An ellipse can be defined as the set of all points whose distances from a given two points sum to the same number.
What shapes are formed when we extend that definition to have three points and the curve specified by all the points whose distance from each of the three points adds to a given number? What about with four specified points? “N” specified points?
Some characteristics I can think of for these shapes include the following: for a greater number of specified points, our allowable sum may need to be larger, to produce a figure of equal area, since more lines must be drawn to it.
With a set of specified points which are colinear, I’m not sure if all of them would resemble an ellipse, but my first intuition is that they would. I have no proof of this idea.
P.S.—I’m not sure if this question falls under the scope of algebraic geometry, so if I should take off that tag, please tell me.
Edit: I’ve specified this algebraically in Desmos, and I’ve seen some solutions. They don’t resemble anything I’ve seen before—like ellipses with corners sometimes. What would these be classified as? Here’s a link to the Desmos graph: https://www.desmos.com/calculator/idimvmdzdw
