LESSON 3.1 · 3:02 · CHAPTER 3 · PATHS ON CURVED SURFACES
The shortest path
A taut fibre on a surface seeks the shortest path, a geodesic. On a surface of revolution a geodesic keeps R sin α constant (Clairaut), so it turns around where R equals that constant, whatever the dome shape.
Predict · at 1:11 in the video
Keep the 20 degree helix on the 150 millimetre cylinder, but swap the round dome for a flatter, elliptical one. Does the opening the fibre turns around change size?
No. The fibre turns around where the angle reaches 90 degrees, and the sine of 90 is one. So it turns at a radius of exactly 51.3 millimetres.
Further reading
This course is our own. These books go deeper.
- Composite Filament Winding, S. T. Peters (ed.), ASM International, 2011
- Filament Winding: a Unified Approach, S. Koussios, Delft University Press, 2004
About the numbers
Simulations in this lesson are our own. Where a number rests on an assumption, the video marks it with an asterisk and says where it comes from.
Up next · 3.2 · 2:46A fibre's angular momentum