Curves in  

Curvature and moving frame
of abstract parametric curve in cartesian coordinates 

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 is equal to zero of course). 

>	Domain(R^2);

 

(3.1.1)

 

>	Forms(e[k]=1);

 

(3.1.2)

 

>	Vectors(E[j]);

 

(3.1.3)

 

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

 

(3.1.4)

 

>	Frame(E[k]);

 

(3.1.5)

 

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

 

(3.1.6)

 

>	Connection(omega);

 

(3.1.7)

 

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); 

(3.2.1)

 

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

(3.2.2)

 

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

(3.2.3)

 

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

(3.2.4)

 

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

(3.2.5)

 

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

(3.2.6)

 

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

 

	(3.2.7)

 

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

 

pi: mapping
	(3.2.8)

 

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);

 

(3.2.9)

 

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

(3.2.10)

 

Result 

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

(3.3.1)

 

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

 

	(3.3.2)

 

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

(3.3.3)

 

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