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# Sage Interactions - Geometry

## 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).

```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))```

## 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)