def __init__(self):
"""F(x,y)=x^2 + y^2 - z = 0"""
Page.__init__(
self,
u"Paraboloide Elíptico<br><br>F(x,y)=(x, y, x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup>)"
)
z = 0.5
par = RevolutionPlot3D(lambda r, t: r**2 + z, (0, 1), (0, 2 * pi))
x, y, z2, u, v, cose, sen, t = createVars(
['x', 'y', 'z', 'u', 'v', 'cos', 'sen', 't'])
mesh1 = Plot3D(lambda x, y, h: h * (x**2 + y**2 + z - .01), (-1, 1),
(-1, 1))
mesh1.addEqn(x**2 + y**2 - z2**2 == 1)
mesh1.addFunction(lambda x, y, h: h * (x**2 + y**2 + z + .01))
mesh1.setLinesVisible(True)
mesh1.setMeshVisible(False)
mesh1.setBoundingBox(zrange=(-1, 1.5))
par.setAmbientColor(_1(145, 61, 74))
par.setDiffuseColor(_1(145, 61, 74))
par.setSpecularColor(_1(145, 61, 74))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(par)
self.addChild(mesh1)
self.addChild(baseplane)
def __init__(self):
"""F(x,y)=x^2 + y^2 - z = 0"""
Page.__init__(self, u"Paraboloide Elíptico<br><br>F(x,y)=(x, y, x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup>)")
z = 0.5
par = RevolutionPlot3D(lambda r, t: r ** 2 + z, (0, 1.42), (0, 2 * pi))
x, y, z2, u, v, cose, sen, t = createVars(['x', 'y', 'z', 'u', 'v', 'cos', 'sen', 't'])
mesh1 = Plot3D(lambda x, y, h: h * (x ** 2 + y ** 2 + z - .01), (-1, 1), (-1, 1))
#mesh1.addEqn(x**2+y**2 - z2**2 == 1)
mesh1.addFunction(lambda x, y, h: h * (x ** 2 + y ** 2 + z + .01))
mesh1.setLinesVisible(True)
mesh1.setMeshVisible(False)
mesh1.setBoundingBox(zrange=(-1, 3.0))
par.setAmbientColor(_1(145, 61, 74))
par.setDiffuseColor(_1(145, 61, 74))
par.setSpecularColor(_1(145, 61, 74))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(par)
self.addChild(mesh1)
self.addChild(baseplane)
def __init__(self):
"x^4 + 2x^2y^2 + y^4 -z = 0"
Page.__init__(
self,
u"Superficie cuártica<br><br>F(x,y)=(x,y,x<sup>4</sup>+2x<sup>2</sup>y<sup>2</sup>+y<sup>4</sup>)"
)
# cuart = Plot3D(lambda x,y: x**4 + 2*x**2*y**2 + y**4 + 1, (-1,1),(-1,1))
cuart = RevolutionPlot3D(lambda r, t: r**4 + 1, (0, 1), (0, 2 * pi))
# cuart.setScaleFactor((1,1,.6))
mesh1 = Plot3D(lambda x, y, h: h * (x**4 + 2 * x**2 * y**2 + y**4 + 1),
(-1, 1), (-1, 1))
mesh1.setLinesVisible(True)
mesh1.setMeshVisible(False)
mesh1.setBoundingBox(zrange=(-1, 2))
# cuart.setAmbientColor(_1(168,211,8))
cuart.setDiffuseColor(_1(168, 211, 8))
cuart.setSpecularColor(_1(168, 211, 8))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(cuart)
self.addChild(mesh1)
self.addChild(baseplane)
def __init__(self):
Page.__init__(self, u"Hélice circular, curvatura y torsión<br><br>(cos s/√2, sen s/√2, s/√2)")
self.camera_position = (10, -10, 10)
self.showAxis(False)
tmin = -2 * pi
tmax = 2 * pi
npuntos = 300
self.addChild(Cylinder(_1(185, 46, 61), tmax - tmin, 2))
## ============================================
# 1 implica primer derivada, 2 implica segunda derivada
def param1hc(t):
return 2*Vec3(cos(t), sin(t), t/3.0)
def param2hc(t):
return 2*Vec3(-sin(t), cos(t), 1/3.0)
def param3hc(t):
return 2*Vec3(-cos(t), -sin(t), 0)
def param4hc(t):
return 2*Vec3(sin(t)/3.0, -cos(t)/3.0, 1.0)
espiral = Curve3D(param1hc, (tmin*1.5, tmax*1.5, npuntos), color=_1(255, 255, 255))
tangente = espiral.attachField("tangente", param2hc).setLengthFactor(1).setWidthFactor(.6)
tangente.setRadius( 0.06 )
tangente.setDiffuseColor( _1(20,240,20) )
normal = espiral.attachField("normal", param3hc).setLengthFactor(1).setWidthFactor(.6)
normal.setRadius( 0.06 )
