Circle Bundle

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/AlgebraicGeometry/CircleBundle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CircleBundle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Descriptions and illustrations by
Todd Rowland
Wolfram Research, Inc.

Introduction

At any point on a surface, there is a circle of unit tangent vectors.  The collection of these circles forms the frame bundle on the surface, and it is a circle bundle.  There is a well-defined rotation of all the circles, even though there may not be a nonvanishing vector field on the surface.  The following animations show the unit circles at a few selected points, embedded in the surface as the points of a fixed distance,using the Riemannian metric (actually a good approximation of it).

Circle Bundle on the Sphere

<<Graphics`Shapes`;
<<Graphics`Colors`;
<<Geometry`Polytopes`;

c[p_,v_,r_,t_]:=
Module[{w=Cross[p,v]},
p+r Cos[t] v + r Sin[t] w]

(*
cpe[p_,s_,clr_List]:=Module[{v},
v=NullSpace[{p}][[1]];
v/=(v.v)^(1/2);
{clr[[1]], Line[Table[c[p,v,.2,t+s],{t,0,2 Pi/5, Pi/10}]],
clr[[2]], Line[Table[c[p,v,.2,t+s],{t,2Pi/5,4 Pi/5, Pi/10}]],
clr[[3]], Line[Table[c[p,v,.2,t+s],{t,4Pi/5,6 Pi/5, Pi/10}]],
clr[[4]], Line[Table[c[p,v,.2,t+s],{t,6Pi/5,8 Pi/5, Pi/10}]],
clr[[5]], Line[Table[c[p,v,.2,t+s],{t,8Pi/5,10 Pi/5, Pi/10}]]}
]
*)

doclrs[sol_, pts_List, clrs_List]:=
Table[Show[Graphics3D[{Thickness[.015],Map[(cpe[#1,s,clrs]&),pts],sol,White,Sphere[1,25,25]}], Lighting -> False],
{s,0,2 Pi-Pi/4, Pi/4}]

dd= Map[(N[#1(#1.#1)^(-1/2)]&), Vertices[Dodecahedron]];

cpe[p_,s_,clr_List]:=Module[{v},
v=NullSpace[{p}][[1]];
v/=(v.v)^(1/2);
Table[{Hue[t/2/Pi],Line[{c[p,v,.2,t+s],c[p,v,.2,t+s+Pi/10]}]},{t,0,2 Pi-Pi/10, Pi/10}]
]

(* The first color is the color of the sphere, and the others are on the circle. *)

Export[gifs<>"circbuns.gif",
doclrs[RGBColor[.2,.13,.22], dd, {RGBColor[0.51,0.72,0.9],RGBColor[0.8,0.64,0.9],RGBColor[0.07,0.85,0.6],RGBColor[0.5,0.6,0.43],RGBColor[0.3,0.2,0.96]}]]

Circle Bundle on the Torus

<<Graphics`Shapes`;
<<Graphics`Colors`;

vecs[{x_,y_}]={{(3+Cos[y])^(-1),0},{0,1}};
c3[p_,r_,t_]:=p+r Cos[t] vecs[p][[1]]+r Sin[t] vecs[p][[2]]

(*
ctor[p_,f_,r_, s_,clr_List]:=
{clr[[1]], Line[Table[f[c3[p,r,t+s]],{t,0,2 Pi/5, Pi/10}]],
clr[[2]], Line[Table[f[c3[p,r,t+s]],{t,2Pi/5,4 Pi/5, Pi/10}]],
clr[[3]], Line[Table[f[c3[p,r,t+s]],{t,4Pi/5,6 Pi/5, Pi/10}]],
clr[[4]], Line[Table[f[c3[p,r,t+s]],{t,6Pi/5,8 Pi/5, Pi/10}]],
clr[[5]], Line[Table[f[c3[p,r,t+s]],{t,8Pi/5,10 Pi/5, Pi/10}]]}

*)

f1[{x_,y_}]:= {Cos[x](3+Cos[y]),Sin[x](3+Cos[y]), Sin[y]}
f2[{x_,y_}]:= {Cos[x](3+1.02Cos[y]),Sin[x](3+1.02Cos[y]), 1.02Sin[y]}

(*colors of the circles*)
them ={RGBColor[0.35,0.86,0.11],RGBColor[0.88,0.4,0.32],RGBColor[0.35,0.99,0.68],RGBColor[0.7,0.3,0.56],RGBColor[0.98,0.66,0.3]};

Clear[too3];
too3:=
Module[{pts=Table[2 Pi{n/8,n/5 },{n,24}]},
Table[
Show[Graphics3D[{Thickness[.015],Map[ctor[#1,f2,.4, s,them]&,pts],
White,Torus[3,1,20,20]}], Lighting-> False],
{s,0,2Pi-Pi/10,Pi/10}]]

ctor[p_,f_,r_,s_,clr_List]:=Module[{v},
v=NullSpace[{p}][[1]];
v/=(v.v)^(1/2);
Table[{Hue[t/2/Pi],Line[{f[c3[p,r,t+s]],f[c3[p,r,t+s+Pi/10]]}]},{t,0,2 Pi-Pi/10, Pi/10}]
]

Export[gifs<>"circbunt.gif",too3]