def __init__(self):
#u"""x^2 + y^2 + z^2 = 1"""
super(EsferaCasquetes, self).__init__(u"Otro atlas de la esfera")
r = .998
esf = ParametricPlot3D(
lambda t, f:
(r * sin(t) * cos(f), r * sin(t) * sin(f), r * cos(t)),
(0, pi, 70), (0, 2 * pi, 70))
esf.setDiffuseColor(_1(99, 136, 63))
esf.setSpecularColor(_1(99, 136, 63))
pars = [
lambda u, v, t1: (u, v, 1.5 - t1 * (1.5 - sqrt(1 - u**2 - v**2))),
lambda u, v, t2: (u, v, -1 - t2 * (-1 + sqrt(1 - u**2 - v**2))),
lambda u, v, t3: (u, 1.5 - t3 * (1.5 - sqrt(1 - u**2 - v**2)), v),
lambda u, v, t4: (u, -1.5 - t4 *
(-1.5 + sqrt(1 - u**2 - v**2)), v),
lambda u, v, t5: (1.5 - t5 * (1.5 - sqrt(1 - u**2 - v**2)), u, v),
lambda u, v, t6, : (-1.5 - t6 *
(-1.5 + sqrt(1 - u**2 - v**2)), u, v)
]
d = .7
colores = [(0, 0, 1), (0, 0, 1), (0, 1, 0), (0, 1, 0), (1, 0, 0),
(1, 0, 0)]
planos = [
ParametricPlot3D(par, (-d, d, 40), (-d, d, 40)).setLinesVisible(
True).setMeshVisible(False).setMeshDiffuseColor(colores[i])
for i, par in enumerate(pars)
]
baseplane = BasePlane()
baseplane.setHeight(-1.005)
baseplane.setRange((-2, 2, 7))
self.addChild(esf)
for p in planos:
self.addChild(p)
self.addChild(baseplane)
## no queremos los controles
for i, plano in enumerate(planos):
plano.parameters['t%d' % (i + 1)].hide()
anims = [
plano.parameters['t%d' % (i + 1)].asAnimation()
for i, plano in enumerate(planos)
]
self.setupAnimations(anims)
def __init__(self):
super(Catenoide, self).__init__(
u"Cortamos una catenoide para obtener parte de una helicoide")
self.camera_position = (8.2, 8.2, 8.2)
self.camera_viewAll = False
def param(u, v, t):
t2 = pi / 2 - t
x = cos(t2) * sinh(v) * sin(u) + sin(t2) * cosh(v) * cos(u)
y = -cos(t2) * sinh(v) * cos(u) + sin(t2) * cosh(v) * sin(u)
z = u * cos(t2) + v * sin(t2)
return x, y, z
cat = ParametricPlot3D(param, (-pi, pi, 60), (-2, 2))
ht = cat.getParameter('t')
ht.timeline.setDuration(3000)
ht.updateRange((0, pi / 2, 0))
cat.setVerticesPerColumn(2)
cat.setAmbientColor(_1(4, 73, 143))
cat.setDiffuseColor(_1(4, 73, 143))
cat.setSpecularColor(_1(4, 73, 143))
s = Slider(rangep=('z', 2, 60, 2, 59),
func=cat.setVerticesPerColumn,
duration=3000,
parent=self)
self.addChild(cat)
params = [s, ht]
## no queremos los controles
for p in params:
p.hide()
anims = [p.asAnimation() for p in params]
self.setupAnimations(anims)
def __init__(self):
super(Helicoide, self).__init__(
u"Isometría local entre una helicoide y una catenoide")
self.camera_position = (8.2, 8.2, 8.2)
self.camera_viewAll = False
def param(u, v, t):
x = cos(t) * sinh(v) * sin(u) + sin(t) * cosh(v) * cos(u)
y = -cos(t) * sinh(v) * cos(u) + sin(t) * cosh(v) * sin(u)
z = u * cos(t) + v * sin(t)
return x, y, z
helic1 = ParametricPlot3D(param, (-pi, pi, 60), (-2, 2))
ht = helic1.getParameter('t')
ht.timeline.setDuration(3000)
ht.updateRange((0, pi / 2, 0))
helic1.setVerticesPerColumn(2)
helic1.setAmbientColor(_1(202, 78, 70))
helic1.setDiffuseColor(_1(202, 78, 70))
helic1.setSpecularColor(_1(202, 78, 70))
s = Slider(rangep=('z', 2, 60, 2, 59),
func=helic1.setVerticesPerColumn,
duration=3000,
parent=self)
self.addChild(helic1)
params = [s, ht]
## no queremos los controles
for p in params:
p.hide()
anims = [p.asAnimation() for p in params]
self.setupAnimations(anims)
def __init__(self):
super(Mobius,
self).__init__(u"No orientabilidad de la Banda de Möbius")
# self.camera_position = (3.0, 2.8, 2.8)
def par(u, v):
return cos(u) + v * cos(u / 2) * cos(u), sin(u) + v * cos(
u / 2) * sin(u), v * sin(u / 2)
mobius = ParametricPlot3D(par, (-pi, pi, 60), (-.5, .5, 14))
mobius.setTransparency(0.5)
def curva(t):
return par(t, 0)
def puntos(u):
return Vec3(
cos(u) * sin(u / 2.0),
sin(u / 2.0) * sin(u), -cos(u / 2.0))
cm = Curve3D(curva, (-pi, pi, 200), color=_1(255, 255, 255), width=3)
aceleracion_cm = cm.attachField(
"aceleracion", puntos).setLengthFactor(1).setWidthFactor(.5)
aceleracion_cm.animation.setDuration(12000)
self.addChild(mobius)
self.addChild(cm)
self.addChild(Arrow((-1, 0, 0), (0, 0, 0), 0.02))
self.setupAnimations([aceleracion_cm])
def createEllipsoid(a,b,c):
par_e = lambda u, v: Vec3(a * sin(v) * cos(u), b * sin(u) * sin(v), c * cos(v))
e = ParametricPlot3D(par_e, (0, 2*pi, 100), (0, pi, 100))
e.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
e.setTransparency(0.5)
e.setDiffuseColor(_1(40, 200, 120))
return e
def createHyperboloid(a,b):
par_h = lambda u, v: Vec3(u, v, u**2/(a**2)-v**2/(b**2))
h = ParametricPlot3D(par_h, (-4, 4, 100), (-4, 4, 100))
h.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
h.setTransparency(0.5)
h.setDiffuseColor(_1(220, 200, 80))
return h
def createTorus(r1,r2):
par_t = lambda u, v: Vec3((r1 + r2 * cos(v)) * cos(u), (r1 + r2 * cos(v)) * sin(u), r2 * sin(v))
tor = ParametricPlot3D(par_t, (-pi, pi, 100), (-pi, pi, 100))
tor.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
tor.setTransparency(0.5)
tor.setDiffuseColor(_1(180, 220, 200))
return tor
def __init__(self):
Page.__init__(self, u"Campo en la esfera con sólo una singularidad")
def make_circulo(t):
return partial(par_esfera, t)
par_esfera = lambda t, f: 0.99 * Vec3(
sin(t) * cos(f),
sin(t) * sin(f), cos(t))
esf = ParametricPlot3D(par_esfera, (0, pi, 100), (0, 2 * pi, 120))
esf.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
esf.setTransparency(0.4)
esf.setDiffuseColor(_1(68, 28, 119))
VisibleCheckBox("esfera", esf, True, parent=self)
self.addChild(esf)
def par_curva(c, t):
t = tan(t / (4 * pi))
den = c**2 + t**2 + 1
return Vec3(2 * c / den, 2 * t / den, (c**2 + t**2 - 1) / den)
def par_tang(c, t):