normal.setDiffuseColor( _1(240,120,20) )
binormal = espiral.attachField("binormal", param4hc).setLengthFactor(1).setWidthFactor(.6)
binormal.setRadius( 0.06 )
binormal.setDiffuseColor( _1(20,120,240) )
self.addChild(espiral)
self.setupAnimations([ AnimationGroup([tangente, normal, binormal], (10000,0,len(espiral)-1)) ])
def __init__(self):
Page.__init__(self, u"Otro 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)*sin(u), -cos(u)*sin(u)*sin(v), 1-cos(u)**2)
parab = ParametricPlot3D(par_esfera, (0,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: par_esfera(t,c)
def make_tang(c):
return lambda t: esfera_u(t,c)
tangentes = []
curves = []
ncurves = 70
for c in range(0,ncurves+1):
## -1 < ct < 1
ct = c/float(ncurves) * 2*pi
curve = Curve3D(make_curva(ct),(-(pi-.02),-.02,100), 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):
"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"Planos osculador, normal y rectificante")
tmin = -2 * pi
tmax = 2 * pi
## ============================================
sq2 = 2 ** 0.5
inv_sq2 = (1. / sq2)
def helix(s):
s_times_sq2 = inv_sq2 * s
return Vec3(cos(s_times_sq2), sin(s_times_sq2), s_times_sq2)
def tangent(s):
s_div_sq2 = s / sq2
return Vec3(-inv_sq2 * sin(s_div_sq2), inv_sq2 * cos(s_div_sq2), inv_sq2)
def normal(s):
s_div_sq2 = s / sq2
return Vec3(-cos(s_div_sq2), -sin(s_div_sq2), 0)
def bi_normal(s):
s_div_sq2 = s / sq2
return Vec3(inv_sq2 * sin(s_div_sq2), -inv_sq2 * cos(s_div_sq2), inv_sq2)
curve = Curve3D(helix, (tmin, tmax, 100), _1(206, 75, 150), 2)
self.addChild(curve)
#=======================================================================
# Vectors
#=======================================================================
field_tangent = curve.attachField("tangent", tangent).show()
field_normal = curve.attachField("normal", normal).show()
field_binormal = curve.attachField("binormal", bi_normal).show()
field_tangent.setDiffuseColor( _1(255, 0, 0) )
field_normal.setDiffuseColor( _1(255, 255, 0) )
field_binormal.setDiffuseColor( _1(0, 0, 255) )
#=======================================================================
# Planes
#=======================================================================
def get_points(v1, v2): return v2.p1, v1.p2, v2.p2
color = (.5, .5, .5)
plane_osculating = Plane(color, *get_points(field_tangent, field_normal))
plane_normal = Plane(color, *get_points(field_normal, field_binormal))
plane_rectifying = Plane(color, *get_points(field_binormal, field_tangent))
self.addChildren([plane_osculating, plane_normal, plane_rectifying])
def update_planes(n):
plane_osculating.setPoints(*get_points(field_tangent, field_normal))
plane_normal.setPoints(*get_points(field_normal, field_binormal))
plane_rectifying.setPoints(*get_points(field_binormal, field_tangent))
r = (5000, 0, len(curve) - 1)
animation = Animatable(update_planes, r)
self.setupAnimations([
AnimationGroup([field_tangent, field_normal, field_binormal, animation], r)
])
def __init__(self):
u"""^2 + y^2 = z^2"""
Page.__init__(self, u"Esfera, parametrización por proyecciones estereográficas")
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.setAmbientColor(_1(99,136,63))
esf.setDiffuseColor(_1(99, 136, 63))
esf.setSpecularColor(_1(99, 136, 63))
def proyZm1(u, v, t1):
"""proy desde el polo norte al plano z=-1"""
den = u ** 2 + v ** 2 + 4
x = u - t1 * (u - 4 * u / den)
y = v - t1 * (v - 4 * v / den)
z = -1 - t1 * (-2 + 8 / den)
