Original scientific paper

https://doi.org/10.31896/k.29.6

A Geometric Construction of a Family of Keplerian Ellipses

Georg Glaeser ; University of Applied Arts Vienna, University of Applied Arts Vienna, Vienna, Austria *

* Corresponding author.

Abstract

We investigate a one-parameter family of Keplerian ellipses in a plane sharing a fixed focus F_1 and passing through a prescribed point P with identical instantaneous speed. By means of a purely geometric construction – the reflection of the ray F_1P in the tangent at P – the second focus F_2 is located on a circle f_2, yielding simple loci for the centers M and the secondary vertices C,D (both circles) and for the principal vertices A,B (conchoids of a circle). The family admits an envelope, itself an ellipse whose semiaxes are obtained in closed form. The configuration provides a direct geometric interpretation of the vis-viva relation: All members share the same semimajor axis a, and thus, the same orbital period. When rotated about the axis F_1P, the envelope ellipses form a family of confocal ellipsoids of revolution, thus connecting the planar Kepler construction with the classical geometry of quadrics.

Keywords

Keplerian ellipses; envelope ellipse,; conchoid; limacon; vis-viva relation,; energy equation

Hrčak ID:

341893

URI

https://hrcak.srce.hr/341893

Publication date:

19.12.2025.

Article data in other languages: croatian

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