This research deals with the development analytical-geometric framework for the prediction of a satellite's entry into and exit from a planet's umbra and penumbra. The core of the method lies in developing the general approach for conic model, the intersection of which with the satellite's orbital plane forms a "shadow ellipse"—a second-order curve. By deriving the parameters of this shadow ellipse and solving for its intersection with the satellite's Keplerian orbit, the problem is reduced to finding the roots of a universal fourth-degree equation. This solution directly yields the true anomalies at the eclipse boundaries, from which the exact eclipse duration can be computed using Kepler's equation.

The method's efficacy is demonstrated through application to a high-eccentricity orbit of a prospective Venus orbiter, relevant to missions like Venera-D. Validation against high-fidelity numerical simulations in a full ephemeris model confirms the approach's accuracy, with results showing close agreement. Furthermore, the concept of an "invariant surface $S(\Delta T_{ED}, \Omega, \omega, i)$ is introduced, providing a powerful tool for visualizing and predicting long-term eclipse dynamics over multiple orbital revolutions.

Beyond calculating eclipses by the solid body, the method is successfully extended to a critical practical application: determining the parameters for atmospheric occultation experiments. It enables the fast and still reliable calculation of the time a satellite spends in the region where the planetary atmosphere occults the Sun, as well as the determination of the latitude and longitude of the perpendicular to the satellite-Sun line on the planet's surface. This demonstrates the versatility and direct utility of the approach in the design and planning of advanced scientific missions to Venus and other planetary bodies.