return (x, y, z)
def proyZ1(u, v, t2):
"""proy desde el polo sur al plano z=1"""
den = u ** 2 + v ** 2 + 4
x = u - t2 * (u - 4 * u / den)
y = v - t2 * (v - 4 * v / den)
z = 1 - t2 * (2 - 8 / den)
return (x, y, z)
stereo = ParametricPlot3D(proyZm1, (-3, 3, 70), (-3, 3, 70))
stereo.setLinesVisible(True)
stereo.setMeshVisible(False)
stereo.setMeshDiffuseColor(_1(117, 55, 79))
stereo2 = ParametricPlot3D(proyZ1, (-3, 3, 70), (-3, 3, 70))
stereo2.setLinesVisible(True)
stereo2.setMeshVisible(False)
stereo2.setMeshDiffuseColor(_1(80, 87, 193))
stereo2.setTransparency(0.5)
stereo2.setTransparencyType(8)
baseplane = BasePlane()
baseplane.setHeight(-1.005)
baseplane.setRange((-4, 4, 7))
self.addChild(esf)
self.addChild(stereo2)
self.addChild(stereo)
self.addChild(baseplane)
params = [stereo,stereo2]
## no queremos los controles
for i,p in enumerate(params):
p.parameters['t%d' % (i+1)].hide()
anims = [p.parameters['t%d' % (i+1)].asAnimation() for i,p in enumerate(params)]
self.setupAnimations(anims)
def __init__(self):
Page.__init__(self, u"Exponencial")
self.showAxis(True)
self.axis_z.setVisible(False)
def curve(t): return Vec3(exp(t) * cos(t), exp(t) * sin(t), exp(t))
def derivada(t): return Vec3(exp(t) * cos(t) - exp(t) * sin(t), exp(t) * cos(t) + exp(t) * sin(t), exp(t))
curva1 = Curve3D(curve, (-pi, 1 * pi, 200), width=2)
self.addChild(curva1)
curva1.derivative = derivada
curva1.tangent_vector.show()
self.setupAnimations([curva1.tangent_vector])
def __init__(self):
Page.__init__(self, u"Campo con un ciclo límite en el plano")
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, (-3,3,20),(-3,3,20))
parab.setTransparency(0.4)
parab.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
parab.setDiffuseColor(_1(68, 28, 119))
self.addChild(parab)
# Esta familia de curvas NO es solucion de un sistema de ecuaciones
# diferenciales de orden 1 (se intersectan)...
# pero se parece a la solucion del sistema presentado
def make_curva(c):
return lambda t: Vec3( e**(-(c*t+c))*cos(t), e**(-(c*t+c))*sin(t), 0.02 )
def make_tang(c):
return lambda t: Vec3( -c*e**(-(c*t+c))*cos(t) - e**(-(c*t+c))*sin(t),
-c*e**(-(c*t+c))*sin(t) + e**(-(c*t+c))*cos(t), 0.02 )
tangentes = []
ncurves = 7
steps = 80
for c in range(0,ncurves):
## -1 < ct < 1
ct = c/20.0 - float(ncurves-1)/40.0
curva = Curve3D(make_curva(ct),(0,2*pi,steps), width=1)
if ct == 0:
curva = Curve3D(make_curva(ct),(0,2*pi,steps), width=1, color=_1(20, 240, 40))
curva.attachField("tangente", make_tang(ct)).setLengthFactor(.5).setWidthFactor(.25).add_tail(0.025)
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])
critic = Sphere( center=Vec3(0,0,0), radius=0.025, color=_1(240,10,20) )
self.addChild(critic)
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):
Page.__init__(self, u"Elipsoide")
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, parent=self)
Slider(rangep=('v', -pi/2,pi/2,0,20),func=tp.setV,parent=self)
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"Curva cúbica alabeada<br><br>α(t)=(t,t<sup>2</sup>,t<sup>3</sup>)")
self.camera_position = (5, 5, 5)
self.camera_viewAll = True
self.setupPlanes()
c = lambda t: Vec3(t, t ** 2, t ** 3)
altura = -1
curva = Curve3D(c, (-1, 1, 100), width=5, nvertices=1)
lyz = curva.project(x=altura, color=(0, 1, 1), width=3, nvertices=1)
lxz = curva.project(y=altura, color=(1, 0, 1), width=3, nvertices=1)
