Math formula for a football shaped object

Jewelry Discussion
1

While we are discussing math formulas for shapes, I wonder if
someone can point me in the right direction? I am looking for a
method of calculating the panels needed for a football shaped
object… Think 5 panels with an elongated football shape. I want
the panels to be curved.

I figure that my desired circumference should be divided by 5 to get
the width of each panel, but how do I calculate the lengthwise arc of
each panel? Am I looking for “sections of a curve?” Or am I looking
for some variation of a cone? (I have this calc.) I have a pattern to
create a ball out of paper; the problem is that I do not want a
ball… I do not know what to look up to get to this and would rather
not go by trial and error.

If you know what sort of formula to look up could you help me out,
please?

Thanks so much!
Mary Ferrulli Barker

2 
 If you know what sort of formula to look up could you help me out,
 please?

I’d do it the easy way, model a football shaped object, in my CAD
package and export it to Pepakura, this makes paper folding
templates.

Regards Charles A.

3

The (American) football shape is a special case of ellipsoid called
a prolate spheroid, which is an ellipse rotated about it’s major
axis.

Draw the football shape (ellipse), to scale, with the long axis
horizontal.

Draw the long and short axis diameters to divide the shape into 4
equal parts, or quadrants.

Choose one of the quadrants to work with - you can ignore the other
three. I’ll assume you chose the top right-hand quadrant.

Divide the quadrant into vertical strips by drawing vertical lines
from the horizontal axis to the circumference of the ellipse. They
don’t have to be equal distances apart, but the more you have the
more accurate will be the final result. I would suggest 6 or 7. The
narrowest strips should correspond with the part of the ellipse that
curves the most, ie. the right hand end.

Each of the (vertical) lines represents the radius of a circle on
the football shape.

From the left, measure the length of each line, starting with the
left edge of the quadrant - the last one is, of course, length zero.
Then (here’s the hard part) measure the length of the ellipse
circumference between each line. Try using a pair of dividers to
"walk along" each section. Be as accurate as you can 'cos this
determines the length of each panel.

You should now have three measurements for each strip: the length of
each vertical side, and the length of the section of the ellipse
circumference.

For each vertical line, calculate a C_Length equal to 1/10 of the
circumference of the corresponding circle - ie. half the part of the
circle that crosses one of your 5 panels. If the length of a
vertical line is L, the C_Length will be 2PIL/10 = 0.628*L
(approx). You are now ready to start drawing one of you 5 panels. You
start by drawing one quadrant of a panel, as follows.

Draw a horizontal line with a length equal to the sum of the ellipse
circumference lengths of the strips you drew. Starting at the left
end of the line, draw a vertical line upwards equal to the first
C_length. Then space along the horizontal line a distance equal to
the ellipse length of the first strip, then draw another vertical
line equal to the second C_Length. Space along by the length of the
ellipse circumference of the second strip, then draw another vertical
line equal to the length of the next C_Length.

Continue to the end of the horizontal line. The last C_Length is, of
course, zero.

Now draw a nice curve through the tips of each of the vertical
lines. Make sure the curve is horizontal at the top of the first
C_Line, and vertical at the last (zero length) one. That curve, plus
the horizontal line and the first C_Length line is the outline of one
quadrant of one of your 5 panels. Mirror copy it to make one half of
a panel, then mirror copy again to make the entire panel.

I hope this helps. Email me off-line if you need better
instructions. Regards, Gary Wooding

4

I think I’d do this without the math other than figuring
circumference. All you need is a piece of paper, a ruler, and a
compass. First get the circumference, (2r times pi)

Example; A football shape 2 inches by 3 inches. Radius = 1 inch
times 2=2, times 3.14= 6.28 inches in circumference, divide by 5 =
1.25 inches per segment (Rounded).

Draw this circle on paper. Draw a line through the center of the
circle 4 or 5 times the diameter of the circle and call it A. Draw a
line perpendicular to the first, through the center of the circle
and call this B. The A line will be the length of the shape so mark
off from the center 1.5 inches on each side of the center, (1.5 by
1.5 inches = 3 inches). Using your compass, on the B line as a guide
adjust so as to intersect the point where the B line crosses the
circle and then to the length marks on the A line at the same time.
Takes some playing around but will work perfectly, even on a small
scale. I imagine that there is a way to do this geometrically but
this is a practical way without all the math.

Best of luck,
Tom

5

Scalene ellipsoids and prolate spheroids! Don’t worry, they’re not
contagious.

Check this out in Wikipedia Spheroid - Wikipedia

Also I found more calculators here

I used these this spring to quote a customer on how many diamonds I
might need for earrings that would be 6.6 mm long spheroids. It was
too many and she bailed on the job. Oh well.

Good luck.

Greg Brooks, co-owner
Ostling & Brooks, Ltd.
www.ostlingbrooks.com

6

Just as an exercise I just modeled a football shaped object, opened
it in Pepakura, and asked it to make me a single template, it took 5
minutes.

If I wanted to get fancy it would have taken a little longer.

Regards Charles A.

P.S. The idea I’m hoping people are imagining is that any shape you
can model, Pepakura can make you a fold-able template for it…
complex hollow ring anyone?