The St. Louis Science Center Planetarium's HYPERBOLOID OF ONE SHEET exterior curved surface was conceived by Mr. Chih-Chen Jen, AIA. His wife played an integral part in the early design concept when she held parts of the model in the air while the design sketches were completed. This model was constructed by stretching parallel strings from the edge of one disk to the edge of another equal radius disk. Then twist the disks. (Example)
A mathematical model of a hyperboloid of revolution of one sheet is a hyperboloid of one sheet where a cross section perpendicular to the axis of revolution is a circular region and a cross section through the axis is a hyperbola of the form x2/a2 - y2/b2 = 1. If this hyperbola, or at least one side, is turned about the y axis it will generate, or carve out, the surface. Each point on one side of this hyperbola will generate a circle on the surface when the point is rotated about the y axis.
a) Find the volume enclosed by the rotation of a side of the planetarium hyperbola about the y axis. (Disk Method) Pick a point at top of the 72-gon outer windows for the lower limit of your integration. Find a point where the roof top intersects the planetarium's hyperboloid for the upper limit of integration. Then your volume will help with air conditioning and heating calculations.
b) Find the surface area of the planetarium's hyperboloid of one sheet. Rotate a short snip of arc length about the y axis with limits of integration that get the entire surface. That surface area will help you purchase enough covering material to surface the outside.
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