lxy = curva.project(z=altura, color=(1, 1, 0), width=3, nvertices=1)
curvas = [curva, lxy, lxz, lyz]
self.showAxis(False)
self.addChildren(curvas)
self.setupAnimations([ AnimationGroup(curvas, (5000,0,len(curva)-1)) ])
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"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"Cilindro")
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, parent=self)
Slider(rangep=('t', -1,1,0,20),func=tp.setV,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 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"Hélice circular, curvatura y torsión<br><br>(cos s/√2, sen s/√2, s/√2)"
)
self.camera_position = (10, -10, 10)
self.showAxis(False)
tmin = -2 * pi
tmax = 2 * pi
npuntos = 300
self.addChild(Cylinder(_1(185, 46, 61), tmax - tmin, 2))
## ============================================
# 1 implica primer derivada, 2 implica segunda derivada
def param1hc(t):
return 2 * Vec3(cos(t), sin(t), t / 3.0)
def param2hc(t):
return 2 * Vec3(-sin(t), cos(t), 1 / 3.0)
def param3hc(t):
return 2 * Vec3(-cos(t), -sin(t), 0)
def param4hc(t):
return 2 * Vec3(sin(t) / 3.0, -cos(t) / 3.0, 1.0)
espiral = Curve3D(param1hc, (tmin * 1.5, tmax * 1.5, npuntos),
color=_1(255, 255, 255))
tangente = espiral.attachField(
"tangente", param2hc).setLengthFactor(1).setWidthFactor(.6)
tangente.setRadius(0.06)
tangente.setDiffuseColor(_1(20, 240, 20))
normal = espiral.attachField(
"normal", param3hc).setLengthFactor(1).setWidthFactor(.6)
normal.setRadius(0.06)
normal.setDiffuseColor(_1(240, 120, 20))
binormal = espiral.attachField(
"binormal", param4hc).setLengthFactor(1).setWidthFactor(.6)
binormal.setRadius(0.06)
binormal.setDiffuseColor(_1(20, 120, 240))
self.addChild(espiral)
self.setupAnimations([
AnimationGroup([tangente, normal, binormal],
(10000, 0, len(espiral) - 1))
])
def __init__(self):
"x^2 + y^2 = z^2"
Page.__init__(self, u"Semicono de revolución<br><br>F(θ,ρ)=(θ,ρ,π/4)")
cono = RevolutionPlot3D(lambda r, t: r + 1, (0, 1), (0, 2 * pi))
cono1 = RevolutionPlot3D(lambda r, t, h: h * (r + 1), (0.05, 1), (0, 2 * pi)) #@UndefinedVariable
cono1.setLinesVisible(True)
cono1.setMeshVisible(False)
cono.setDiffuseColor(_1(149, 24, 82))
cono.setSpecularColor(_1(149, 24, 82))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(cono)
self.addChild(cono1)
self.addChild(baseplane)
def __init__(self):
Page.__init__(self, u"Exponencial")
self.showAxis(True)
self.axis_z.setVisible(False)
def curve(t):
return Vec3(exp(t) * cos(t), exp(t) * sin(t), exp(t))
def derivada(t):
return Vec3(
exp(t) * cos(t) - exp(t) * sin(t),
exp(t) * cos(t) + exp(t) * sin(t), exp(t))
curva1 = Curve3D(curve, (-pi, 1 * pi, 200), width=2)
self.addChild(curva1)
curva1.derivative = derivada
curva1.tangent_vector.show()
self.setupAnimations([curva1.tangent_vector])
def __init__(self):
"x^3 - 3xy^2 - z = 0"
Page.__init__(
self,
u"Silla del mono<br><br>F(x,y)=(x, y, x<sup>3</sup> - 3xy<sup>2</sup>)"
)
silla = Plot3D(lambda x, y: x**3 - 3 * x * y**2 + 2.5, (-1, 1),
(-1, 1))
silla.setAmbientColor(_1(151, 139, 125))
silla.setDiffuseColor(_1(151, 139, 125))
silla.setSpecularColor(_1(151, 139, 125))
# silla.setShininess(1)
# plano.setScaleFactor((1,1,.6))
def cVec(pto):
"pto: Vec3"
return pto * 1.1
silla.addVectorField(cVec)
# def setXscale(t):
# scale.scaleFactor = (1,1,t)
# Slider(
# rangep=('z', .2, 1, 1, 20),
# func=setXscale,
# parent=self
# )
#
silla1 = Plot3D(lambda x, y, h: h * (x**3 - 3 * x * y**2 + 2.5),
