physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

and

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the troposkein (from )In English language scientific literature it has been variously spelled as troposkein, troposkien, troposkin, troposkine. is the curve an idealized

rope A rope is a group of yarns, Plying, plies, fibres, or strands that are plying, twisted or braided together into a larger and stronger form. Ropes have high tensile strength and can be used for dragging and lifting. Rope is thicker and stronger ...

assumes when anchored at its ends and spun around its long axis at a constant

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

. This shape is similar to the shape assumed by a

skipping rope A skipping rope (or jump rope) is a tool used in a sport involving rhythmic jumping over a rope swung underfoot and overhead. The activity, practiced both recreationally and competitively, encompasses disciplines such as freestyle routines—f ...

, 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 Closed form may refer to: Mathematics * Closed-form expression, a finitary expression * Closed differential form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior deri ...

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 In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") ...

. 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 ...

Notes

References

* Ashwill, T., Leonard, T. (1986). "Developments in Blade Shape Design for a Darrieus Vertical Axis Wind Turbine", ''Sandia Report'', 86(1085). Plane curves {{elementary-geometry-stub