Abstract
The infinite set of cycloids is created. Each cycloid of this set is defined as a movement trajectory of a point when this point circulates on the convex closed contour of arbitrary form when this contour moves rectilinearly without rotation on the plane with a velocity equal to the tangential velocity of a point on circulation contour. The classical cycloid is elements of this set. The differential equation of a cycloid set is derived and its solution in quadratures is received. The inverse problem when for the given cycloid it is necessary to fine the form of a circulation contour is solved. The problem of differential equation of the second order with boundary conditions about a bend of big curvature of an elastic rod of infinite length is solved in quadratures. Geometry of the loop which is formed at such bend is investigated. It is discovered that at movement of an elastic loop on a rod when the form and the size of a loop don’t change, each point of a loop moves on a trajectory which named by us the cycloid and which represents a circumference arch.
Highlights
The work stated below is not generalization by someone before the executed researches and not development by someone before the offered methods
This paper contains two independent, but connected with each other, parts. These parts are united that both they can be classified as the problems of differential geometry about infinite curves with a loop which is formed by self-crossing of branches of these curves
Geometry of movement trajectory of a point at its circulation on rectilinearly moving contour is investigated. We have named such trajectories cycloids by analogy to a classical cycloid which is a point trajectory of a circumference when the circumference rolls on a straight line without sliding
Summary
The work stated below is not generalization by someone before the executed researches and not development by someone before the offered methods. This paper contains two independent, but connected with each other, parts. These parts are united that both they can be classified as the problems of differential geometry about infinite curves with a loop which is formed by self-crossing of branches of these curves. Difference of parts is caused by the loops investigated below are made of materials of essentially different properties
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