Ellipses

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 r=\frac{1}{1-k\cos\theta} gives rise to ellipses if 0<|k|<1
    ellipse1
  • An ellipse arises out of squashing or stretching a unit circle horizontally or vertically (or both)ellipse4
    which means that algebraically…the rectangular equation is (\frac{x}{\square})^2+(\frac{y}{\triangle})^2=1
  • An ellipse arises out of looking at a circle straight on (so it looks like a circle) and then tilting that circle.ellipse3
  • Ellipses can be created by taking a cone (or cylinder) and slicing it at a variety of anglesellipse2This is equivalent to shining a flashlight at a wall at an angle:ellipse5
  • 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)ellipse6
    [
    not drawn to scale, obvi.]
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13 thoughts on “Ellipses

    • I don’t know if you mean you don’t see the value of drawing perfect ellipses (meaning they need to be drawn accurately/precisely)… or that you don’t see the value of drawing ellipses by hand at all (meaning: we have calculators and computer programs that can do that).

  1. Great time of yr to go outside with sidewalk chalk and string to draw ellipses. Have you ever tried it? Use one kids’ legs as foci, tie string into circle, and another kid pulls string taut with chalk and marks out a great ellipse. Possible to do hyperbola but trickier. They can play w/width between foci, difft string length.

    • Awww cute! I didn’t do that. I’m stupid and just taped pieces of strings of different length to large whiteboards, and then briefly had kids draw them on the whiteboards with dry erase markers.

      But next year… yes.

      The hyperbola in the same way? Perhaps, though I have my doubts.

  2. Sam

    I stole your idea from a recent post when my Precalc Honors class started our conics unit. We used Desmos and a slider and came up with some wacky shapes as we modified. The kids were engaged, were curious about the results of their guesses and even got a little competitive. Super cool. It was also a nice way to remind them that trig has not gone away even though we were safely back in a world devoid of trig functions.

    Love Reilly’s idea of sidewalk chalk. If spring ever does arrive for real here in PA we might go out and do that. I used to do shoestrings and thumbtacks on my door

  3. Or: if A is a nonsingular 2×2 matrix, and v = (cos(t), sin(t)), then the set Av will be an ellipse (for t in [0,2pi]). Probably beyond where you want to go but it could be an excuse to introduce matrix-vector multiplication.

  4. I’ve used Geometer’s Sketchpad to construct ellipses by using the locus of a line perpendicular to a line segment, where one of the endpoints of the line segment is on a circle.

  5. This is a really terrific video/animation (and part of series on conics) that demonstrates ways to construct ellipses:

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