June 19, 2009
Hemisphere geometry

Considering building a small hemisphere for projection, I wondered about the geometry of the sections of a dome. This tutorial, while applying to parachutes, has the formulas for the sections of the hemisphere - a triangle from the pole to the equator has more than 180 degrees total in its angles. I knew that, but not the formulas.

Hemisphere Parachute Design

And here's someone who tried to do the same thing for a dome.

Building an inexpensive usable projection dome. - Observatory Central

His numbers for a 10' dome are below.

10 foot diameter with 10 gores. Circumference 31.416 feet = 376.99 inches 1/4 circumference = 94.25 inches which is the length of each gore from spring line to zenith.

From the base (spring line) in inches = B (inches)
Width of gore in inches (1/2 on either side of a perpendicular line running from the center of the base [spring line] to top of gore) = W

B ~ W

0 ~ 37.699
5 ~ 37.568
10 ~ 37.177
15 ~ 36.527
20 ~ 35.624
25 ~ 34.474
30 ~ 33.084
35 ~ 31.465
40 ~ 29.628
45 ~ 27.585
50 ~ 25.35
55 ~ 22.94
60 ~ 20.37
65 ~ 17.659
70 ~ 14.825
72 ~ 13.662
74 ~ 12.483
76 ~ 11.291
78 ~ 10.086
80 ~ 8.87
82 ~ 7.6437
84 ~ 6.4092
86 ~ 5.1676
88 ~ 3.9202
90 ~ 2.668
92 ~ 1.4138
94.25 ~ point

These are the exact widths for the dome described and DO NOT allow any extra material for either overlap (if welding plastic) or a seam (if sewing fabric). You must decide what type of support structure you want. In my small experimental dome I allowed for extra material so that support ribs could be contained.

Also note that the widths of each gore get rather small as you approach the zenith. A central support core usually occupies this space and I would suggest ending the dome material before you reach this point. A circle of white poster board would cover this "hole" in the dome and the entire area where the most sewing would normally be done.

If you go back to the very beginning of this thread you will see a small experimental dome with this type of support. Actually my dear wife has sewn together a much larger one approximately 18 feet in diameter which I plan on using 1/2 inch plastic water pipe for supports. It is actually all put together in four sections of four gores each as I'm not at all sure if an entire dome could be handled at once. My thought was to erect each quarter and then have small velcro patches to hold the four joints together. All of my free time has been used up working on my projectors and I have not attacked the dome. I will need to do this just to see how well it works.

Anyway, I would add an extra two inches to both sides of each gore to allow for sewing at the proper point and then making a loop in which to slide a support strut. I will take some pictures next week to show you what I mean. Right now taxes call (ugh).


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Ron Walker

Posted by Dave at June 19, 2009 06:09 PM