Curves in  

Curvature and torsion of space curves  

Problem: 

Find curvature, torsion, tangent, binormal and principal normal vectors of the following space curve:    

Solution: 

Load atlas package: 

>	restart: with(atlas):

 

Space 

First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. 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^3);

 

(2.1.1)

 

>	Forms(e[k]=1);

 

(2.1.2)

 

>	Vectors(E[j]);

 

(2.1.3)

 

>	Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

 

(2.1.4)

 

>	Frame(E[k]);

 

(2.1.5)

 

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

 

(2.1.6)

 

>	Connection(omega);

 

(2.1.7)

 

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

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

Curve 

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

(2.2.1)

 

Declare constants a and b:
Constants(a,b); 

(2.2.2)

 

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

(2.2.3)

 

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

(2.2.4)

 

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

(2.2.5)

 

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

(2.2.6)

 

Declare mapping of the curve into :
Mapping(pi,C,R^3,
                  x=a*cos(tau),
                  y=a*sin(tau),
                  z=b*tau); 

 

	(2.2.7)

 

One can also calculate metric induced on the curve by the mapping. 

>	Metric(G = g &/ pi);

 

(2.2.8)

 

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

(2.2.9)

 

Extract tangent normalized vector field: 

>	T=Inv[basis][0];

 

(2.2.10)

 

Extract normal normalized vector field: 

>	N=Inv[basis][1];

 

(2.2.11)

 

Extract binormal normalized vector field: 

>	B=Inv[basis][2];

 

(2.2.12)

 

Extract curvature of the curve: 

>	k=Inv[curvatures][1];

 

(2.2.13)

 

Extract torsion of the curve: 

>	chi=Inv[curvatures][2];

 

(2.2.14)

 

>