(-1, 1), (-1, 1)) #@UndefinedVariable
# silla1.setScaleFactor((1,1,.6))
silla1.setLinesVisible(True)
silla1.setMeshVisible(False)
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(silla)
self.addChild(silla1)
self.addChild(baseplane)
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"Curva cúbica alabeada<br><br>α(t)=(t,t<sup>2</sup>,t<sup>3</sup>)"
)
self.camera_position = (5, 5, 5)
self.camera_viewAll = True
self.setupPlanes()
c = lambda t: Vec3(t, t**2, t**3)
altura = -1
curva = Curve3D(c, (-1, 1, 100), width=5, nvertices=1)
lyz = curva.project(x=altura, color=(0, 1, 1), width=3, nvertices=1)
lxz = curva.project(y=altura, color=(1, 0, 1), width=3, nvertices=1)
lxy = curva.project(z=altura, color=(1, 1, 0), width=3, nvertices=1)
curvas = [curva, lxy, lxz, lyz]
self.showAxis(False)
self.addChildren(curvas)
self.setupAnimations(
[AnimationGroup(curvas, (5000, 0, len(curva) - 1))])
def __init__(self):
"x^2 - y^2 - z = 0"
Page.__init__(self, u"Paraboloide Hiperbólico<br><br>F(x,y)=(x, y, x<sup>2</sup>-y<sup>2</sup>)")
z = 1.5
parab = Plot3D(lambda x, y: x ** 2 - y ** 2 + z, (-1, 1), (-1, 1))
parab1 = Plot3D(lambda x, y, h: h * (x ** 2 - y ** 2 + z), (-1, 1), (-1, 1)) #@UndefinedVariable
parab1.setLinesVisible(True)
parab1.setMeshVisible(False)
parab.setAmbientColor(_1(145, 61 , 74))
parab.setDiffuseColor(_1(127, 119, 20))
parab.setSpecularColor(_1(145, 61 , 74))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(parab)
self.addChild(parab1)
self.addChild(baseplane)
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):
"x^2 + y^2 = z^2"
Page.__init__(
self,
u"Semicono de revolución<br><br>F(θ,ρ)=(θ,ρ,π/4)"
)
cono = RevolutionPlot3D(lambda r, t: r + 1, (0, 1), (0, 2 * pi))
cono1 = RevolutionPlot3D(lambda r, t, h: h * (r + 1), (0.05, 1),
(0, 2 * pi)) #@UndefinedVariable
cono1.setLinesVisible(True)
cono1.setMeshVisible(False)
cono.setDiffuseColor(_1(149, 24, 82))
cono.setSpecularColor(_1(149, 24, 82))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(cono)
self.addChild(cono1)
self.addChild(baseplane)
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"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):
"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):
"x^3 - 3xy^2 - z = 0"
Page.__init__(self, u"Silla del mono<br><br>F(x,y)=(x, y, x<sup>3</sup> - 3xy<sup>2</sup>)")
silla = Plot3D(lambda x, y: x ** 3 - 3 * x * y ** 2 + 2.5, (-1, 1), (-1, 1))
silla.setAmbientColor(_1(151, 139, 125))
silla.setDiffuseColor(_1(151, 139, 125))
silla.setSpecularColor(_1(151, 139, 125))
# silla.setShininess(1)
# plano.setScaleFactor((1,1,.6))
def cVec(pto):
"pto: Vec3"
return pto * 1.1
silla.addVectorField(cVec)
# def setXscale(t):
# scale.scaleFactor = (1,1,t)
# Slider(
# rangep=('z', .2, 1, 1, 20),
# func=setXscale,
# parent=self
# )
#
silla1 = Plot3D(lambda x, y, h: h * (x ** 3 - 3 * x * y ** 2 + 2.5), (-1, 1), (-1, 1)) #@UndefinedVariable
# silla1.setScaleFactor((1,1,.6))
silla1.setLinesVisible(True)
silla1.setMeshVisible(False)
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(silla)
self.addChild(silla1)
self.addChild(baseplane)
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):
"x^4 + 2x^2y^2 + y^4 -z = 0"
Page.__init__(self, u"Superficie cuártica<br><br>F(x,y)=(x,y,x<sup>4</sup>+2x<sup>2</sup>y<sup>2</sup>+y<sup>4</sup>)")
# cuart = Plot3D(lambda x,y: x**4 + 2*x**2*y**2 + y**4 + 1, (-1,1),(-1,1))
cuart = RevolutionPlot3D(lambda r, t: r ** 4 + 1, (0, 1.4), (0, 2 * pi))
