Calculating Gore Shape

    Before constructing my domed observatory, I was unable to find any literature or calculations for gore shape.  The following text describes my method and calculations.  Although this method is difficult the resulting gores fit perfectly.  If there is a simpler method I hope someone will point it out.

UPDATE: A young engineering student provided an Excel spreadsheet that calculates the values but asked that I not post it on my site. I can however, provide you with the values if you ask.

    The following line drawing is a very simplified illustration of the pattern I created to measure the dimensions required for the calculations.  My observatory is ten feet in diameter.  For larger or smaller observatories only the X and Y axes will have to be changed.  One note for construction design is that a gore for a ten foot diameter dome measures exactly 95 5/8 inches long, only 3/8 inch short of standard eight foot building material.  However, one or more feet will be removed from the gore when the aperture is cut out.  I only mention this because larger domes may require gores (even considering the removal of the pointed end) to be longer than eight feet.  Splicing material would introduce greater construction difficulty and the potential for leaks. 

    To begin I taped together several large sheets of graph paper being careful to keep the lines perfectly square.  For the ten foot diameter dome the resulting graph paper should be slightly larger than 5 feet by 5 feet and have horizontal one-inch lines.  Draw a five foot line for the Y axis and a perpendicular five foot line for the X axis.  Now draw the Z axis as precisely as possible.   A string and a pencil is one method.  The X, Y and Z axes are easy to draw accurately in U.S. Standards (feet, inches).  However, I decided to record the remaining measurements in millimeters to counteract misshapen gores due to the additive effects of a possible miscalculation as well as inaccuracies introduced by saw cuts.  Now the fun begins...

    Because the shape of the gore presents no linearity (at least to me) I found I had to measure points along the curved Z axis from the origin (noted above) to the top of the Y axis.  The important points to record are the intersections of the one-inch horizontal lines and the Z axis.   It is important to remember that the measurement of each intersection is from the origin, not the preceding intersection.  Otherwise the measurements would have to be added together.  As the curve of the Z axis begins to flatten near the top, prior to intersecting at the top of the Y axis, the distance between the one-inch lines becomes greater and introduces less precision in the shape.  To counteract this problem I switched to one-half inch increments at 48 1/2 inches up the Y axis.   This is illustrated in the line drawing above by comparing the Z axis length between the horizontal lines at point A and point B.   I recorded a total of 76 measurements along the Z axis (I'll call these the Z axis measurements (ARM)).  It just gets better...

    Next I measured the length (in millimeters) from the Y axis to the Z axis of each one-inch horizontal line. I began at zero, or the X axis, and recorded the measurement up to the the 48 inch mark then every one-half inch up to the 60 inch mark.  This essentially gave me the radius of the hemisphere at one inch increments or less from bottom to top.  I ended up with 76 measurements.  Find your calculator...

    Next I calculated the circumferences from the 76 measurements above.   This is simply 2rp or 2 times the radius times 3.14159.  Now determine the number of gores you want.  I made 32 gores.   The more gores the more round the dome appears but more cutting and trimming is required.  Divide each circumference by twice the number of gores you desire (I'll call this C2G).  I had 76 C2G values.  Sharpen your pencil...

    Next I made a template from quarter inch plywood.  Draw a straight line from the top of the plywood to the bottom  approximately one foot in from the edge (maybe a little more if you're making fewer gores).  Now plot all points from the Z axis measurements (ARM) along this line.  I started plotting from the top to the bottom.  This allows any excess plywood to be at the bottom of the hemisphere for finishing, drip edge or whatever technique you want to use at the bottom of the domed portion of the observatory.  Remember the largest number (ARM value) will be at the top or what will become the point of the gore.  Draw a perpendicular line intersecting at each plotted Z axis measurement (ARM).  The length of this perpendicular line can be either (1) exactly the value of the corresponding C2G (above) on each side of the line, or (2) a long perpendicular line can be drawn through the center line and points corresponding to C2G values can be plotted on the perpendicular line on each side of the center line.  Now simply connect the ends of the parallel lines or connect the plotted points and you have the gore.  I hope!  See illustration below for clarity. 

The table below is values for a 10' diameter dome made from 32 gores with 76 points (ARM) measured along the "Z" axis. An "IMPORTANT" note - the "Measured" or "Formula" values must be divided by 2 to obtain the C2G value in the illustration above.  Half of the "Measured" or "Formula" value will go on each side of the centerline.