Curves in R2

Curvature and moving frame of abstract parametric curve in cartesian coordinates

What we do?

  • Find curvature and moving frame of some abstract plane curve defined by parametric equations:

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).
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Declare some forms:
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Declare some vectors:
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Declare coframe:
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.

Abstract parametric curve

Define the curve as a manifold:
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Define two functions on the curve:
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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:
Declare frame of the curve:
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Declare mapping of the curve into R2:
Let us see the mapping attributes:
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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.
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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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