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Curves in R2

Curvature and moving frame of abstract parametric curve in polar coordinates

What we do?

  • Find curvature of the abstract plane curve defined by parametric equation: r = ()

Solution:

Connection[id] id - variable - connection identifier.
Mapping[f,m,n,y1→f1(x1,x2...xm),y2→f2(x1,x2...xm),...,
yn→fn(x1,x2...xm)]		f - variable - the mapping identifier i.e. f : m n, m - variable - first domain identifier, n - variable - second domain identifier.
Invariants[f]f - mapping identifier.
Coframe[id1→ expr1,id2→ expr2,...idn→ exprn]id - identifier for indexed variable - the coframe 1-forms, n - dimension of working manifold (a variable or integer), idi
expri - rules where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.

Necessary functions.

Load the Atlas package:
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Plane (cartesian coordinate system)

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.
Just for right simplification:
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Plane (polar coordinate system)

To solve the problem we have to change coordinate system on manifold R2 from Cartesian to polar. We can do it easily just by definition of another Eucledean domain.
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Visualize the mapping:
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Let us see mapping attributes:
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Now we can calculate metric induced by the mapping:
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Calculation of the corresponding connection:
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Now we can continue.

Abstract curve

Define the curve as a manifold:
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Declare function = (t):
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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 vectors for curve's frame:
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Declare mapping of the curve into E2:
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Let us see the domain attributes:
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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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