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Curves in R2

Curvature and moving frame of epicycloid

What we do?

Find curvature and moving frame for epicycloid:

Solution:

Domain[manifold]manifold - string - a manifold name or a name of a manifold domain.
Metric[id→expr]id - variable - metric identifier, expr - expression - metric declaration.
Connection[id]id - variable - connection identifier.

Necessary functions.

Load the Atlas package:
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Just for right simplification:
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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).
Declare domain:
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Declare some forms:
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Declare some vectors:
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Declare coframe:
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Declare frame:
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Declare a flat metric:
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Calculate the connection of the metric:
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Now the working space is defined completely and we can start to solve the problem.

Epicycloid

Define the curve as a manifold:
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Declare constants a and m:
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Declare 1-form for curve's coframe:
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Declare vectors for curve's frame:
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Declare coframe on the curve:
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Declare frame of the curve:
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Declare mapping of the curve into R2:
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Visualize the mapping:
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Calculate metric on the curve using Pullback- operator:
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Calculate invariants of the mapping:
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Result

The curve curvature:
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The curve moving frame:
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Check the "orthonormality":
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