t = tan(t / (4 * pi))
den = (c**2 + t**2 + 1)**2
return Vec3(-2 * c * (2 * t) / den,
(2 * (c**2 + t**2 + 1) - 4 * t**2) / den, 4 * t / den)
def make_curva(c):
return partial(par_curva, c)
def make_tang(c):
return partial(par_tang, c)
tangentes = []
for c in range(-10, 11):
ct = tan(c / (2 * pi))
curva = Curve3D(make_curva(ct), (-20, 20, 80), width=1)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(1).setWidthFactor(.1)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
self.addChild(curva)
def animaTangentes(n):
for tang in tangentes:
tang.animateArrow(n)
a1 = Animation(animaTangentes, (10000, 0, 79))
self.setupAnimations([a1])
def __init__(self):
Page.__init__(self, u"Elipsoide<br><br>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> + z<sup>2</sup>/c<sup>2</sup> = 1")
param = lambda u,v: (cos(u)*cos(v), 1.5*cos(v)*sin(u), 2*sin(v))
elipsoide = ParametricPlot3D(param, (-pi, pi), (-pi/2,pi/2))
col = _1(84,129,121)
elipsoide.setAmbientColor(col).setDiffuseColor(col).setSpecularColor(col)
par1 = lambda u,v: Vec3(-sin(u)*cos(v), 1.5*cos(u)*cos(v), 0)
par2 = lambda u,v: Vec3(-cos(u)*sin(v), -1.5*sin(u)*sin(v), 2*cos(v))
tp = TangentPlane2(param,par1,par2,(0,0),_1(252,250,225))
self.addChild(elipsoide)
self.addChild(tp)
Slider(rangep=('u', -pi,pi,0,20),func=tp.setU, duration=8000, parent=self)
Slider(rangep=('v', -pi/2,pi/2,0,20),func=tp.setV, duration=8000, parent=self)
def __init__(self):
"F(x,y) = (x, y, x + y - 6)"
#u"""l plano x + y + z - 2.5 = 0"""
Page.__init__(self, u"Plano<br><br>F(x,y) = (x, y, x + y - 6)")
plane = lambda x, y: -x - y
p1 = lambda x, y, t1: (x, y, (1 - t1) * (-x - y) - 2 * t1)
p2 = lambda x, y, t2: (x, (1 - t2) * y - 2 * t2, -x - y)
p3 = lambda x, y, t3: ((1 - t3) * x - 2 * t3, y, -x - y)
r = (-1, 1, 15)
plano = Plot3D(plane, (-1, 1), (-1, 1))
plano.setTransparencyType(8)
plano1 = ParametricPlot3D(p1, r, r)
plano2 = ParametricPlot3D(p2, r, r)
plano3 = ParametricPlot3D(p3, r, r)
planos = [plano1, plano2, plano3]
for p in planos:
p.linesVisible = True
p.meshVisible = False
plano1.setMeshDiffuseColor((1, 0, 0))
plano2.setMeshDiffuseColor((0, 1, 0))
plano3.setMeshDiffuseColor((0, 1, 1))
plano.diffuseColor = _1(29, 214, 216)
plano.transparency = 0.5
plano.setAmbientColor(_1(29, 214, 216))
self.setupPlanes((-2, 2, 7))
self.addChildren([plano, plano1, plano2, plano3])
## no controls
for i, plano in enumerate(planos):
plano.parameters['t%d' % (i + 1)].hide()
self.setupAnimations([
plano.parameters['t%d' % (i + 1)].asAnimation()
for i, plano in enumerate(planos)
])
def __init__(self):
Page.__init__(
self,
u"Hélice circular reflejada<br><br>(cos s/√2, sen s/√2, -s/√2)"
)
self.camera_position = (10, -10, 10)
self.showAxis(False)
tmin, tmax, npuntos = (-2 * pi, 2 * pi, 200)
self.addChild(Cylinder(_1(7, 83, 150), tmax - tmin, 2))
def param1hr(t):
return 2 * Vec3(cos(t), sin(t), -t / 3.0)
def param2hr(t):
return 2 * Vec3(-sin(t), cos(t), -1 / 3.0)
def param3hr(t):
return 2 * Vec3(-cos(t), -sin(t), 0)
espiral = Curve3D(param1hr, (tmin * 1.5, tmax * 1.5, npuntos),
color=_1(240, 10, 120))
def param1hc_der(t):
return 2 * Vec3(cos(t), sin(t), t / 3.0)
espiral_der = Curve3D(param1hc_der, (tmin * 1.5, tmax * 1.5, npuntos),
color=_1(20, 240, 240))
tangente = espiral.attachField(
"tangente", param2hr).setLengthFactor(1).setWidthFactor(.6)
tangente.setRadius(0.06)
tangente.setDiffuseColor(_1(20, 240, 20))
normal = espiral.attachField(
"normal", param3hr).setLengthFactor(1).setWidthFactor(.6)
normal.setRadius(0.06)
normal.setDiffuseColor(_1(240, 120, 20))
self.addChild(espiral)
self.addChild(espiral_der)
plano_xy_par = lambda u, v: Vec3(u, v, 0)
plano_xy = ParametricPlot3D(plano_xy_par, (-4, 4, 20), (-4, 4, 20))
plano_xy.setDiffuseColor(_1(200, 200, 200))
plano_xy.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
plano_xy.setTransparency(0.85)
self.addChild(plano_xy)
self.addChild(Line([(-4, 0, 0), (4, 0, 0)], color=(0.8, 0.8, 0.5)))
self.addChild(Line([(0, -4, 0), (0, 4, 0)], color=(0.8, 0.8, 0.5)))
self.setupAnimations(
[AnimationGroup([tangente, normal], (10000, 0, len(espiral) - 1))])
def __init__(self):
Page.__init__(self, u"Otro campo en el toro sin singularidades")
a = 1
b = 0.5
def toroParam1(u, v):
return ((a + b * cos(v)) * cos(u), (a + b * cos(v)) * sin(u),
b * sin(v))
def toro_u(u, v):
return Vec3(-(a + b * cos(v)) * sin(u), (a + b * cos(v)) * cos(u),
0)
def toro_v(u, v):
return Vec3(-b * sin(v) * cos(u), -b * sin(v) * sin(u), b * cos(v))
parab = ParametricPlot3D(toroParam1, (0, 2 * pi, 150),
(0, 2 * pi, 100))
parab.setTransparency(0.4)
parab.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
def make_curva(c):
return lambda t: toroParam1(t, c)
def make_tang(c):
return lambda t: toro_u(t, c)
tangentes = []
ncurves = 50
for c in range(0, ncurves + 1):
## -1 < ct < 1
ct = c / float(ncurves) * 2 * pi
curva = Curve3D(make_curva(ct), (0, 2 * pi, 100), width=1)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
self.addChild(curva)
def animaTangentes(n):
for tang in tangentes:
tang.animateArrow(n)
a1 = Animation(animaTangentes, (6000, 0, 99))
self.setupAnimations([a1])
def __init__(self):
Page.__init__(self, u"Cilindro<br><br>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> = 1")
param = lambda u,t: Vec3(cos(u),sin(u),t)
cilindro = ParametricPlot3D(param, (0, 2*pi), (-1,1))
col = _1(177,89,77)
cilindro.setAmbientColor(col).setDiffuseColor(col).setSpecularColor(col)
def par1(u,t): return Vec3(-sin(u),cos(u),0)
def par2(u,t): return Vec3(0,0,1)
tp = TangentPlane2(param,par1,par2,(0,0),_1(252,250,225))
