Plane curves

Copyright © 2004-2008  DigiArea Group . All rights reserved.

Description:

This worksheet illustrates how to use atlas  package to solve problems in elementary differential geometry. As an example we calculate curvatures  and moving frame  of  the following plane curves:
 

We assume that astroid  is given as a parametric curve in Cartesian coordinate system and cardioid  is given as a parametric curve in polar coordinate system. For abstract  plane curve we will do calculations in both systems.  

Solution:

Load atlas package:

restart:
with(atlas):

Just for right simplification:
`atlas/simp`:=proc(a) factor(simplify(a,trig)) end:

Description of the total space

First of all we have to describe the space we are working in. The space is 2-dimensional Euclidean (flat) space i.e. a plane. To define the space we declare domain , forms , vectors , coframe , frame , flat metric  and calculate connection  (it equals to zero of cause).

Define Euclidean space  as a manifold :
Domain(R^2);

Declare 1-forms  for the space coframe:
Forms(e[k]=1);

Declare vectors  for the space frame:
Vectors(E[j]);

Declare coframe  on the space:
Coframe(e[1]=d(x),e[2]=d(y));

Declare frame  on the space:
Frame(E[k]);

Declare flat metric  on the space:
Metric(g[R]=d(x)&.d(x)+d(y)&.d(y));

Calculate connection  of the metric:
Connection(Omega);

Astroid

Define the curve as a manifold:
Domain(A);

Declare constant :
Constants(a);

Declare 1-form for curve's coframe
Forms(w[i]=1);

Declare vectors for curve's frame:
Vectors(W[k]);

Declare coframe on the curve:
Coframe(w[1]=d(t[a]));

Declare frame of the curve:
Frame(W[l]);

Declare mapping of the curve into :
Mapping(pi,A,R^2,x=a*cos(t[a])^3,
                 y=a*sin(t[a])^3);

Calculate metric on the curve using &/  operator:
Metric(g[A] = g[R] &/ pi);

It is obvious that the metric gives squared differential of the curve's arc :

Calculate invariants of the mapping:
Inv[pi]:=Invariants(pi);

Extract tangent  normalized vector field:

T:=Inv[pi][basis][0];

Extract normal  normalized vector field:

N:=Inv[pi][basis][1];

Extract curvature  of the curve:

k:=Inv[pi][curvatures][1];

Let us check the "orthonormality" of the moving frame vectors T and N. To do this we use metric tensor field :

'g[R](T,T)'=simplify(g[R](T,T));
'g[R](N,N)'=simplify(g[R](N,N));  

'g[R](T,N)'=g[R](T,N);

Abstract curve (Cartesian)

Let a parametric curve be given

Define the curve as a manifold:
Domain(A[c]);

Define two functions on the curve:
Functions(xi=xi(t),eta=eta(t));

Declare 1-form for curve's coframe
Forms(v[i]=1);

Declare vectors for curve's frame:
Vectors(V[k]);

Declare coframe on the curve:
Coframe(v[1]=d(t));

Declare frame of the curve:
Frame(V[l]);

Declare mapping of the curve into :
Mapping(psi,A[c],R^2,x=xi,
                     y=eta);

Now we can calculate metric induced on the curve by the mapping.
Metric(g[A[c]] = g[R] &/ psi);

Calculate invariants of the mapping:
Inv[A[c]]:=Invariants(psi);

Change coordinates on

To continue with cardioid and abstract polar curves we have to change coordinate system  on manifold  from Cartesian  to polar . We can do it easily just by definition of another Euclidean domain .

Define Euclidean space  as a manifold :
Domain(E^2);

Declare 1-forms  for the space coframe:
Forms(u[k]=1);

Declare vectors  for the space frame:
Vectors(U[j]);

Declare coframe  on the space:
Coframe(u[1]=d(r),u[2]=d(phi));

Declare frame  on the space:
Frame(U[k]);

Declare mapping  of  into :
Mapping(f,E^2,R^2,x=r*cos(phi),
                  y=r*sin(phi));

Now we can calculate metric  induced by the mapping:
Metric(g[E] = g[R] &/ f);

Calculation of the corresponding connection :
Connection(Gamma);

Let us see the result:
eval(Gamma);

Cardioid

Define the curve as a manifold:
Domain(C);

Declare 1-form for curve's coframe
Forms(theta[i]=1);

Declare vectors for curve's frame:
Vectors(Theta[k]);

Declare coframe on the curve:
Coframe(theta[1]=d(tau[c]));

Declare frame of the curve:
Frame(Theta[l]);

Declare mapping of the curve into :
Mapping(alpha,C,E^2,r=a*(1+cos(tau[c])),
                    phi=tau[c]);

Calculate metric on the curve using `&/` operator:
Metric(g[C] = g[E] &/ alpha);

Calculate invariants of the mapping:
Inv[C]:=Invariants(alpha);

Abstract curve (polar)

Let a parametric curve be given by equation  in polar coordinates:

Define the curve as a manifold:
Domain(A[p]);

Declare function :
Functions(rho=rho(tau));

Declare 1-form for curve's coframe
Forms(zeta[i]=1);

Declare vectors for curve's frame:
Vectors(ZETA[k]);

Declare coframe on the curve:
Coframe(zeta[1]=d(tau));

Declare frame of the curve:
Frame(ZETA[l]);

Declare mapping of the curve into :
Mapping(lambda,A[p],E^2,r=rho,
                        phi=tau);

Calculate metric on the curve using `&/` operator:
Metric(g[A[p]] = g[E] &/ lambda);

Calculate invariants of the mapping:
Inv[A[p]]:=Invariants(lambda);