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 Post subject: Math/Geometry questionPosted: Mon Sep 27, 2010 1:18 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Hello Guys,

I've got a math/geometry problem for you. On the surface of a sphere I have both a great circle and a small circle.

On my sphere however the two cross each other. To move from one crossing point to the other along the great circle one rotates about the axis of the great circle by 60 degrees. If I know the angle between the axis of the great circle and the axis of the small circle how do I determine the angle one rotates through going between the two crossing points along the small circle? It's something greater then 60 degrees and less then 180 degrees depending on the angle but the function is failing me.

If theta is the angle between the two axes and the function I'm after is f(theta) then I have:

f(0) = 60
f(90) = 180

What is the general f(theta)?

I could use this in my POV-Ray model of the Mixup Master Skewb.

Thanks,
Carl

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 Post subject: Re: Math/Geometry questionPosted: Mon Sep 27, 2010 1:37 am

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
Take the radius of the great circle and the angle you have to rotate (so 60 deg) and use that to find the distance between the two points
Then just set that as the base of an isosceles trangle with the planar radius of the small circle as the legs.

Edit:
Oh wait, I'm messing up what you're given.
If you're doing this as a function of the angle between the axes, then you could:

Draw the sphere as a circle, draw the great/small circles as lines (because they're being viewed directly from the side)
Find the point where the side-on circle's lines intersect.
Take the distance from the axis of the small circle to the point you just found.
You now have the height of an isosceles triangle formed by radii in the small circle.
Some basic trig later, and you've got the angle

Edit 2

Actually, your function should take 3 arguments, Radius of the sphere, radius of the small circle, and angle between the axes.

Code:
def f(rSphere,rSmall,theta):
hSmall = (rSphere**2 - rSmall**2)**.5
x = hSmall * tan(90 - theta)
alpha =acos( x / rSmall)
return alpha*2

What each line does:
First find the length of the axis of the small circle
Then find the height of the isosceles triangle
Find half the peak angle using trig
double it and return

 Attachments: circles.gif [ 4.92 KiB | Viewed 839 times ]
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 Post subject: Re: Math/Geometry questionPosted: Mon Sep 27, 2010 9:42 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
Edit 2

Actually, your function should take 3 arguments, Radius of the sphere, radius of the small circle, and angle between the axes.

Code:
def f(rSphere,rSmall,theta):
hSmall = (rSphere**2 - rSmall**2)**.5
x = hSmall * tan(90 - theta)
alpha =acos( x / rSmall)
return alpha*2
Thanks. I think I can get what I need now. However in the general case I don't think you need both radii. You just need the ratio of the two radii because if you scale everything up or down it shouldn't change the angle I'm after. And the particular problem I had I was given the piece of info that the two crossing points were 60 degrees apart, lets call this angle phi. The ratio of the two radii should be a function of theta and phi so I should be able to find f(theta,phi). And I think I can take it from here...

Thanks,
Carl

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 Post subject: Re: Math/Geometry questionPosted: Tue Sep 28, 2010 11:29 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Got it!!!

The Slidey pieces are 30 degrees apart along the Great Circle.... but 31 and a fraction degrees apart along the conical paths they are moving along in this animation. I used 30 in the first pass at this animation which produced an odd jump at the end. I suspect most wouldn't have even noticed it but it was bugging me. Thus the question was asked above.

Thanks again,
Carl

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 Post subject: Re: Math/Geometry questionPosted: Tue Sep 28, 2010 12:07 pm

Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Wow Carl.... what are you up to???
Nice puzzle... errr... I mean... animation!

Pantazis

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 Post subject: Re: Math/Geometry questionPosted: Tue Sep 28, 2010 12:29 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
kastellorizo wrote:
Wow Carl.... what are you up to???
It's the Mixup Master Skewb or maybe the Slice-turn only Mixup Master Skewb talked about here:

http://twistypuzzles.com/forum/viewtopic.php?f=9&t=18855

Enjoy,
Carl

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