Dec 19, 2008

How to Make a Puppet Pattern, Part II

The Ellipsoid

Go to Part I if you want to see how to make a sphere. Now, we're going to make an ellipsoid. The formulas are more difficult, but overall, the process is almost identical.

I worked this up last night in my apartment. The circumference (or perimeter--whatever you want to call it) of an ellipse is a really crazy formula:
where (denoted in the C equation as ) is the eccentricity of the ellipse.

"Whoa!" you say. "What are these a's and b's you're talking about?"

Well, my dear friend. Look to the left. You can see in my little drawing that I have denoted both a and b. We are going to make this figure, which is a 'prolate' spheroid (a 'stretched' sphere). The darker edge is one half of the ellipse (oriented vertically). Can you see that b is the radius of a circle in the middle? Good. Now, what do those mean?

a is half of the ultimate height of your spheroid (A is the total height). So, if I want my puppet's spheroid head to be 1.5 feet tall, then a is 1.5/2 = 0.75 feet. b is half of the width of the spheroid (B is the total width; the diameter of the central circle).

That equation above, though, is ridiculous. No one wants to spend the time solving for eccentricity. That is where Srinivasa Ramanujan comes in. He estimated (way more accurately than we care to be) that the perimeter of an ellipse is roughly:
That is (though it may not appear so) way easier than the real formula. Look at that picture above again. See the dark curve marked C/2? That, you can see, will be the height of our pattern. That amount is simply half of the circumference, which means it is roughly the amount to the right: [4.5(a+b) - 1.5*sqrt((3a+b)(a+3b))]. That is by far the most difficult part of this process.

The width of each piece will, like the sphere, be one-eighth (1/8) the circumference of the center circle. Remember that b is the radius of the center circle? That means the circumference is 2πb. Estimate that π is 3, and we get roughly 6b. Now divide that by eight, and we get .75b, as you can see in the formula to the left.

Now, you simply need to follow the instructions for the sphere. Choose your height-- A=2a, your width-- B=2b for the final product. Now, draw a vertical line equal to 4.5(a+b) - 1.5*sqrt((3a+b)(a+3b)). Through the midpoint of that line, draw a horizontal line equal to .75b. Now, copy the angle made by connecting the end of the horizontal line to the end of the vertical line (M), to make angle N. Make an isoceles triangle (as you can see to the right), and use its tip as a center for your compass to creat a curve connecting the top of the vertical line, the right end of the horizontal line, and the bottom of the vertical line.
Do that again on the other side, and you get a figure that looks like this.
Just to demonstrate that the equation really isn't that hard, take my example from before. Say that I want my height to be 1.5 feet and my width to be 1 foot. Using the formula, we discover that the height is simply 2.27 feet and the width is .375 feet. I can now follow the rest of the instructions, cut out eight foam pieces, and creat my ellipsoid!
So--that's how you make a puppet pattern for a sphere or an ellipsoid. I know that it may not make complete sense. Please feel free to write questions in the comment field, and I'll be happy to answer them ASAP


FisticuffDC said...

I've got to say...I'm glad it just works out for me when I do it by eye, because math for me was never a beautiful subject. I love seeing it put to good use though. Had my teachers in high school used examples like this, where we could actually build something, I would have been at the top of the class!

When you get a chance, you should put this into action and show the results. Keep it up. While, I don't fully understand it all, it's really fun to follow. It may convince me to give it a try next time.

Lauren said...

Aaaand....cue ginormous nerddom.

It's ok, we love you anyway