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Newton's Ellipse

Newton showed this construction in Book 1, Section 4, Lemma 15, of Principia. On March 13, 1964, Feynman resurrected the construction and used it in a lecture, "The Motion of Planets Around the Sun". The lecture is detailed in a book with audio CD, Feynman's Lost Lecture, by David and Judith Goodstein. In the lecture, Feynman used the diagram and differential geometry to prove the planetary laws of motion.
In the construction, the green dot is the primary focus of the ellipse about which the planet orbits; the blue dot is the secondary focus. The black dot is on a circle at a distance in radius equal in length to the major axis of an ellipse. A line is drawn from the blue dot to the black dot and its perpendicular bisector is constructed. The point where this perpendicular bisector intersects the line from the green dot (primary focus) to the black dot (circle) is a point on the ellipse. The perpendicular bisector is tangent to the ellipse at this point. A further interesting point of the construction is that the length of the line from the blue dot (secondary focus) to the black dot (circle) is proportional to the velocity of the orbiting planet at this point. In the Demonstration, half this length is represented by the black vector traveling in the direction of the planet.
All of the possible ellipses with the given major axis are contained in the circle. You can adjust the eccentricity and rotation of the ellipse.

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The construction is done in the complex plane, using the vector properties of complex numbers to construct the lines to the points. The equation of the circle is , where is the semi-major axis of the ellipse; the equation of the ellipse is , where is the radius of the ellipse from the primary focus at that point. The construction can be redone so that the secondary focus is outside the circle, creating a hyperbola, or on the circle, where it should create a parabola. A simple geometric proof can be found in chapter 9 section 5 of Jeremy Tatum's online text Celestial Mechanics.
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