
Take 2 circles of the same radius in 3 dimensions, such that the circles lie in parallel planes and the line joining their centers is perpendicular to the planes. If we create a surface by joining corresponding points on the two circles, we get a cylinder. If we instead we join points on the two circles which are 120 degrees apart, say, we get a hyperboloid.
We’d like to find the curve which is observed when we view the surface. Although this is a 3 dimensional problem, we can work in 2 dimensions. We can model the circles in perspective as ellipses. Parametric location on an ellipse is equivalent to angular location on the 3D circle which the ellipse represents. Finally, the envelope of the 2D family of lines represents the profile of the 3D surface.
To create the model, we first draw an ellipse, give it an equation, and translate it by a vector perpendicular to its axis. We then join a point at parametric location t on one circle to a point at parametric location t+2n/3 on the other.
Taking the envelope as t varies gives us an image of the hyperboloid. The envelope of the lines gives us its profile as a curve: We can from the equation that this locus is a hyperbola. We can find its foci by creating a hyperbola and giving it the equation of the locus.
What do you observe as you reduce the numeric value of b? As our model approaches one where we are looking sideon at the hyperboloid rather than obliquely, what value does b approach? What would be the equation of the outline of the curve be in this case? What would the location of the focus be?
Expressions

