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When we cover quadric surfaces in a multivariable calculus class, one of the things that students struggle with the most is the saddle point in a hyperbolic paraboloid. Part of this is because hyperbolic paraboloids are a bit tricky to draw well, but another reason is that saddle points can be tricky to understand at first because of the different directions of curvature.
First, consider the red curve. This a parabola that opens downward. Traces in parallel planes are also parabolas that open downward. So in this direction, the curvature is downward.
Next, consider the blue curve. This is a parabola that opens upward. Traces in parallel planes are also parabolas that open upward. So in this direction, the curvature is upward.
This creates a unique point in the middle that is the vertex of both a downward- and an upward-opening parabola. We call this a saddle point.
When I was in Tehran, we visited the Azadi Tower, and I immediately noticed that the underside of the tower resembled a hyperbolic paraboloid! The downward-opening parabola in red is fairly easy to see in the photo; however, the upward-opening parabola is a bit difficult to see because of the angle. Part of it is colored here in blue, but the rest curves up on the back side of the building, which makes it harder to follow.
You can read more about the tower at http://www.miimdesigns.com/freedom-tower
Sample Problems
Sample Problems
1. Research other instances of hyperbolic surfaces and saddle points in architecture. In particular, look into saddle roofs. Which sorts of buildings have saddle roofs? Why might this be the case?
2. What are some benefits of designing a building with a saddle roof? For instance, how would rainwater drain off of a saddle roof?
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