# Project Details

### Description

Closed space curves with

non-vanishing curvature

defines via the Frenet

formulas some closed curves

on the unit 2-sphere, called

the spherical indicatrices.

By using Gauss-Bonnet's

Formula after cutting a

spherical curve into simple

closed sub-curves an index on

the unit 2-sphere is found.

This spherical index may be

seen as a spherical analogy

to the winding number of

closed plane curves. The

spherical index has the

property that the integral

over the unit 2-sphere

of this index, equals the

integrated geodesic

curvature of the spherical

curve. Using this result on

the spherical indicatrices of

a space curve, we obtain

almost similar proofs of some

(generalizations of)

classical theorems. The

spherical index gives both

upper and lower bounds on the

total curvature and

total torsion of space curves.

non-vanishing curvature

defines via the Frenet

formulas some closed curves

on the unit 2-sphere, called

the spherical indicatrices.

By using Gauss-Bonnet's

Formula after cutting a

spherical curve into simple

closed sub-curves an index on

the unit 2-sphere is found.

This spherical index may be

seen as a spherical analogy

to the winding number of

closed plane curves. The

spherical index has the

property that the integral

over the unit 2-sphere

of this index, equals the

integrated geodesic

curvature of the spherical

curve. Using this result on

the spherical indicatrices of

a space curve, we obtain

almost similar proofs of some

(generalizations of)

classical theorems. The

spherical index gives both

upper and lower bounds on the

total curvature and

total torsion of space curves.

Status | Finished |
---|---|

Effective start/end date | 01/01/1996 → 31/01/1999 |