# cuart.setScaleFactor((1,1,.6))
mesh1 = Plot3D(lambda x, y, h: h * (x ** 4 + 2 * x ** 2 * y ** 2 + y ** 4 + 0.9), (-1, 1), (-1, 1))
mesh1.setLinesVisible(True)
mesh1.setMeshVisible(False)
mesh1.setBoundingBox(zrange=(-1, 6))
# cuart.setAmbientColor(_1(168,211,8))
cuart.setDiffuseColor(_1(168, 211, 8))
cuart.setSpecularColor(_1(168, 211, 8))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(cuart)
self.addChild(mesh1)
self.addChild(baseplane)
def __init__(self):
"x^2 - y^2 - z = 0"
Page.__init__(
self,
u"Paraboloide Hiperbólico<br><br>F(x,y)=(x, y, x<sup>2</sup>-y<sup>2</sup>)"
)
z = 1.5
parab = Plot3D(lambda x, y: x**2 - y**2 + z, (-1, 1), (-1, 1))
parab1 = Plot3D(lambda x, y, h: h * (x**2 - y**2 + z), (-1, 1),
(-1, 1)) #@UndefinedVariable
parab1.setLinesVisible(True)
parab1.setMeshVisible(False)
parab.setAmbientColor(_1(145, 61, 74))
parab.setDiffuseColor(_1(127, 119, 20))
parab.setSpecularColor(_1(145, 61, 74))
baseplane = BasePlane()
baseplane.setHeight(0)
baseplane.setRange((-2, 2, 7))
self.addChild(parab)
self.addChild(parab1)
self.addChild(baseplane)
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)
def __init__(self):
Page.__init__(self, u"Planos osculador, normal y rectificante")
tmin = -2 * pi
tmax = 2 * pi
## ============================================
sq2 = 2**0.5
inv_sq2 = (1. / sq2)
def helix(s):
s_times_sq2 = inv_sq2 * s
return Vec3(cos(s_times_sq2), sin(s_times_sq2), s_times_sq2)
def tangent(s):
s_div_sq2 = s / sq2
return Vec3(-inv_sq2 * sin(s_div_sq2), inv_sq2 * cos(s_div_sq2),
inv_sq2)
def normal(s):
s_div_sq2 = s / sq2
return Vec3(-cos(s_div_sq2), -sin(s_div_sq2), 0)
def bi_normal(s):
s_div_sq2 = s / sq2
return Vec3(inv_sq2 * sin(s_div_sq2), -inv_sq2 * cos(s_div_sq2),
inv_sq2)
curve = Curve3D(helix, (tmin, tmax, 100), _1(206, 75, 150), 2)
self.addChild(curve)
#=======================================================================
# Vectors
#=======================================================================
field_tangent = curve.attachField("tangent", tangent).show()
field_normal = curve.attachField("normal", normal).show()
field_binormal = curve.attachField("binormal", bi_normal).show()
#=======================================================================
# Planes
#=======================================================================
def get_points(v1, v2):
return v2.p1, v1.p2, v2.p2
color = (.5, .5, .5)
plane_osculating = Plane(color, *get_points(field_tangent,
field_normal))
plane_normal = Plane(color, *get_points(field_normal, field_binormal))
plane_rectifying = Plane(color,
*get_points(field_binormal, field_tangent))
self.addChildren([plane_osculating, plane_normal, plane_rectifying])
def update_planes(n):
plane_osculating.setPoints(
*get_points(field_tangent, field_normal))
plane_normal.setPoints(*get_points(field_normal, field_binormal))
plane_rectifying.setPoints(
*get_points(field_binormal, field_tangent))
r = (5000, 0, len(curve) - 1)
animation = Animatable(update_planes, r)
self.setupAnimations([
AnimationGroup(
[field_tangent, field_normal, field_binormal, animation], r)
])
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):
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)
def __init__(self, parent=None):
Page.__init__(self, u"Paralelos y círculos máximos de la esfera")
self.showAxis(False)
pmin = 0
pmax = 2 * pi
r2 = 3.
