EVOLVENT OF SPHERE AS A MODEL OF SNOW HUT 'IGLOO'

Abstract

The conditional evolvent of sphere as a conditional evolvent of curvilinear helicoid, approximating it, is proposed. The evolvent is a part of plane bounded by two curves resembling a clothoid or a Cornu spiral. The spiral method of constructing the evolvent of a sphere is a mathematical basis for determination of form and sizes of blocks from which a snow hut 'igloo' is building. This is due to the fact that the most common way to build a snow hut 'igloo' is a spiral method, consisting in the fact that snow bricks are stacked on helicoid with a rectangular cross section and axial line in the form of a helix belonging to a sphere. The snow hut 'igloo' is represented as a sphere approximating its outer surface, and snow blocks, which are used in the spiral method of its construction, is represented as elements of a curvilinear helical surface that approximates the sphere. A spherical helix is a spatial curve formed by uniform motion of a point along the meridian of the sphere, while the meridian performs a uniform rotational motion around the axis of the sphere. Two helix lines are built on the surface of the sphere. The part of surface of the sphere, enclosed between two helix lines, is separated and a curvilinear helical surface is obtained, approximating the sphere. This helical surface can be represented as a surface formed by rotation of an arc of a circle around its vertical axis of symmetry and simultaneous rotation around an axis perpendicular to its plane. The approximation of the sphere is performed by a family of cylindrical surfaces, for which the meridional sections of the sphere serve as directrixes. It is assumed that when the part of the sphere bounded by meridional sections is replaced with a part of a cylindrical surface, the helix on the sphere is converted into a helix on a cylindrical surface. The conditional evolvent of a sphere is constructed as a set of evolvents of cylindrical surfaces that approximate it. After this, a conditional evolvent of a curvilinear helical surface is constructed as a set of evolvents of its parts built on a conditional evolvent of a sphere.