tp.localOriginSphere.hide()
tp.localYAxis.setColor(col).setWidth(2).show()
Slider(rangep=('u', 0,2*pi,0,20),func=tp.setU, duration=8000, parent=self)
Slider(rangep=('t', -1,1,0,20),func=tp.setV, duration=4000, parent=self)
self.addChild(cilindro)
self.addChild(tp)
def __init__(self):
Page.__init__(
self,
u"Campo sin singularidades en el plano<br><br>(x,y) → (1,0)")
par_plano = lambda u, v: Vec3(u, v, 0)
def plano_u(u, v):
return Vec3(1, 0, 0)
def plano_v(u, v):
return Vec3(0, 1, 0)
parab = ParametricPlot3D(par_plano, (-1, 1, 20), (-1, 1, 20))
parab.setTransparency(0.4)
parab.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
def make_curva(c):
return lambda t: par_plano(t, c)
def make_tang(c):
return lambda t: plano_u(t, c)
tangentes = []
ncurves = 30
steps = 70
for c in range(0, ncurves + 1):
## -1 < ct < 1
ct = c / float(ncurves) * 2 - 1
curva = Curve3D(make_curva(ct), (-1, 1, steps), width=1)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
self.addChild(curva)
def animaTangentes(n):
for tang in tangentes:
tang.animateArrow(n)
a1 = Animation(animaTangentes, (6000, 0, steps - 1))
self.setupAnimations([a1])
def __init__(self):
Page.__init__(
self,
u"Otro campo en el paraboloide hiperbólico sin singularidades<br><br>(x,y) → (0, 1, x)"
)
par_parab = lambda x, y: Vec3(x, y, x * y)
par_tang = lambda x, y: Vec3(0, 1, x)
parab = ParametricPlot3D(par_parab, (-1, 1), (-1, 1))
parab.setTransparency(0.4)
parab.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
def make_curva(c):
return partial(par_parab, c)
def make_tang(c):
return partial(par_tang, c)
tangentes = []
for c in range(0, 21):
## -1 < ct < 1
ct = 2 * c / 20.0 - 1
curva = Curve3D(make_curva(ct), (-1, 1, 50), width=1)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
self.addChild(curva)
def animaTangentes(n):
for tang in tangentes:
tang.animateArrow(n)
a1 = Animation(animaTangentes, (6000, 0, 49))
self.setupAnimations([a1])
def __init__(self):
Page.__init__(self, u"Campo en la esfera con dos singularidades")
par_esfera = lambda u, v: Vec3(
sin(u) * cos(v),
sin(u) * sin(v), cos(u))
def esfera_u(u, v):
return Vec3(cos(u) * cos(v), cos(u) * sin(v), -sin(u))
def esfera_v(u, v):
return Vec3(-sin(u) * sin(v), cos(v) * sin(u), 0)
parab = ParametricPlot3D(par_esfera, (0, 2, 150), (0, 2 * pi, 100))
parab.setTransparency(0.4)
parab.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
def make_curva(c):
return partial(par_esfera, c)
def make_tang(c):
return partial(esfera_v, c)
tangentes = []
curves = []
ncurves = 70
for c in range(0, ncurves + 1):
## -1 < ct < 1
ct = c / float(ncurves) * pi
curve = Curve3D(make_curva(ct), (0, 2 * pi, 100), width=1)
tangent = curve.attachField(
"tangente",
make_tang(ct)).setLengthFactor(.4).setWidthFactor(.1).show()
tangentes.append(tangent)
curves.append(curve)
self.addChildren(curves)
self.setupAnimations([AnimationGroup(tangentes, (6000, 0, 99))])
def __init__(self):
Page.__init__(
self,
u"Construcción de un vector del campo de Morse sobre el toro")
def coreTorusAt(p):
dyz = sqrt(p[1]**2 + p[2]**2)
return Vec3(0.0, a * p[1] / dyz, a * p[2] / dyz)
def unitNormalToTorusAt(p):
core = coreTorusAt(p)
p_core = p - core
dp_core = p_core.length()
return p_core / dp_core
def projAtTorus(p):
core = coreTorusAt(p)
p_core = p - core
factor = 1.005 * b / p_core.length(
) #un poco más de 1 para que se vea mejor...
return core + factor * p_core
def valMorseFieldAt(p):
n = unitNormalToTorusAt(p)
gdotn = -g * n[2]
return Vec3(gdotn * n[0], gdotn * n[1], g + gdotn * n[2])
def nextPoint(p, dt):
return projAtTorus(p + dt * valMorseFieldAt(p))
class CurveVectorField:
def __init__(self, c):
self.curve = c
def basePoint(self, t):
return self.curve[int(t)]
def endPoint(self, t):
return self.curve[int(t)] + valMorseFieldAt(self.curve[int(t)])
class CurveNormalField:
def __init__(self, c):
self.curve = c
def basePoint(self, t):
return self.curve[int(t)]
def endPoint(self, t):
return self.curve[int(t)] + unitNormalToTorusAt(
self.curve[int(t)])
class CurveGravityField:
def __init__(self, c):
self.curve = c
def basePoint(self, t):
return self.curve[int(t)]
def endPoint(self, t):
return self.curve[int(t)] + Vec3(0, 0, g)
curves = []
vectorial_fields_curves = []
vectorial_fields_curves_bk = []
dtheta = pi / 10.0
nrot = -4
points_down_curve = []
points_up_curve = []
q = Vec3(b * cos(nrot * dtheta), a + b * sin(nrot * dtheta), 0.0)
# calculo empezando enmedio del toro
for n in range(0, 100):
p = projAtTorus(q)
v = valMorseFieldAt(p)
if v.length() < 0.01:
break
points_down_curve.append(p)
points_up_curve.append(Vec3(p[0], p[1], -p[2]))
q = nextPoint(p, 0.05)
#Tangent Plane
p = projAtTorus(q)
p[2] = -p[2]
v = valMorseFieldAt(p)
u = v.cross(unitNormalToTorusAt(p))
tangent_plane = Plane(_1(200, 200, 200), p, v + p, u + p)
points_down_curve.reverse() # recorrer de arriba a enmedio
points_down_curve.pop(
) # quitar los puntos de enmedio, repetidos en las listas
points_down_curve.extend(points_up_curve) # unir listas
points_down_curve.reverse()
curve = Line(points_down_curve, width=2.5)
curves.append(curve)
cvf = CurveVectorField(curve)
vectorial_fields_curves_bk.append(cvf)
arrow = AnimatedArrow(cvf.basePoint, cvf.endPoint)
arrow.setDiffuseColor(_1(220, 40, 200))
arrow.setWidthFactor(0.48)
arrow.add_tail(0.025)
vectorial_fields_curves.append(arrow)
cnf = CurveNormalField(curve)
vectorial_fields_curves_bk.append(cnf)
arrown = AnimatedArrow(cnf.basePoint, cnf.endPoint)
arrown.setDiffuseColor(_1(220, 240, 20))
arrown.setWidthFactor(0.4)
