Differences between revisions 10 and 11
 ⇤ ← Revision 10 as of 2009-05-21 08:07:31 → Size: 3053 Editor: pang Comment: ← Revision 11 as of 2009-05-23 11:22:36 → ⇥ Size: 3572 Editor: pang Comment: slight improvements over previous version of Evolutes Deletions are marked like this. Additions are marked like this. Line 35: Line 35: by Pablo Angulo. Computes the evolute of a plane curve given in parametric coordinates. The curve must be parametrized from the interval [0,2pi]. The following animation was done with similar code:http://www.uam.es/personal_pdi/ciencias/pangulo/adjuntos/evoluta.gif by Pablo Angulo. Computes the evolute of a plane curve given in parametric coordinates. The curve must be parametrized from the interval [0,2pi]. Line 44: Line 43: def _( gamma1=input_box(default=sin(t)), gamma2=input_box(default=1.3*cos(t)),     rango_angulos=range_slider(0,360,paso_angulo,(0,45),label='Draw lines for these angles') ):    print rango_angulos def _( gamma1=input_box(default=sin(t)), gamma2=input_box(default=1.3*cos(t)),    draw_normal_lines=True,     rango_angulos=range_slider(0,360,paso_angulo,(0,90),label='Draw lines for these angles'),     draw_osculating_circle=True,     t0=input_box(default=pi/3,label='parameter value for the osculating circle'),     auto_update=False ): Line 55: Line 58: radio=np^3/sqrt(np^2*npp^2-pe^2) curvatura=(gammap[1]*gammapp[0]-gammap[0]*gammapp[1])/norma(gammap)^3    radio=1/curvatura Line 57: Line 62: Line 60: Line 65: grafica=curva+evoluta Line 61: Line 67: f=2*pi/360    lineas=sum(line2d([(gamma[0](t=i*f), gamma[1](t=i*f)), (centros[0](t=i*f), centros[1](t=i*f)) ], thickness=1,rgbcolor=(1,0.8,0.8)) for i in range(rango_angulos[0],rango_angulos[1]+paso_angulo,paso_angulo)) if draw_normal_lines:        f=2*pi/360        lineas=sum(line2d( [ (gamma[0](t=i*f), gamma[1](t=i*f)),                              (centros[0](t=i*f), centros[1](t=i*f)) ],                            thickness=1,rgbcolor=(1,0.8,0.8))                    for i in range(rango_angulos[0], rango_angulos[1]+paso_angulo, paso_angulo))        grafica+=lineas        if draw_osculating_circle and 0

Sage Interactions - Geometry

Intersecting tetrahedral reflections

by Marshall Hampton. Inspired by a question from Hans Schepker of Glass Geometry.

```#Pairs of tetrahedra, one the reflection of the other in the internal face, are joined by union operations:
p1 = Polyhedron(vertices = [[1,1,1],[1,1,0],[0,1,1],[1,0,1]])
p2 = Polyhedron(vertices = [[1/3,1/3,1/3],[1,1,0],[0,1,1],[1,0,1]])
p12 = p1.union(p2)
p3 = Polyhedron(vertices = [[0,0,1],[0,0,0],[0,1,1],[1,0,1]])
p4 = Polyhedron(vertices = [[2/3,2/3,1/3],[0,0,0],[0,1,1],[1,0,1]])
p34 = p3.union(p4)
p5 = Polyhedron(vertices = [[1,0,0],[1,0,1],[0,0,0],[1,1,0]])
p6 = Polyhedron(vertices = [[1/3,2/3,2/3],[1,0,1],[0,0,0],[1,1,0]])
p56 = p5.union(p6)
p7 = Polyhedron(vertices = [[0,1,0],[0,0,0],[1,1,0],[0,1,1]])
p8 = Polyhedron(vertices = [[2/3,1/3,2/3],[0,0,0],[1,1,0],[0,1,1]])
p78 = p7.union(p8)
pti = p12.intersection(p34).intersection(p56).intersection(p78)
@interact
def tetra_plot(opac = slider(srange(0,1.0,.25), default = .25)):
p12r = p12.render_wireframe()+p12.render_solid(opacity = opac)
p34r = p34.render_wireframe()+p34.render_solid(rgbcolor = (0,0,1),opacity = opac)
p56r = p56.render_wireframe()+p56.render_solid(rgbcolor = (0,1,0),opacity = opac)
p78r = p78.render_wireframe()+p78.render_solid(rgbcolor = (0,1,1),opacity = opac)
ptir = pti.render_wireframe()+pti.render_solid(rgbcolor = (1,0,1),opacity = .9)
show(p12r+p34r+p56r+p78r+ptir, frame = False)```

Evolutes

by Pablo Angulo. Computes the evolute of a plane curve given in parametric coordinates. The curve must be parametrized from the interval [0,2pi].

```var('t');
def norma(v):
return sqrt(sum(x^2 for x in v))
paso_angulo=5

@interact
def _( gamma1=input_box(default=sin(t)), gamma2=input_box(default=1.3*cos(t)),
draw_normal_lines=True,
rango_angulos=range_slider(0,360,paso_angulo,(0,90),label='Draw lines for these angles'),
draw_osculating_circle=True,
t0=input_box(default=pi/3,label='parameter value for the osculating circle'),
auto_update=False ):

gamma=(gamma1,gamma2)
gammap=(gamma[0].derivative(),gamma[1].derivative())
normal=(gammap[1]/norma(gammap), -gammap[0]/norma(gammap))
gammapp=(gammap[0].derivative(),gammap[1].derivative())

np=norma(gammap)
npp=norma(gammapp)
pe=gammap[0]*gammapp[0]+gammap[1]*gammapp[1]
curvatura=(gammap[1]*gammapp[0]-gammap[0]*gammapp[1])/norma(gammap)^3

curva=parametric_plot(gamma,(t,0,2*pi))
evoluta=parametric_plot(centros,(t,0,2*pi), color='red')
grafica=curva+evoluta

if draw_normal_lines:
f=2*pi/360
lineas=sum(line2d( [ (gamma[0](t=i*f), gamma[1](t=i*f)),
(centros[0](t=i*f), centros[1](t=i*f)) ],
thickness=1,rgbcolor=(1,0.8,0.8))
for i in range(rango_angulos[0], rango_angulos[1]+paso_angulo, paso_angulo))
grafica+=lineas

if draw_osculating_circle and 0<t0<2*pi:
punto=point((gamma[0](t=t0), gamma[1](t=t0)), rgbcolor=hue(0),pointsize=30)