ... Deletions are marked like this. Additions are marked like this. Line 6: Line 6: == Intersecting tetrahedral reflections == == Intersecting tetrahedral reflections FIXME == Line 8: Line 8: {{{ {{{#!sagecell Line 12: Line 12: p12 = p1.union(p2) p12 = p1.convex_hull(p2) Line 15: Line 15: p34 = p3.union(p4) p34 = p3.convex_hull(p4) Line 18: Line 18: p56 = p5.union(p6) p56 = p5.convex_hull(p6) Line 21: Line 21: p78 = p7.union(p8) p78 = p7.convex_hull(p8) Line 36: Line 36: {{{ {{{#!sagecell Line 52: Line 52: normal=(gammap[1]/norma(gammap), -gammap[0]/norma(gammap)) np=norma(gammap) Line 54: Line 54: np=norma(gammap) Line 57: Line 55: pe=gammap[0]*gammapp[0]+gammap[1]*gammapp[1] normal=(gammap[1]/np, -gammap[0]/np) Line 80: Line 79: show(grafica,aspect_ratio=1,xmin=-2,xmax=2,ymin=-2,ymax=2)}}} show(grafica,aspect_ratio=1,xmin=-2,xmax=2,ymin=-2,ymax=2)}}} Line 82: Line 82: == Geodesics on a parametric surface ==by Antonio Valdés and Pablo Angulo. This example was originally composed of two interacts:  - the first allowing the user to introduce a parametric surface, and draw it. - the second drawing a geodesic within the surface. The separation was so that after the first interact, the geodesic equations were "compiled", thus making the second interact faster.This still looks as a good idea to me, so please read the original code at https://malabares.cancamusa.net/home/pub/14/But the following is fixed so that there is only one interact, and sagecell works. There might be another way yto {{{#!sagecellfrom scipy.integrate import odeintu, v, t, du, dv = var('u v t du dv')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@interactdef _(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'),      init_point = input_grid(1, 2, default = [[-pi/4,pi/8]], label = 'coordinates of \ninitial point'),      init_vector = input_grid(1, 2, default = [[1,0]], label = 'coordinates of \ninitial vector'),      int_s = slider(0, 10, 1/10,                            default = pi/2,                            label = 'geodesic interval'),      sliding_color = checkbox(True,'change color along the geodesic')):    int_u = int_u[0]    int_v = int_v[0]    u_0, v_0 = init_point[0]    V_u, V_v = init_vector[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]))       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]]            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])}}}{{attachment:geodesics1.png}}{{attachment:geodesics2.png}}== Dimensional Explorer ==By Eviatar BachRenders 2D images (perspective or spring-layout) and 3D models of 0-10 dimensional hypercubes. It also displays number of edges and vertices.{{{#!sagecell@interactdef 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))}}}{{attachment:dimensions.png}}== 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).{{{#!sagecellfrom collections import defaultdictvar('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 idef random_line(cota):    theta = random()*pin    k = 2*cota*random() - cota    return sin(theta)*x + cos(theta)*y + kdef crofton_exact(curva, t0, t1, L, M):    forget()    assume(t>t0)    assume(t

# Sage Interactions - Geometry

## Intersecting tetrahedral reflections FIXME

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. This example was originally composed of two interacts:

• - the first allowing the user to introduce a parametric surface, and draw it. - the second drawing a geodesic within the surface.

The separation was so that after the first interact, the geodesic equations were "compiled", thus making the second interact faster.

This still looks as a good idea to me, so please read the original code at https://malabares.cancamusa.net/home/pub/14/ But the following is fixed so that there is only one interact, and sagecell works. There might be another way yto

## 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.

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