#arrown.add_tail( 0.025 )
vectorial_fields_curves.append(arrown)
cgf = CurveGravityField(curve)
vectorial_fields_curves_bk.append(cgf)
arrowg = AnimatedArrow(cgf.basePoint, cgf.endPoint)
arrowg.setDiffuseColor(_1(10, 240, 20))
arrowg.setWidthFactor(0.4)
#arrowg.add_tail( 0.025 )
vectorial_fields_curves.append(arrowg)
self.addChildren(curves)
self.addChildren(vectorial_fields_curves)
self.addChild(tangent_plane)
def setSyncParam(t):
for i in range(0, len(vectorial_fields_curves)):
#curve = curves[i]
if t < len(curves[0].getPoints()):
vec_field = vectorial_fields_curves[i]
vec_field.animateArrow(t)
q = (curves[0])[int(t)]
p = projAtTorus(q)
v = valMorseFieldAt(p)
u = v.cross(unitNormalToTorusAt(p))
tangent_plane.setPoints(p, v + p, u + p)
Slider(rangep=('t', 0, 198, 1, 199),
func=setSyncParam,
duration=8000,
parent=self)
# T(u,v)
def toroParam1(u, v):
return (b * sin(u), (a + b * cos(u)) * cos(v),
(a + b * cos(u)) * sin(v))
def toroParam(u, v):
return Vec3(b * sin(u), (a + b * cos(u)) * cos(v),
(a + b * cos(u)) * sin(v))
paratoro = ParametricPlot3D(toroParam1, (0, 2 * pi, 150),
(0, 2 * pi, 100))
paratoro.setTransparency(0.25)
paratoro.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
paratoro.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
paratoro.setDiffuseColor(_1(68, 28, 119))
self.addChild(paratoro)
def __init__(self):
Page.__init__(self, u"Campo de Morse sobre el toro")
a = 2.0
b = 1.0
g = -1.25
# T(u,v)
def toroParam1(u, v):
return (b * sin(u), (a + b * cos(u)) * cos(v),
(a + b * cos(u)) * sin(v))
def toroNormal(u, v):
coef = b * (a + b * cos(u))
return Vec3(coef * sin(u),
coef * cos(u) * cos(v),
coef * cos(u) * sin(v))
def toroMorse(u, v):
#coef = -b * ( a + b * cos(u) )
coef2 = -g * cos(u) * sin(v)
return Vec3(coef2 * sin(u),
coef2 * cos(u) * cos(v), g + coef2 * cos(u) * sin(v))
paratoro = ParametricPlot3D(toroParam1, (0, 2 * pi, 150),
(0, 2 * pi, 100))
paratoro.setTransparency(0.25)
paratoro.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
paratoro.setDiffuseColor(_1(68, 28, 119))
self.addChild(paratoro)
def make_curva(c):
return lambda t: toroParam1(c, t)
def make_curva2(c):
return lambda t: toroParam1(c, -t)
def make_tang(c):
return lambda t: toroMorse(c, t)
def make_tang2(c):
return lambda t: toroMorse(c, -t)
tangentes = []
tangentes2 = []
ncurves = 12
for c in range(0, ncurves + 1):
## -1 < ct < 1
ct = c / float(ncurves) * 2 * pi
#curva = Curve3D(make_curva(ct),(-pi/2,pi/2,100), width=0.5)
curva = Curve3D(make_curva(ct), (pi / 2, 3 * pi / 2, 100),
width=0.5)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(1).setWidthFactor(.5)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
###
ct2 = c / float(ncurves) * 2 * pi
#curva2 = Curve3D(make_curva2(ct2),(pi/2,3*pi/2,100), width=0.5)
curva2 = Curve3D(make_curva2(ct2), (-pi / 2, pi / 2, 100),
width=0.5)
curva2.attachField(
"tangente",
make_tang2(ct2)).setLengthFactor(1).setWidthFactor(.5)
curva2.fields['tangente'].show()
tangentes2.append(curva2.fields['tangente'])
self.addChild(curva)
self.addChild(curva2)
def animaTangentes(n):
for tang in tangentes + tangentes2:
tang.animateArrow(int(n))
a1 = Animation(animaTangentes, (6000, 0, 99), times=1)
self.setupAnimations([a1])
Slider(rangep=('u', 0, 99, 0, 100), func=animaTangentes, parent=self)
import sys
from math import pi, sin, cos
from PyQt4 import QtGui
from superficie.plots import ParametricPlot3D
from superficie.viewer import MinimalViewer
app = QtGui.QApplication(sys.argv)
viewer = MinimalViewer()
a = 1
b = 0.5
c = .505
def torus(u, v):
return (a + b * cos(v)) * cos(u), (a + b * cos(v)) * sin(u), b * sin(v)
plot = ParametricPlot3D(torus, (0, 2 * pi, 150), (0, 2 * pi, 100))
viewer.addChild(plot).viewAll().resize(400, 400)
viewer.show()
sys.exit(app.exec_())
def __init__(self):
Page.__init__(
self,
u"Campo en el paraboloide hiperbólico con una singularidad<br><br>(x,y) → (0, 1, -k/x<sup>2</sup>)"
)
par_parab = lambda x, y: Vec3(x, y, x * y)
par_tang = lambda x, y: Vec3(0, 1, x)
parab = ParametricPlot3D(par_parab, (-1, 1), (-1, 1))
parab.setTransparency(0.4)
parab.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
def make_curva(c):
#return partial(par_parab,c)
return lambda x: Vec3(x, c / x, c * 1.01)
def make_curva_negy(c):
#return partial(par_parab,c)
return lambda x: Vec3(x, -c / x, -c * 0.99)
def make_tang(c):
#return partial(par_tang,c)
return lambda x: Vec3(x, -c / (x**2), 0.0) / (sqrt(x**2 + c**2 /
(x**4)))
def make_tang_negy(c):
#return partial(par_tang,c)
return lambda x: Vec3(x, c / (x**2), 0.0) / (sqrt(x**2 + c**2 /
(x**4)))
tangentes = []
for c in range(1, 10):
## 0 < ct < 1
ct = c / 10.0
curva = Curve3D(make_curva(ct), (ct, 1.0, 50), width=1.5)
curva.attachField(
"tangente",
make_tang(ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente'].show()
tangentes.append(curva.fields['tangente'])
self.addChild(curva)
curva = Curve3D(make_curva_negy(ct), (ct, 1.0, 50), width=1.5)
curva.attachField(
"tangente_negy",
make_tang_negy(ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente_negy'].show()
tangentes.append(curva.fields['tangente_negy'])
self.addChild(curva)
#ct = -1.0 + c/10.0
curva = Curve3D(make_curva(ct), (-ct, -1.0, 50), width=1.5)
curva.attachField(
"tangente2",
make_tang(-ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente2'].show()
tangentes.append(curva.fields['tangente2'])
self.addChild(curva)
curva = Curve3D(make_curva_negy(ct), (-ct, -1.0, 50), width=1.5)
curva.attachField(
"tangente_negy2",
make_tang_negy(-ct)).setLengthFactor(.4).setWidthFactor(.1)
