Differences between revisions 11 and 12
 ⇤ ← Revision 11 as of 2009-05-23 11:22:36 → Size: 3572 Editor: pang Comment: slight improvements over previous version of Evolutes ← Revision 12 as of 2010-07-30 21:08:53 → ⇥ Size: 7450 Editor: pang Comment: Added a new interact showing geodesics on a surface Deletions are marked like this. Additions are marked like this. Line 82: Line 82: == 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')@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')):        global F, Fu, Fv, func, S_plot, int_u, int_v    int_u = _int_u[0]    int_v = _int_v[0]        def F(uu, vv):         X = vector([x, y, z])         return X.subs({u : uu, v : vv})    S_plot = parametric_plot3d( F(u, v),                                 (u, int_u[0], int_u[1]),                                 (v, int_v[0], int_v[1]))    show(S_plot, aspect_ratio = [1, 1, 1])        dFu = F(u, v).diff(u)    dFv = F(u, v).diff(v)        Fu = fast_float(dFu, u, v)    Fv = fast_float(dFv, u, v)        ufunc = function('ufunc', t)    vfunc = function('vfunc', t)        dFtt = F(ufunc, 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]]}}}{{attachment:geodesics1.png}}{{{from scipy.integrate import odeint@interactdef _(u_0 = slider(int_u[0], int_u[1], default = (int_u[0] + int_u[1])/2, label = 'u_0'),      v_0 = slider(int_v[0], int_v[1], default = (int_v[0] + int_v[1])/2, label = 'v_0'),      V_u = slider(-10, 10, default = 1, label = 'V_u'),      V_v = slider(-10, 10, default = 0, label = 'V_v'),       int_s = range_slider(-10, 10, 0.1,                            default = (0, (int_u[1] - int_u[0])/2),                            label = 'geodesic interval') ):                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(int_s[0], int_s[1], 0.01))                geo3D = [F(l,r) for [j, k, l, r] in geo2D_aux]                g_plot = line3d(geo3D, rgbcolor = (1, 0, 0), thickness = 4)                P = F(Point[0], Point[1])        P_plot = point3d((P[0], P[1], P[2]), rgbcolor = (0, 0, 0))        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)                show(g_plot + S_plot + V_plot + P_plot,aspect_ratio = [1, 1, 1])}}}{{attachment:geodesics2.png}}

# Sage Interactions - Geometry

## 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)```

## 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

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)
grafica+=punto+circulo

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

## 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]

def F(uu, vv):
X = vector([x, y, z])
return X.subs({u : uu, v : vv})

S_plot = parametric_plot3d( F(u, v),
(u, int_u[0], int_u[1]),
(v, int_v[0], int_v[1]))
show(S_plot, aspect_ratio = [1, 1, 1])

dFu = F(u, v).diff(u)
dFv = F(u, v).diff(v)

Fu = fast_float(dFu, u, v)
Fv = fast_float(dFv, u, v)

ufunc = function('ufunc', t)
vfunc = function('vfunc', t)

dFtt = F(ufunc, 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]]```

```from scipy.integrate import odeint

@interact
def _(u_0 = slider(int_u[0], int_u[1], default = (int_u[0] + int_u[1])/2, label = 'u_0'),
v_0 = slider(int_v[0], int_v[1], default = (int_v[0] + int_v[1])/2, label = 'v_0'),
V_u = slider(-10, 10, default = 1, label = 'V_u'),
V_v = slider(-10, 10, default = 0, label = 'V_v'),
int_s = range_slider(-10, 10, 0.1,
default = (0, (int_u[1] - int_u[0])/2),
label = 'geodesic interval') ):

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(int_s[0], int_s[1], 0.01))

geo3D = [F(l,r) for [j, k, l, r] in geo2D_aux]

g_plot = line3d(geo3D, rgbcolor = (1, 0, 0), thickness = 4)

P = F(Point[0], Point[1])
P_plot = point3d((P[0], P[1], P[2]), rgbcolor = (0, 0, 0))
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)

show(g_plot + S_plot + V_plot + P_plot,aspect_ratio = [1, 1, 1])```

interact/geometry (last edited 2013-11-25 10:30:11 by pang)