How is an ellipse mathematically defined based on the sum of distances relative to its two fixed points?
Answer
As the set of all points where the sum of the distances to the two fixed points (the foci) is constant.
The formal geometric definition of an ellipse relies entirely on its two foci. An ellipse is precisely defined as the locus of points in a plane such that the sum of the distances from any point on the curve to the two fixed points (the foci) remains perpetually constant. When applying this to a planetary orbit, one focus holds the Sun, and the other focus remains empty space. This constancy of the sum of distances dictates the oval shape and the relationship between the semi-major axis and the focal distance.
#Videos
Why Are Planetary Orbits Elliptical? - YouTube
Related Questions
Where is the Sun located according to Kepler's First Law of Planetary Motion regarding the ellipse?What term quantifies the degree to which a planet's orbit deviates from being a perfect circle?What path shape results if a celestial object's eccentricity ($e$) is 1 or greater?What is defined as the longest radius of an ellipse, representing half its longest diameter?What term describes the point in an orbit where a planet moves at its maximum orbital velocity?What is the name given to the farthest point a planet reaches from the Sun during its revolution, where its speed is minimum?Which physical conservation law necessitates the variation in planetary speed between perihelion and aphelion?What orbital parameter, approximately 0.205 for Mercury, results in the highest eccentricity among major solar system planets?What did Isaac Newton provide that explained the underlying physical cause for elliptical orbits described by Kepler?How is an ellipse mathematically defined based on the sum of distances relative to its two fixed points?How do the orbital eccentricities of typical comets compare to those of major planets like Earth in terms of path stretching?