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/sci/ - Science & Math

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File: 8 KB, 392x235, spherical20greta20circle20picture.png [View same] [iqdb] [saucenao] [google]
4341391 No.4341391 [Reply] [Original]

Let's say you're on a planet that's a perfect sphere with radius r. If you keep walking in what seems to you to be a straight line, you'll eventually trace out a great circle (of radius r) and end up where you started.

What if you want to trace out a circle of radius r/2? How should you walk?

>> No.4341618

bump

>> No.4341650

You will be walking on the plane that acts as a "perpendicular bisector" of the radius of the circle in terms of the center of the sphere.

>> No.4341656
File: 5 KB, 231x231, SphericalCap_1001.gif [View same] [iqdb] [saucenao] [google]
4341656

The area of a spherical cap is S = 2*pi*r*h , where h is as in the pic.
You want to have a = r/2 , i.e. R² = (R-h)² + r²/4
Solve for h and you're done.

>> No.4341662

>>4341656
R should be read as r

>> No.4341675

(r/2)^2 = x^2 + y^2

r^2 - (r/2)^2 = z^2 -> z = sqrt(3)/2 * r

move z off center and walk?

>> No.4341698

>>4341656
Actually knowing the area of the spherical cap is useless here I don't know why I brought that up. You just gotta solve the equation, and then knowing a=r/2 and h you know where the small circle you're supposed to walk on is located.

>> No.4341839

you will need to walk curving constantly to one side.

on great circle: you walk straight ahead

going around the pole: walk turning to the side very sharply

on "small circle": walk turning to the side gradually

to figure out how sharply to turn, need differential geometry

>> No.4342705
File: 38 KB, 500x376, bump.jpg [View same] [iqdb] [saucenao] [google]
4342705

>>4341839
Intriguing.

>> No.4343268

circle of radius r/2?

move to 60° latitude, go east or west.