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Journal for Geometry and Graphics 23 (2019), No. 1, 099--114
Copyright Heldermann Verlag 2019

A Task About Spheres and Cones, Applicable in Jet Theory

Viktor Mileikovskyi
Kyiv National University of Construction and Architecture, Povitroflotskyi pr. 31, r. 288, Kyiv 03037, Ukraine

A task about spheres and cones is solved. The task is about tangent spheres around an x-axis. Three right circular cones with the same apex on the x-axis at x=0 contain corresponding endpoints of diameters of the spheres, perpendicular to x, and centres of the spheres, respectively. Any three adjacent spheres are mutually tangent. The diameters of the spheres are proportional to x. The x-axis intersects all spheres. The goal of the task is to find geometric relations between the involved cones and to give rules for plotting the scheme. The solution allows a transformation of the sphere sequence onto itself by rotation and homothety.
In 2D, the right circular cones degenerate to angles formed by two rays starting at the apex, while the cones' axis degenerates to the angle bisector. The spheres degenerate to circles. Therefore, the task has a 2D analogue. If one of the angles is known, the others can be found uniquely. The radii of the circles form a geometric progression.
The 3D task has more freedom. If the opening angle of the largest cone is known, it is possible to find the minimum of the opening angle of the intermediate cone passing through the sphere centres. This task allows a simplified geometric simulation of large eddies in free jets.

Keywords: Cone, sphere, tangency, large eddy, jet.

MSC: 51M04; 52C17, 76F40

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