What are the ways we can generate ellipses?
We’ve been working with ellipses. I have talked about some of these this year. Others I haven’t. But I like this list for future reference.
- The polar equation gives rise to ellipses if
- An ellipse arises out of squashing or stretching a unit circle horizontally or vertically (or both)
which means that algebraically…the rectangular equation is
- An ellipse arises out of looking at a circle straight on (so it looks like a circle) and then tilting that circle.
- Ellipses can be created by taking a cone (or cylinder) and slicing it at a variety of anglesThis is equivalent to shining a flashlight at a wall at an angle:
- The set of points from two points (called foci) which have a set sum of distances from these two pointsand for a cool video illustrating this (alongside the reflective property of ellipses):
- Drop a planet in space near a massive object, and give it an initial push (velocity)
[not drawn to scale, obvi.]