curva.fields['tangente_negy2'].show()
tangentes.append(curva.fields['tangente_negy2'])
self.addChild(curva)
def animaTangentes(n):
for tang in tangentes:
tang.animateArrow(n)
a1 = Animation(animaTangentes, (5000, 0, 49))
self.setupAnimations([a1])
self.addChild(
Line([(-1, 0, 0.01), (1, 0, 0.01)], color=(1, 1, 1)).setWidth(1.5))
self.addChild(
Line([(0, -1, 0.01), (0, 1, 0.01)], color=(1, 1, 1)).setWidth(1.5))
def __init__(self):
super(Hiperboloide,self).__init__(u'Secciones normales de un paraboloide hiperbólico<br><br>x<sup>2</sup>/4 - y<sup>2</sup>/9 = z')
self.showAxis(False)
hyperboloid = createHyperboloid(2,3)
self.addChild(hyperboloid)
normal = Arrow((0,0,0), (0,0,1), 0.03)
self.addChild(normal)
class Parabole(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
return Vec3(cos(pi_2*self.param)*s, sin(pi_2*self.param)*s, (cos(pi_2*self.param)*s)**2/4 - (sin(pi_2*self.param)*s)**2/9)
def setParam(self, t):
self.param = t
parabole_obj = Parabole()
curve = Curve3DParam(parabole_obj, (-4.0, 4.0, 200), color=(0.9, 0.2, 0.1), width=6)
normal_plane_function = lambda u, v: (cos(pi_2*th)*v, sin(pi_2*th)*v, u)
normal_plane_function.func_globals['th']=0.0
normal_plane = ParametricPlot3D(normal_plane_function, (-4.1, 4.1), (-4.1, 4.1))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
#normal_plane.animation = normal_plane.parameters['th'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(0,0,0)
def endTangentPoint(t):
s = pi_2*t #/1000.0
return Vec3(cos(s), sin(s), 0)
def endCurvaturePoint(t):
# ||curve'(0)||**2
s = pi_2*t #/1000.0
# vn = 1.0
# ||curve''(0)||
nn = ((cos(s))**2)/2 - 2*((sin(s))**2)/9
return Vec3(0, 0, 2.0*nn)
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
curvature_arrow = AnimatedArrow(basePoint, endCurvaturePoint)
curvature_arrow.setDiffuseColor(_1(220,200,20))
objects = [curve, normal_plane, tangent_arrow, curvature_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['th']=t
normal_plane.updateAll()
curve.setParam(t)
tangent_arrow.animateArrow(t)
curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,1.0,0,20),func=setSyncParam, duration=4000, parent=self)
def __init__(self):
Page.__init__(self, u"Toro")
a = 1
b = 0.5
def toroParam1(u, v):
return ((a + b * cos(v)) * cos(u), (a + b * cos(v)) * sin(u),
b * sin(v))
toro = ParametricPlot3D(toroParam1, (0, 2 * pi, 150), (0, 2 * pi, 100))
toro.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
toro.setTransparency(.4)
# delta = 0
# p_eli = Sphere((.9571067805, .9571067805, .35+delta),0.02,visible=True)
# p_eli.setColor( _1(194,38,69))
# p_eli.setShininess(1)
#
# p_par = Sphere ((-0.7071067810, 0.7071067810, 0.5+delta),0.02,visible=True)
# p_par.setColor( _1(240,108,21))
# p_par.setShininess(1)
#
# p_hyp = Sphere ((0, -0.6464466095, .3535+delta),0.02,visible=True)
# p_hyp.setColor( _1(78,186,69))
# p_hyp.setShininess(1)
def toro_u(u, v):
return Vec3(-(a + b * cos(v)) * sin(u), (a + b * cos(v)) * cos(u),
0)
def toro_v(u, v):
return Vec3(-b * sin(v) * cos(u), -b * sin(v) * sin(u), b * cos(v))
## plano parabólico
ptopar = (0, pi / 2)
plane_par = TangentPlane2(toroParam1, toro_u, toro_v, ptopar,
_1(252, 250, 225))
plane_par.baseplane.setTransparency(0)
def curvaPlana(t):
return plane_par.planeParam(cos(t), sin(t) + 1)
curva = Curve3D(curvaPlana, (-pi, 0, 30), color=(1, 0, 0), width=2)
self.addChild(toro)
self.addChild(plane_par)
self.addChild(curva)
def animaCurva1(n):
def curva(t):
return (t * 2 * pi, pi / 2)
plane_par.setLocalOrigin(curva(n / 100.))
def animaCurva2(n):
def curva(t):
return (0, pi / 2 - t * (2 * pi + pi / 2))
plane_par.setLocalOrigin(curva(n / 100.))
def animaCurva3(n):
def curva(t):
return (t * 2 * pi, 0)
plane_par.setLocalOrigin(curva(n / 100.))
a1 = Animation(animaCurva1, (6000, 0, 100))
a2 = Animation(animaCurva2, (6000, 0, 100))
a3 = Animation(animaCurva3, (6000, 0, 100))
self.setupAnimations([a1, a2, a3])
def __init__(self):
"""x^2 - y^2 - z = 0"""
Page.__init__(self, u"Paraboloide hiperbólico<br><br>x<sup>2</sup>/a<sup>2</sup> - y<sup>2</sup>/b<sup>2</sup> = z")
z = 1.5
def fn(x, y):
return x ** 2 - y ** 2 + z
def polar(function):
def polar_fn(r, t):
x = r * cos(t)
y = r * sin(t)
return x, y, function(x, y)
return polar_fn
paraboloid = ParametricPlot3D(polar(fn), (.001, 1, 20), (0, 2 * pi, 60))
paraboloid. \
setAmbientColor(_1(145, 61, 74)). \
setDiffuseColor(_1(127, 119, 20)). \
setSpecularColor(_1(145, 61, 74))
base_plane = BasePlane()
base_plane.setHeight(0)
base_plane.setRange((-2, 2, 7))
## the hiperbolic paraboloid in parametric form
def fn_par(x,y): return Vec3(x, y, x ** 2 - y ** 2 + z)
## its derivatives
def fn_x(x,y): return Vec3(1, 0, 2 * x)
def fn_y(x,y): return Vec3(0, 1, -2 * y)
tangent_plane = TangentPlane2(fn_par, fn_x, fn_y, (0, 0), _1(252, 250, 225))
tangent_plane.setRange((-1.2, 1.2, 7))
self.addChild(paraboloid)
self.addChild(base_plane)
self.addChild(tangent_plane)
def spiral(t):
c = t / (2 * pi)
t2 = t * 2
return c * cos(t2), c * sin(t2)
animate_points = 200
def animate_plane(n):
tangent_plane.setLocalOrigin(spiral(2 * pi * n / float(animate_points)))
def animate_plane_2(t):
tangent_plane.setLocalOrigin(spiral(t))
a1 = Animation(animate_plane, (10000, 0, animate_points))
Slider(('t', 0, 2 * pi, 0, animate_points), animate_plane_2, duration=10000, parent=self)
self.setupAnimations([a1])
def __init__(self):
super(Toro3,self).__init__(u'Secciones normales de un toro en un punto hiperbólico')
self.showAxis(False)
torus = createTorus(r1, r2)
self.addChild(torus)
normal = Arrow((r1-r2,0,0), (r1-r2-1,0,0), 0.03)
