Having Fun with Ellipsoids!

We are currently working on understanding strange things like ellipsoids and hyperbolic paraboloids and all sorts of strange things like that in my Calc 3 class. It seems like it wouldn’t be that hard, but it is pretty hard to draw these things correctly! This week we got a take home quiz! Thank goodness! I’m posting one example where we are supposed to name the shape and then draw it and talk about the plane sections that intersect it. I don’t know if I’m doing it correctly or not. Guess I will find out!

Basically, how you know it is an equation for an ellipsoid is that an ellipsoid has a standard form for the equation which is basically (x/a)^2 + (y/b)^2 + (z/c)^2 = 1. Now there are many variations on that, so that it might look a bit different, but if it is too different it will be a different shape, like a hyperboloid of one sheet or something else like that. The a’s give you the ellipse size along the x-axis, the b’s give you the ellipse parts for the y-axis and the c’s give you the ellipse information for the z-axis. It is kind of hard to explain in words! lol I guess what you first need to know is that a regular old ellipse is drawn on the x and y axis where the a number gives you information on the number of steps away from the origin along the x-axis and the b number gives  you the number of steps away from the origin along the y-axis. I guess I better post an example.

In the example below #1b, you look at the bottom numbers to give you the size of the ellipse. The x-axis number is 5, so it is five units wide on each side. The y-axis number is 10, so it is 10 units wide on each side of the origin. The x-3 and y-3 has to do with shifting the graph away from the origin up and over 3 units each.

Anyway, from what I can gather, to draw the ellipsoid (not ellipse) like in example 2a, you need to draw the various ellipses that make up the ellipsoid in order to get the final shape, so you set x, y and z equal to zero for three different cases to get your ellipse equations. Blah blah blah. This probably makes no sense to anyone, but I thought I’d share! lol

Hyperbola (top) and Ellipse (bottom)

Ellipsoid Example