Abstract
The analysis of a surface generated by quasi-rotation of a straight line around a circle is provided in the present paper. The considered case features a straight generatrix belonging to the plane of a circular axis of quasi-rotation and intersecting it in two points. A geometric method of determination of a point belonging to a surface given its projection on the axis plane is demonstrated. Geometric construction of the curves of intersection between the considered surface and a conic surface is presented. A method of determination of points belonging to the considered surface as points belonging to the curve of intersection of two conic surfaces is acquired. Step-by-step constructions illustrating the solution of the problem of determination of a plane tangent to the considered surface in a given point are provided. The problem is solved through the methods of descriptive geometry. Every construction is performed according to an analytic algorithm, not involving approximate methods of determination of the sought points. The construction is carried out in a CAD system through the use of tools “straight line by two points” and “circle by center and point”. The presented solution to the defined problem is connected to the solution to the problem of determination of the rays reflected from the considered surface. The results of the paper expose the geometric properties of surfaces of quasi-rotation. The provided constructions can serve as the basis for the research of optical properties of the considered surfaces.
Highlights
Quasi-rotation is a geometric correspondence between a point located in common plane with a conic, which is the axis of quasi-rotation, and, in general case, four circles located in planes perpendicular to the plane of the conic
Further to the research of geometrical properties of the surfaces of quasi-rotation and their possible practical applications, the solution to the task of construction of a plane tangent to a surface of quasi-rotation in an arbitrary point is considered in the present paper
A surface generated upon quasi-rotation of a straight line around a circle as the axis is considered in the present paper
Summary
The point L’ constitutes the result of quasi-rotation of the point L around the axis i on 180° angle around the closest center of rotation S’; k’ is the circle correspondent to this quasirotation centered at the point S’. The geometric properties of surfaces of quasi-rotation were considered in paper [5]. Further to the research of geometrical properties of the surfaces of quasi-rotation and their possible practical applications, the solution to the task of construction of a plane tangent to a surface of quasi-rotation in an arbitrary point is considered in the present paper
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