self.addChild(normal)
class TorusCurve(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
param1 = pi_2*self.param #/1000.0
cosp = cos(param1)
sinp = sin(param1)
v = s
btor = 2.0*v**2 - 20.0
ctor = v**4 + (16.0 - 36.0*sinp**2)*v**2 + 64.0
dis = btor**2 - 4.0*ctor
if dis < 0.0:
#v = 0.0
u = 4.0
else:
inside = -btor-sqrt(dis)
if inside < 0.0:
#v = 0.0
u = 4.0
else:
u = sqrt( inside/2.0 )
return Vec3(u, sinp*v, -cosp*v)
def setParam(self, t):
self.param = t
torusc_obj = TorusCurve()
curve = Curve3DParam(torusc_obj, (-2.0, 2.0, 200), color=(0.9, 0.2, 0.1), width=6)
curve.setBoundingBox((-0.05,2.95),(-4.1,4.1),(-1.1,1.1))
normal_plane_function = lambda u, v: (u, sin(pi_2*tt3)*v, -cos(pi_2*tt3)*v)
normal_plane_function.func_globals['tt3']=t
normal_plane = ParametricPlot3D(normal_plane_function, (-4.1, 4.1), (-4.1, 4.1))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
#normal_plane.animation = normal_plane.parameters['tt3'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(r1-r2,0,0)
def endTangentPoint(t):
# ||curve'(0)||
s = pi_2*t #/1000.0
#vn = sqrt((b*sin(s))**2 + (c*cos(s))**2)
return Vec3(r1-r2, -sin(s), cos(s))
def endCurvaturePoint(t):
# ||curve'(0)||**2
#s = pi_2*t/1000.0
s = t #/1000.0
#vn = (b*sin(s))**2 + (c*cos(s))**2
# ||curve''(0)||
# nn = a
return Vec3(r1-2*s*r2, 0, 0)
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
curvature_arrow = AnimatedArrow(basePoint, endCurvaturePoint)
curvature_arrow.setDiffuseColor(_1(220,200,20))
objects = [curve, normal_plane, tangent_arrow, curvature_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['tt3']=t
normal_plane.updateAll()
curve.setParam(t)
tangent_arrow.animateArrow(t)
curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,1.0,0,40),func=setSyncParam, duration=4000, parent=self)
def __init__(self):
super(Elipsoide2,self).__init__('Secciones normales de un elipsoide en el punto (0,0,1)<br><br>x<sup>2</sup>/9 + y<sup>2</sup>/4 + z<sup>2</sup> = 1')
self.showAxis(False)
ellipsoid = createEllipsoid(a, b, c)
self.addChild(ellipsoid)
normal = Arrow((0,0,c), (0,0,c+1), 0.03)
self.addChild(normal)
class Ellipse(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
param1 = pi_2*self.param #/1000.0
return Vec3(a*cos(param1)*sin(s), b*sin(param1)*sin(s), c*cos(s))
def setParam(self, t):
self.param = t
ellipse_obj = Ellipse()
curve = Curve3DParam(ellipse_obj, (-3.14, 3.14, 200), color=(0.9, 0.2, 0.1), width=6)
normal_plane_function = lambda u, v: (a*cos(pi_2*t2)*v, b*sin(pi_2*t2)*v, c*u)
normal_plane_function.func_globals['t2']=0.0
normal_plane = ParametricPlot3D(normal_plane_function, (-1.1, 1.1), (-1.1, 1.1))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
#normal_plane.animation = normal_plane.parameters['t2'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(0,0,c)
def endTangentPoint(t):
# ||curve'(0)||
s = pi_2*t #/1000.0
vn = sqrt((a*cos(s))**2 + (b*sin(s))**2)
return Vec3(a*cos(s)/vn, b*sin(s)/vn, c)
def endCurvaturePoint(t):
# ||curve'(0)||**2
s = pi_2*t #/1000.0
vn = (a*cos(s))**2 + (b*sin(s))**2
# ||curve''(0)||
# nn = c
return Vec3(0, 0, c-2.0*c/vn)
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
curvature_arrow = AnimatedArrow(basePoint, endCurvaturePoint)
curvature_arrow.setDiffuseColor(_1(220,200,20))
objects = [curve, normal_plane, tangent_arrow, curvature_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['t2']=t
normal_plane.updateAll()
curve.setParam(t)
tangent_arrow.animateArrow(t)
curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,1,0,20),func=setSyncParam, duration=4000, parent=self)
def __init__(self):
Page.__init__(self, u"Toro<br><br>x<sup>4</sup> + y<sup>4</sup> + z<sup>4</sup><br> + 2x<sup>2</sup>y<sup>2</sup> + 2y<sup>2</sup>z<sup>2</sup> + 2z<sup>2</sup>x<sup>2</sup><br> - 10x<sup>2</sup> - 10y<sup>2</sup> + 6z<sup>2</sup> + 9 = 0")
a = 1
b = 0.5
def toroParam1(u, v):
return (a + b * cos(v)) * cos(u), (a + b * cos(v)) * sin(u), b * sin(v)
toro = ParametricPlot3D(toroParam1, (0, 2 * pi, 150), (0, 2 * pi, 100))
toro.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
toro.setTransparency(.4)
# delta = 0
# p_eli = Sphere((.9571067805, .9571067805, .35+delta),0.02,visible=True)
# p_eli.setColor( _1(194,38,69))
# p_eli.setShininess(1)
#
# p_par = Sphere ((-0.7071067810, 0.7071067810, 0.5+delta),0.02,visible=True)
# p_par.setColor( _1(240,108,21))
# p_par.setShininess(1)
#
# p_hyp = Sphere ((0, -0.6464466095, .3535+delta),0.02,visible=True)
# p_hyp.setColor( _1(78,186,69))
# p_hyp.setShininess(1)
def toro_u(u, v):
return Vec3(-(a + b * cos(v)) * sin(u), (a + b * cos(v)) * cos(u), 0)
def toro_v(u, v):
return Vec3(-b * sin(v) * cos(u), -b * sin(v) * sin(u), b * cos(v))
## plano parabólico
ptopar = (0, pi / 2)
b2 = b - .00025
## trick: the tangent plane is located in a torus of diameter slightly smaller than the torus; so the
## intersection is visible to the naked eye
def toroParam_delta(u, v):
return (a + b2 * cos(v)) * cos(u), (a + b2 * cos(v)) * sin(u), b2 * sin(v)
plane_par = TangentPlane2(toroParam_delta, toro_u, toro_v, ptopar, _1(252, 250, 225))
plane_par.baseplane.setTransparency(0)
plane_par.setRange((-.5, .5, 7))
self.addChild(toro)
self.addChild(plane_par)
def animaCurva1(n):
def curva(t): return (t * 2 * pi, pi / 2)
plane_par.setLocalOrigin(curva(n / 100.))
def animaCurva2(n):
def curva(t): return (0, pi / 2 - t * (2 * pi + pi / 2))
plane_par.setLocalOrigin(curva(n / 100.))
def animaCurva3(n):
def curva(t): return (t * 2 * pi, 0)
plane_par.setLocalOrigin(curva(n / 100.))