l = -1
def puntos2(t):
return Vec3(-cos(t), -sin(t), 0)
def make_circulo(t):
return partial(par_esfera, t)
par_esfera = lambda t, f: Vec3(sin(t) * cos(f), sin(t) * sin(f), cos(t))
par_circulo = lambda f: Vec3(sin(t) * cos(f), sin(t) * sin(f), cos(t))
par_circulo_der = lambda f: Vec3(-cos(f) * sin(t), -sin(t) * sin(f), 0)
par_circulo_maximo = make_circulo(pi / 2)
esf = ParametricPlot3D(par_esfera, (0, pi, 100), (0, 2 * pi, 120))
esf.setTransparencyType(SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
esf.setTransparency(0.3).setDiffuseColor(_1(68, 28, 119)).setSpecularColor(_1(99, 136, 63))
VisibleCheckBox("esfera", esf, True, parent=self)
self.addChild(esf)
cm = Curve3D(par_circulo_maximo, (pmin, pmax, 200), color=_1(255, 255, 255))
self.addChild(cm)
aceleracion_cm = cm.attachField("aceleracion", puntos2).show().setLengthFactor(.98).setWidthFactor(.3)
tini=1.0472
par_circulo.func_globals['t'] = tini
par = Curve3D(par_circulo, (pmin, pmax, 200), color=_1(255, 221, 0))
self.addChild(par)
aceleracion_par = par.attachField("aceleracion", par_circulo_der).show().setLengthFactor(1).setWidthFactor(.3)
circle_2 = SimpleSphere(Vec3(0, 0, cos(tini)), radius=.02)
circle_2_tr = circle_2.getByName("Translation")
self.addChild(circle_2)
self.addChild(SimpleSphere(Vec3(0, 0, 0), radius=.02))
## los meridianos
sep = SoSeparator()
mer = Curve3D(lambda t: (0, .99 * cos(t), .99 * sin(t)), (pmin, pmax, 100), color=_1(18, 78, 169))
for i in range(24):
sep.addChild(rot(2 * pi / 24))
sep.addChild(mer.root)
self.addChild(sep)
# the sphere rotation axis
self.addChild(Line([(0, 0, -1.2), (0, 0, 1.2)], width=2))
def test(t):
par_circulo.func_globals['t'] = t
par.updatePoints()
circle_2_tr.translation = (0, 0, cos(t))
Slider(('t', 0.1, pi-.1, tini, 100), test, duration=4000, parent=self)
self.setupAnimations([aceleracion_cm, aceleracion_par])
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):
"""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):
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, parent=None):
Page.__init__(self, u"Paralelos y círculos máximos de la esfera")
self.showAxis(False)
pmin = 0
pmax = 2 * pi
r2 = 3.
