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Thursday, December 8, 2016

Spherical Triangles Explained

What is the sum of the angles on a spherical triangle?

Here is a simple version that ignores the technical details:

Tiny triangle on huge ball, triangle sum will be close to 180° but never exactly 180°
Along the equator, cuts ball exactly in half, is exactly 540° (3x180)
Go all the way around and you get (3x360)-180 is just under 900°
And the ball radius matters to figuring out the actual angles but doesn't change the minimum or maximum possible values.

Here are the technical details along with a (hopefully easy) to understand rationale:

The first point I would like to make is that we first have to define what we're going to count as the angles.  On a simple plane we're ALWAYS talking about the INTERIOR angles of the triangle but a sphere doesn't technically have a natural INTERIOR / EXTERIOR - so you have to define what you want to measure.  Usually people just take the smaller of two and know there is an inverse as well but you can calculate both sum of angles just as easily.  So we will do that.

First of all the SIZE of the sphere is irrelevant to the minimum and maximum values.  Just as the center of a circle has 360° around it regardless of the size each point on the sphere has only 360° immediately around it.  However, the size of sphere does affect the actual angular values for a triangle of some size so you DO NEED that information.  It's not totally irrelevant any more than the length of the sides are irrelevant -- it just doesn't change the minimum or maximum values.

Why?

Imagine drawing a 1" x 1" x 1" triangle on one of those rubber balls you get from the vending machine - let's say it's a 4" circumference ball.  It would go all way from the North Pole, down to the Equator, all the way 1/4 around the Equator, and then back up to the North Pole.  Each turn would be a full 90° turn, so that's 270° total.  This is trivial to verify on your own, just make each line 1/4 of the total circumference of whatever sphere you have handy.

Now, draw the same 1"x1"x1" triangle on one of those GIANT beach balls (or any other large sphere).  It's going to look almost exactly like a normal 180° triangle - because it barely curves around the ball at all.  If the ball is a perfect sphere it WILL be slightly more than 180° but so slight you couldn't tell with your eyes.

But no matter WHAT the size of the sphere is, the minimum is 180° and the maximum is 540° or 900° depending on which side of the triangle you are counting.   I will cover each case in detail...

Now let's draw a "triangle" made of 3 points, but all three points are EXACTLY on the Equator of our ball.  That means each of those "angles" is 180° and we have 3 of them so that's 540° total.  Easy.

Exact same answer regardless of the SIZE of the ball BUT, hopefully obviously, the length of the line would have to be longer to make it all the way around.  Ok so far?

But what happens when you move one of those points a little "further" past the equator?  Do you only count the "interior" as being the smaller angles or do you allow it to continue past 540°?  You can do both of course.

This next bit will explore that further....

Ok, let's start with a microscopic, itty bitty, triangle on a HUGE GIANT sphere billions of miles around and nanometers for each leg of our triangle.   That's going to be so close to 180° we could never measure the difference.  So that is the minimum size (180°) and no matter how big our sphere or tiny our triangle it will always be every so slightly more than 180° total.

Let's make the triangle bigger and bigger and bigger... the sum of the angles will grow... they will hit our 270° mark, grow and grow bigger and now we hit 540°... but let's allow it to keep "going" past this part but STILL count the sum of angles on the SAME SIDE of the triangle.  Now our 3 points get closer and closer and closer together on the other side... eventually they are microscopically close and the EXTERIOR angle is back very very close to 180° right?

So what is the OPPOSITE of 180°?

Well, each point has 360° around it so that 3 x 360° -- and we subtract out the 180° which leaves just shy of 900°.  So that is the maximum angular sum for a triangle and the size of the triangle doesn't change that.  It DOES change how long your lines need to be to get to almost 900° but it doesn't change the maximum value.

So the answer is >180° and <=540° or <900° depending on which angles you want to count.

These values hold regardless of the size of the sphere BUT you do need to know the spherical radius along with the lengths of the sides to figure out the actual angles.