a1 = Animation(animaCurva1, (6000, 0, 100))
a2 = Animation(animaCurva2, (6000, 0, 100))
a3 = Animation(animaCurva3, (6000, 0, 100))
self.setupAnimations([a1, a2, a3])
def __init__(self):
super(Elipsoide3,self).__init__(u'Secciones normales de un elipsoide un punto umbílico<br><br>x<sup>2</sup>/9 + y<sup>2</sup>/4 + z<sup>2</sup> = 1')
self.showAxis(False)
ellipsoid = createEllipsoid(a, b, c)
self.addChild(ellipsoid)
# umbilic point U = (px, 0, pz)
#px = sqrt((a**2-b**2)/(a**2-c**2))
#pz = sqrt((c**2)*(b**2-c**2)/(a**2-c**2))
pz = sqrt((c**2-b**2)/(c**2-a**2))
px = sqrt((a**2)*(b**2-a**2)/(c**2-a**2))
# gradient in umbilic point U
nx = 2*px/(a**2)
nz = 2*pz/(c**2)
n = sqrt(nx**2+nz**2)
nx = nx/n
nz = nz/n
#print px, pz, px**2/(a**2)+pz**2/(c**2), nx, nz
normal = Arrow((px,0,pz), (px+nx,0,pz+nz), 0.03)
self.addChild(normal)
curvature_arrow = Arrow((px,0,pz), (px-0.5*nx,0,pz-0.5*nz), 0.05)
curvature_arrow.setDiffuseColor(_1(220,200,20))
self.addChild(curvature_arrow)
class Ellipse(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
param1 = pi_2*self.param #/1000.0
cosp = cos(param1)
sinp = sin(param1)
A = (nx**2)/9.0 + nz**2
C = ((nz**2)/9.0 + nx**2)*cosp**2+(sinp**2)/4.0
B = (16.0/9.0)*nx*nz*cosp
D = 2.0*(nx*px/9.0 + nz*pz)
E = 2.0*(nx*pz - nz*px/9.0)*cosp
#F = 0
ins_sqrt = A**2 + C**2 + B**2 - 2.0*A*C
L1 = (A + C + sqrt(ins_sqrt) )/2.0
L2 = (A + C - sqrt(ins_sqrt) )/2.0
dis = 4.0*A*C-B**2
C1 = (B*E-2.0*C*D)/dis
C2 = (D*B-2.0*A*E)/dis
detq = (A*(E**2) + C*(D**2) - B*D*E) #/4.0
aell = sqrt(dis/(L1*detq))/(1.0+sinp*sinp)
bell = sqrt(dis/(L2*detq))/(1.0+sinp*sinp)
theta = atan( B/(A-C) )/2.0
u = aell*cos(s)*cos(theta) - bell*sin(s)*sin(theta) + C1
v = bell*sin(s)*cos(theta) + aell*cos(s)*sin(theta) + C2
return Vec3(px+u*nx-cosp*v*nz, sinp*v, pz+u*nz+cosp*v*nx)
def setParam(self, t):
self.param = t
ellipse_obj = Ellipse()
curve = Curve3DParam(ellipse_obj, (-3.14, 3.14, 200), color=(0.9, 0.2, 0.1), width=6)
# Rotation Z_Axis=(0,0,1) -> n=(nx,0,nz) (||n||=1)
# Rot(x,y,z)=(x*nz+z*nx,y,-x*nx+z*nz)
normal_plane_function = lambda u, v: (px+u*nx-cos(pi_2*t3)*v*nz, sin(pi_2*t3)*v, pz+u*nz+cos(pi_2*t3)*v*nx)
normal_plane_function.func_globals['t3']=0.0
normal_plane = ParametricPlot3D(normal_plane_function, (-3.3, 0.1), (-1.9, 4.9))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
normal_plane.setBoundingBox((-3.5,3.5),(-2.1,2.1),(-1.5,1.5))
#normal_plane.animation = normal_plane.parameters['t3'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(px,0,pz)
def endTangentPoint(t):
# ||curve'(0)||
s = pi_2*t #/1000.0
vn = sqrt((nz*cos(s))**2 + (sin(s))**2 + (nx*cos(s))**2)
return Vec3(px - nz*cos(s)/vn, sin(s)/vn, pz + nx*cos(s)/vn)
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
objects = [curve, normal_plane, tangent_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['t3']=t
normal_plane.updateAll()
curve.setParam(t)
tangent_arrow.animateArrow(t)
#curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,1,0,20),func=setSyncParam, duration=4000, parent=self)
def __init__(self):
Page.__init__(self, u"Campo de Morse sobre el toro")
def coreTorusAt(p):
dyz = sqrt(p[1]**2 + p[2]**2)
return Vec3(0.0, a * p[1] / dyz, a * p[2] / dyz)
def unitNormalToTorusAt(p):
core = coreTorusAt(p)
p_core = p - core
dp_core = p_core.length()
return p_core / dp_core
def projAtTorus(p):
core = coreTorusAt(p)
p_core = p - core
factor = 1.01 * b / p_core.length(
) #un poco más de 1 para que se vea mejor...
return core + factor * p_core
def valMorseFieldAt(p):
n = unitNormalToTorusAt(p)
gdotn = -g * n[2]
return Vec3(gdotn * n[0], gdotn * n[1], g + gdotn * n[2])
def nextPoint(p, dt):
return projAtTorus(p + dt * valMorseFieldAt(p))
class CurveVectorField:
def __init__(self, c):
self.curve = c
def basePoint(self, t):
return self.curve[int(t)]
def endPoint(self, t):
return self.curve[int(t)] + valMorseFieldAt(self.curve[int(t)])
curves = []
vectorial_fields_curves = []
vectorial_fields_curves_bk = []
dtheta = 2.0 * pi / 20.0
for nrot in range(0, 20):
points_down_curve = []
points_up_curve = []
q = Vec3(b * cos(nrot * dtheta), a + b * sin(nrot * dtheta), 0.0)
# calculo empezando enmedio del toro
for n in range(0, 100):
p = projAtTorus(q)
v = valMorseFieldAt(p)
if v.length() < 0.01:
break
points_down_curve.append(p)
points_up_curve.append(Vec3(p[0], p[1], -p[2]))
q = nextPoint(p, 0.05)
points_down_curve.reverse() # recorrer de arriba a enmedio
points_down_curve.pop(
) # quitar los puntos de enmedio, repetidos en las listas
points_down_curve.extend(points_up_curve) # unir listas
points_down_curve.reverse()
curve = Line(points_down_curve, width=2.5)
curves.append(curve)
cvf = CurveVectorField(curve)
vectorial_fields_curves_bk.append(cvf)
arrow = AnimatedArrow(cvf.basePoint, cvf.endPoint)
arrow.setDiffuseColor(_1(220, 40, 20))
arrow.setWidthFactor(0.25)
arrow.add_tail(0.025)
vectorial_fields_curves.append(arrow)
# la otra mitad del toro... reflejando por el eje Z
points_reflected_curve = []
for p in points_down_curve:
points_reflected_curve.append(Vec3(-p[0], -p[1], p[2]))
curveR = Line(points_reflected_curve, width=2.5)
curves.append(curveR)
cvf = CurveVectorField(curveR)
vectorial_fields_curves_bk.append(cvf)
arrow = AnimatedArrow(cvf.basePoint, cvf.endPoint)
arrow.setDiffuseColor(_1(220, 40, 20))
arrow.setWidthFactor(0.25)
arrow.add_tail(0.025)
vectorial_fields_curves.append(arrow)
# paralelos hasta arriba
points_curve1 = []
q = Vec3(0.25, 0.0, a + b)
for n in range(0, 100):
p = projAtTorus(q)
v = valMorseFieldAt(p)
if v.length() < 0.01:
break
points_curve1.append(p)
q = nextPoint(p, 0.05)
curve1 = Line(points_curve1, width=2.5)
curves.append(curve1)
cvf = CurveVectorField(curve1)
vectorial_fields_curves_bk.append(cvf)
arrow = AnimatedArrow(cvf.basePoint, cvf.endPoint)
arrow.setDiffuseColor(_1(220, 40, 20))
arrow.setWidthFactor(0.25)
arrow.add_tail(0.025)
vectorial_fields_curves.append(arrow)
points_curve2 = []
q = Vec3(-0.25, 0.0, a + b)
for n in range(0, 100):
p = projAtTorus(q)
v = valMorseFieldAt(p)
if v.length() < 0.01:
