Curvature and moving frame
of abstract parametric curve in cartesian coordinates
 

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

Problem:

Find curvature and moving frame of some abstract plane curve defined by parametric  equations:

Curve

Solution:

Load atlas package:
restart:
with(atlas):

Plane

First of all we have to discribe the space we are working in. The space is 2-dimensional Eucledean (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).
Domain(R^2);

Forms(e[k]=1);

Vectors(E[j]);

Coframe(e[1]=d(x),e[2]=d(y));

Frame(E[k]);

Metric(g=d(x)&.d(x)+d(y)&.d(y));

Connection(omega);

Now the working space is defined completely and we can start to solve the problem.

Abstract parametric curve

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

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

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

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

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

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

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

Let us see the mapping attributes:
Who(pi);

pi: mapping

Now we can calculate metric induced on the curve by the mapping. It is obvious that the metric gives squared differential of the curve's arc i.e.

Metric(G = g &/ pi);

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

Result

The curve curvature:
k:=Inv['curvatures'][1];

The curve moving frame:
X:=Inv['basis'][0];
Y:=Inv['basis'][1];

Check the "orthonormality":
'g(X,Y)'=simplify(g(X,Y));