Curves in  

Curvature and moving frame of epicycloid 

Problem: 

Find curvature and moving frame for epicycloid:  

Curve 

Epicycloid for some a and m: 

subs({a=2,m=3},[a*(1+m)*cos(m*t)-a*m*cos((1+m)*t), a*(1+m)*sin(m*t)-a*m*sin((1+m)*t),t=-2*Pi..2*Pi]):
plot[parametric](%,thickness=3,title="Epicycloid: a=2, m=3"); 

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. 

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

 Epicycloid 

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

(3.2.1)

 

Declare constants and :
Constants(a,m); 

(3.2.2)

 

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

(3.2.3)

 

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

(3.2.4)

 

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

(3.2.5)

 

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

(3.2.6)

 

Declare mapping of the curve into :
Mapping(pi,E,R^2,
       x=a*(1+m)*cos(m*tau)-a*m*cos((1+m)*tau),
       y=a*(1+m)*sin(m*tau)-a*m*sin((1+m)*tau)); 

 

	(3.2.7)

 

Calculate metric on the curve using `&/` operator:
Metric(G = g &/ pi); 

(3.2.8)

 

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

(3.2.9)

 

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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