break
points_curve2.append(p)
q = nextPoint(p, 0.05)
curve2 = Line(points_curve2, width=2.5)
curves.append(curve2)
cvf = CurveVectorField(curve2)
vectorial_fields_curves_bk.append(cvf)
arrow = AnimatedArrow(cvf.basePoint, cvf.endPoint)
arrow.setDiffuseColor(_1(220, 40, 20))
arrow.setWidthFactor(0.25)
arrow.add_tail(0.025)
vectorial_fields_curves.append(arrow)
self.addChildren(curves)
self.addChildren(vectorial_fields_curves)
def setSyncParam(t):
for i in range(0, len(vectorial_fields_curves)):
curve = curves[i]
if t < len(curve.getPoints()):
vec_field = vectorial_fields_curves[i]
#vec_field.animateArrow(int(t))
vec_field.animateArrow(t)
Slider(rangep=('t', 0, 198, 1, 199),
func=setSyncParam,
duration=16000,
parent=self)
# T(u,v)
def toroParam1(u, v):
return (b * sin(u), (a + b * cos(u)) * cos(v),
(a + b * cos(u)) * sin(v))
def toroParam(u, v):
return Vec3(b * sin(u), (a + b * cos(u)) * cos(v),
(a + b * cos(u)) * sin(v))
paratoro = ParametricPlot3D(toroParam1, (0, 2 * pi, 150),
(0, 2 * pi, 100))
paratoro.setTransparency(0.25)
paratoro.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
paratoro.setDiffuseColor(_1(68, 28, 119))
self.addChild(paratoro)
critic1 = Sphere(center=Vec3(0, 0, a + b),
radius=0.075,
color=_1(240, 10, 20))
critic2 = Sphere(center=Vec3(0, 0, a - b),
radius=0.075,
color=_1(240, 10, 20))
critic3 = Sphere(center=Vec3(0, 0, -a + b),
radius=0.075,
color=_1(240, 10, 20))
critic4 = Sphere(center=Vec3(0, 0, -a - b),
radius=0.075,
color=_1(240, 10, 20))
self.addChild(critic1)
self.addChild(critic2)
self.addChild(critic3)
self.addChild(critic4)
def __init__(self):
super(Toro1,self).__init__(u'Secciones normales de un toro en un punto elíptico')
self.showAxis(False)
torus = createTorus(r1, r2)
self.addChild(torus)
normal = Arrow((r1+r2,0,0), (r1+r2+1,0,0), 0.03)
self.addChild(normal)
class TorusCurve(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
param1 = pi_2*self.param #/1000.0
cosp = cos(param1)
sinp = sin(param1)
u = s #*rot
btor = 2.0*u**2 + 16.0 - 36.0*sinp**2
v = sqrt( (-btor+sqrt(btor**2 - 4.0*(u**4-20.0*u**2+64.0)) )/2.0 )
return Vec3(u, sinp*v, cosp*v)
def setParam(self, t):
self.param = t
torusc_obj = TorusCurve()
curve = Curve3DParam(torusc_obj, (2.0, 4.0, 200), color=(0.9, 0.2, 0.1), width=6)
class TorusCurve2(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
param1 = pi_2*self.param #/1000.0
cosp = cos(param1)
sinp = sin(param1)
u = s #*rot
btor = 2.0*u**2 + 16.0 - 36.0*sinp**2
v = -sqrt( (-btor+sqrt(btor**2 - 4.0*(u**4-20.0*u**2+64.0)) )/2.0 )
return Vec3(u, sinp*v, cosp*v)
def setParam(self, t):
self.param = t
torusc2_obj = TorusCurve2()
curve2 = Curve3DParam(torusc2_obj, (2.0, 4.0, 200), color=(0.9, 0.2, 0.1), width=6)
normal_plane_function = lambda u, v: (u, sin(pi_2*tt1)*v, cos(pi_2*tt1)*v)
normal_plane_function.func_globals['tt1']=0.0
normal_plane = ParametricPlot3D(normal_plane_function, (-4.1, 4.1), (-4.1, 4.1))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
#normal_plane.animation = normal_plane.parameters['tt1'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(r1+r2,0,0)
def endTangentPoint(t):
# ||curve'(0)||
s = pi_2*t #/1000.0
return Vec3(r1+r2, sin(s), cos(s))
def endCurvaturePoint(t):
# ||curve'(0)||**2
s = t #/1000.0
#vn = (b*sin(s))**2 + (c*cos(s))**2
# ||curve''(0)||
return Vec3(r1+s*(r2-1/r1), 0, 0)
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
curvature_arrow = AnimatedArrow(basePoint, endCurvaturePoint)
curvature_arrow.setDiffuseColor(_1(220,200,20))
objects = [curve, curve2, normal_plane, tangent_arrow, curvature_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['tt1']=t
normal_plane.updateAll()
curve.setParam(t)
curve2.setParam(t)
tangent_arrow.animateArrow(t)
curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,1.0,0,40),func=setSyncParam, duration=4000, parent=self)
def __init__(self):
super(Cilindro,self).__init__('Secciones normales de un cilindro recto circular<br><br>x<sup>2</sup> + z<sup>2</sup> = 1')
self.showAxis(False)
cyl = SoCylinder()
cyl.radius.setValue(1.0)
cyl.height.setValue(4.0)
cyl.parts = SoCylinder.SIDES
light = SoShapeHints()
mat = SoMaterial()
mat.emissiveColor = _1(80, 120, 200)
mat.diffuseColor = _1(80, 120, 200)
mat.transparency.setValue(0.5)
sep = SoSeparator()
sep.addChild(light)
sep.addChild(mat)
sep.addChild(cyl)
self.addChild(sep)
normal = Arrow((0,0,1), (0,0,2), 0.03)
self.addChild(normal)
class CylCurve(object):
def __init__(self, t=0.0):
self.param = t
def __call__(self, s):
#pi_2*self.param/1000.0
return Vec3(cos(s), tan(pi_2*self.param) * cos(s), sin(s))
def setParam(self, t):
self.param = t
cylc_obj = CylCurve()
curve = Curve3DParam(cylc_obj, (-3.14, 3.14, 200), color=(0.9, 0.2, 0.1), width=6)
curve.setBoundingBox((-3.1,3.1),(-3.1,3.1),(-3.1,3.1))
# Ojo! No basta con "acotar", la vista se recalcula con todo el objeto...
normal_plane_function = lambda u, v: (cos(pi_2*tc)*v, sin(pi_2*tc)*v, u)
normal_plane_function.func_globals['tc']=0.0
normal_plane = ParametricPlot3D(normal_plane_function, (-1.1, 1.1), (-2.1, 2.1))
normal_plane.setTransparency(0.75)
normal_plane.setTransparencyType(SoTransparencyType.SCREEN_DOOR)
#normal_plane.animation = normal_plane.parameters['tc'].asAnimation()
VisibleCheckBox("Plano Normal", normal_plane, True, parent=self)
def basePoint(t):
return Vec3(0,0,1)
def endTangentPoint(t):
s = pi_2*t #/1000.0
return Vec3(cos(s), sin(s), 1)
def endCurvaturePoint(t):
return Vec3(0, 0, 0.9-cos(pi_2*t)) #/1000.0
tangent_arrow = AnimatedArrow(basePoint, endTangentPoint)
tangent_arrow.setDiffuseColor(_1(20,10,220))
curvature_arrow = AnimatedArrow(basePoint, endCurvaturePoint)
curvature_arrow.setDiffuseColor(_1(220,200,20))
objects = [curve, normal_plane, tangent_arrow, curvature_arrow]
self.addChildren( objects )
def setSyncParam(t):
normal_plane_function.func_globals['tc']=t
normal_plane.updateAll()
curve.setParam(t)
tangent_arrow.animateArrow(t)
curvature_arrow.animateArrow(t)
Slider(rangep=('t', 0,0.99,0,20),func=setSyncParam, duration=4000, parent=self)