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:
|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.|
Just for right simplification:
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).
Calculate the connection of the metric:
Now the working space is defined completely and we can start to solve the problem.
Abstract parametric curve
Define the curve as a manifold:
Define two functions on the curve:
Declare 1-form for curve's coframe:
Declare vectors for curve's frame:
Declare coframe on the curve:
Declare frame of the curve:
Declare mapping of the curve into R2
Let us see the mapping attributes:
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.
Calculate invariants of the mapping:
Check the "orthonormality":