Curves in R3
Curvature and torsion of space curves
What we do?
Find curvature, torsion, tangent, binormal and principal normal vectors of the following space curve:
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.
Space
First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. To define the space we declare domain, forms, vectors, coframe, frame, flat metric and calculate connection (it is equal to zero of course).
Define Euclidean space as a manifold:
| Out[8]= |  |
Declare 1-forms for the space coframe:
| Out[9]= |  |
Declare vectors for the space frame:
| Out[10]= |  |
Declare coframe on the space:
| Out[11]= |  |
Declare frame on the space:
| Out[12]= |  |
Declare flat metric on the space:
| Out[13]= |  |
Calculate connection of the metric:
| Out[14]= |  |
Now the working space is defined completely and we can start to solve the problem.
Just for right simplification:
Curve
Define the curve as a manifold:
| Out[16]= |  |
Declare constants a and b:
| Out[17]= |  |
Declare 1-form for curve's coframe:
| Out[18]= |  |
Declare vectors for curve's frame:
| Out[19]= |  |
Declare coframe on the curve:
| Out[20]= |  |
Declare frame of the curve:
| Out[21]= |  |
Declare mapping of the curve into
R3:
| Out[22]= |  |
| Out[23]= |  |
One can also calculate metric induced on the curve by the mapping.
| Out[24]= |  |
Calculate invariants of the mapping:
| Out[25]= |  |
Extract tangent normalized vector field:
| Out[26]= |  |
Extract normal normalized vector field:
| Out[27]= |  |
Extract binormal normalized vector field:
| Out[28]= |  |
Extract curvature of the curve:
| Out[29]= |  |
Extract torsion of the curve:
| Out[30]= |  |
- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
