physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the troposkein is the curve an idealized
rope
A rope is a group of yarns, plies, fibres, or strands that are twisted or braided together into a larger and stronger form. Ropes have tensile strength and so can be used for dragging and lifting. Rope is thicker and stronger than similarly ...
assumes when anchored at its ends and spun around its long axis at a constant angular velocity. This shape is similar to the shape assumed by a
skipping rope
A skipping rope (British English) or jump rope (American English) is a tool used in the sport of skipping/jump rope where one or more participants jump over a rope swung so that it passes under their feet and over their heads. There are multi ...
, and is independent of rotational speed in the absence of gravity, but varies with respect to rotational speed in the presence of gravity. The troposkein does not have a closed-form representation; in the absence of gravity, though, it can be approximated by a pair of line segments spanned by a circular arc (tangential to the line segments at its endpoints). The form of a troposkein can be approximated for a given gravitational acceleration, rope density and angular velocity by iterative approximation. This shape is also useful for decreasing the stress experienced by the blades of a Darrieus
vertical axis wind turbine
A vertical-axis wind turbine (VAWT) is a type of wind turbine where the main rotor shaft is set transverse to the wind while the main components are located at the base of the turbine. This arrangement allows the generator and gearbox to be ...
.
References
*Ashwill, T., Leonard, T. (1986). "Developments in Blade Shape Design for a Darrieus Vertical Axis Wind Turbine", ''Sandia Report'', 86(1085).
Plane curves