Hard math, made easy, when calculating a horizontal ellipsoid
What are the areas and volume of a horizontal ellipsoid dome? There isn’t an easy answer. Spherical domes have well-defined formulas. Vertical ellipsoids are symmetrical. But tip an elliptical dome on its side, and measuring it gets a whole lot harder.
Several of my math professors took a crack at the problem — and failed. One tried the newly developed Mathematica application and quit when the formula hit 40 polynomials. It took a physicist to put me on the right track. “Mathematicians want something to prove,” he said. “Physicists just want the job done.”
At the time, I was trying to improve the laborious process of designing interconnected domes. My new friend demonstrated how the law of cosines, vectors, and trigonometry would help me automate this process. Since then, I’ve applied his principles to all ellipsoid calculators I’ve developed over the years, including the new Horizontal Ellipsoid Dome Calculator.
Where a vertical ellipsoid dome has a circular floor and an elliptical cross-section, a horizontal ellipsoid dome has an elliptical floor with an elliptical cross-section along one axis and a circular cross-section along the other axis.
Watermelon is a close approximation to a prolate ellipsoid. Imagine cutting a watermelon lengthwise and laying it horizontally on a table. The watermelon forms an ellipse on the table. Looking at it from the side, it’s also an ellipse. If you look at it from the end, it’s a circle. That’s a horizontal ellipsoid.
Cut the watermelon perpendicular to the table into slices — like you are cutting them for a picnic. Measure the thickness and the arc lengths of each slice. Calculate its area and volume. Measure the next slice, and add those together. Repeat the process across the whole watermelon. Add it all up, and you have an approximation of the surface area and volume of the watermelon.
We figure the surface distance is about seven significant digits, the surface area is about six significant digits, and the overall volume is around five significant digits.
Horizontal ellipsoids are rarely used in dome construction. There’s nothing wrong with the shape, but for many domes — especially small domes — there’s no particular advantage to it, either. Starting with a circular base makes sense for many designs. It’s also what people expect when you say “dome.”
However, we are missing opportunities if we limit ourselves to just drawing circles on the ground. Elliptical domes expand the design palette in ways a circular dome cannot.
The Eye of the Storm dome home, for example, was meant to look like a shell on the beach by the Atlantic Ocean. The prolate shape with giant openings, large decks, and white exterior gives the home a singularly beautiful appearance. More importantly, the broad “side” expanse of the dome faces the ocean giving everyone inside a breathtaking ocean view.
Tupelo, Mississippi, built two prolate ellipsoid tornado shelters on concrete stem walls. Narrow land constrained the capacity of a standard, circular dome design. By “stretching” the dome into a horizontal ellipsoid, it fits on the land and shelters more people.
A chain of horizontal oblate ellipsoids created the caterpillar-shaped manufacturing plant in Italy, Texas. Two recent structures use the elliptical dome set on an angle atop circular stem walls.
The Monolithic Dome industry is still young, and we are just beginning to tap the potential for air-inflated designs. By creating this calculator, we are adding another tool for designing future structures.