Math CentralQuandaries & Queries


Question from Herman:

How do I solve for a segment of a sphere so that the orange peel or pie shaped section is converted to a flat surface with dimensions. I form large diameter domes, elliptical and sphere heads on a press. I enter the diameter say 30 feet with two segments above and below the equator and the total number of segment around the circumference at say 18 so the widest part of the pie shaped section will fit my press. How do I take the upper or lower course above or below the equator and figure the height of the orange peel shape, the chord length at top and bottom, and solve for the right angle at 2 degree increments down the arc length of the pieces so I can layout the flat plates prior to pressing. Thanks for your help.

As you probably realize, if you were making a cylinder or a cone, you would (with many materials anyway) not need the press. This is because those surfaces,unlike a sphere, have no "intrinsic curvature" so you can bend the pieces to shape without deforming them. For a sphere this is impossible. As a result, if you are pressing your piece into a spherical mold, the material will be deformed.

If the segments were being bent into place without any intrinsic deformation (you would not get a perfect sphere of course) then the total height of the segment would be (pi/2)r [you can use 1.5708 r, good to a millimeter] while (with 18 segments) the width of each would be 2r x sin(10 degrees) or about 0.3473 r at the base. Call that width W.

Now, as you go up in 2 degree increments (that is, increments of height/45, or 0.0349 r) the width at the nth site will be cos(2n degrees) W. The first few of these values will be very close to 1:

n=0: width = W (= .3473 r)
n=1: width = 0.9994 W (= .3471 r)
n=2: width = 0.9976 W (= .3465 r)
n=43 width = 0.0698 W (= .0242 r)
n=44 width = 0.0349 W (= .0121 r)
n=45 width = 0 ("north pole")

If you are using a spherical mold and pressing causes your elements to be the wrong shape, you will probably have to resort to ad hoc methods. Try first adjusting the height till it presses to the right final height. Divide the height into 45ths and press another piece; then measure the widths. At each site, compute a new width using the formula

new width = (computed width)^2 / (pressed width of test piece)

and it ought to be pretty close.

Good Hunting!

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