Curvature and moving frame of epicycloid
 

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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 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.

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

   Epicycloid

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

Declare constants  and :
Constants(a,m);

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

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

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

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

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));

Calculate metric on the curve using `&/` operator:
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));