Affiliation:
1. The Crimean Federal University named after V.I. Vernadsky
Abstract
The properties of the first reflection of curves can be investigated by their connections. It is giving us the possibility to construct new reflecting curves and reflecting systems with the demand properties. The studied reflective properties of plane curves as sections of reflecting surfaces are preserved if these sections are generators of surfaces of revolution and rotative surfaces (the normal to the section at a given point coincides with the normal to the surface at the same point) and the source of the incident rays is in the same plane section. Caustics, orthotomics, and reflective curves with a source at an infinity are considered. For a given orthotomics, there is a one-parameter set of reflective curves with a source at an infinity. To select one reflecting curve from this set it is necessary to put straight line, perpendicular to the direction of the incident rays. The shaping of a reflecting curve and its caustics according to its given orthotomics, as well as the shaping of orthotomics according to a given reflective curve with a source at an infinity, is studied, and algorithms for these shaping are proposed. Equations are written for all studied curves. Caustics, orthotomics, and reflective curves with a source at infinity are a promising means of using the methods of geometric optics for modeling natural objects in ergo design.
Publisher
MONOMAX Limited Liability Company
Reference6 articles.
1. Cayley A. A memoir upon caustics// Phil. Trans. Roy. Soc. Lond. 147:273-312; Collected Works. Vol. 2. pp. 336-380 (1857).
2. Glaeser G. Reflection on Spheres and Cylinders of Revolution. Journal for Geometry and Graphics// Volume 3. No. 2, p. 121 – 139 (1999).
3. Bruce J.W., Giblin P.J. Curves and Singularities: A Geometrical Introduction to Singularity Theory. Cambridge University Press. 262 p. (2012).
4. Bruce J.W., Giblin P.J., Gibson C.G. On caustics by reflection, Topology. I.21. 179 – 199 (1981).
5. Podgorny A.L., Dvoretsky A.T. Model of a plane reflective system for parallel incident rays// Proceedings of the Tavriya State Agrotechnical Academy. – Issue 4, volume 15. –Melitopol : TSAA, p.p. 12-15 (2002).