l = -1
def puntos2(t):
return Vec3(-cos(t), -sin(t), 0)
def make_circulo(t):
return partial(par_esfera, t)
par_esfera = lambda t, f: Vec3(
sin(t) * cos(f),
sin(t) * sin(f), cos(t))
par_circulo = lambda f: Vec3(sin(t) * cos(f), sin(t) * sin(f), cos(t))
par_circulo_der = lambda f: Vec3(-cos(f) * sin(t), -sin(t) * sin(f), 0)
par_circulo_maximo = make_circulo(pi / 2)
esf = ParametricPlot3D(par_esfera, (0, pi, 100), (0, 2 * pi, 120))
esf.setTransparencyType(
SoTransparencyType.SORTED_OBJECT_SORTED_TRIANGLE_BLEND)
esf.setTransparency(0.3).setDiffuseColor(_1(68, 28,
119)).setSpecularColor(
_1(99, 136, 63))
VisibleCheckBox("esfera", esf, True, parent=self)
self.addChild(esf)
cm = Curve3D(par_circulo_maximo, (pmin, pmax, 200),
color=_1(255, 255, 255))
self.addChild(cm)
aceleracion_cm = cm.attachField(
"aceleracion",
puntos2).show().setLengthFactor(.98).setWidthFactor(.3)
tini = 1.0472
par_circulo.func_globals['t'] = tini
par = Curve3D(par_circulo, (pmin, pmax, 200), color=_1(255, 221, 0))
self.addChild(par)
aceleracion_par = par.attachField(
"aceleracion",
par_circulo_der).show().setLengthFactor(1).setWidthFactor(.3)
circle_2 = SimpleSphere(Vec3(0, 0, cos(tini)), radius=.02)
circle_2_tr = circle_2.getByName("Translation")
self.addChild(circle_2)
self.addChild(SimpleSphere(Vec3(0, 0, 0), radius=.02))
## los meridianos
sep = SoSeparator()
mer = Curve3D(lambda t: (0, .99 * cos(t), .99 * sin(t)),
(pmin, pmax, 100),
color=_1(18, 78, 169))
for i in range(24):
sep.addChild(rot(2 * pi / 24))
sep.addChild(mer.root)
self.addChild(sep)
# the sphere rotation axis
self.addChild(Line([(0, 0, -1.2), (0, 0, 1.2)], width=2))
def test(t):
par_circulo.func_globals['t'] = t
par.updatePoints()
circle_2_tr.translation = (0, 0, cos(t))
Slider(('t', 0.1, pi - .1, tini, 100),
test,
duration=4000,
parent=self)
self.setupAnimations([aceleracion_cm, aceleracion_par])
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):
"""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, name):
Page.__init__(self, name)
self.camera_position = (0, 0, 13)
self.camera_point_at = [Vec3(0, 0, 0), Vec3(0, 1, 0)]
self.showAxis(True)
self.axis_z.setVisible(False)
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):
u"""^2 + y^2 = z^2"""
Page.__init__(
self, u"Esfera, parametrización por proyecciones estereográficas")
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.setAmbientColor(_1(99,136,63))
esf.setDiffuseColor(_1(99, 136, 63))
esf.setSpecularColor(_1(99, 136, 63))
def proyZm1(u, v, t1):
"""proy desde el polo norte al plano z=-1"""
den = u**2 + v**2 + 4
x = u - t1 * (u - 4 * u / den)
y = v - t1 * (v - 4 * v / den)
z = -1 - t1 * (-2 + 8 / den)
return (x, y, z)
def proyZ1(u, v, t2):
"""proy desde el polo sur al plano z=1"""
den = u**2 + v**2 + 4
x = u - t2 * (u - 4 * u / den)
y = v - t2 * (v - 4 * v / den)
z = 1 - t2 * (2 - 8 / den)
return (x, y, z)
stereo = ParametricPlot3D(proyZm1, (-3, 3, 70), (-3, 3, 70))
stereo.setLinesVisible(True)
stereo.setMeshVisible(False)
stereo.setMeshDiffuseColor(_1(117, 55, 79))
stereo2 = ParametricPlot3D(proyZ1, (-3, 3, 70), (-3, 3, 70))
stereo2.setLinesVisible(True)
stereo2.setMeshVisible(False)
stereo2.setMeshDiffuseColor(_1(80, 87, 193))
stereo2.setTransparency(0.5)
stereo2.setTransparencyType(8)
baseplane = BasePlane()
baseplane.setHeight(-1.005)
baseplane.setRange((-4, 4, 7))
self.addChild(esf)
self.addChild(stereo2)
self.addChild(stereo)
self.addChild(baseplane)
params = [stereo, stereo2]
## no queremos los controles
for i, p in enumerate(params):
p.parameters['t%d' % (i + 1)].hide()
anims = [
p.parameters['t%d' % (i + 1)].asAnimation()
for i, p in enumerate(params)
]
self.setupAnimations(anims)
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, name):
Page.__init__(self, name)
self.camera_position = (0, 0, 13)
self.camera_point_at = [SbVec3f(0, 0, 0), SbVec3f(0, 1, 0)]
self.showAxis(True)
self.axis_z.setVisible(False)
def __init__(self, parent=None):
Page.__init__(self, "Loxodroma")
self.creaLoxodroma()
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"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):
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, parent=None):
Page.__init__(self, "Loxodroma")
self.creaLoxodroma()