Indeed, the following generates uniform points on the unit sphere, where “uniform” means that each bit of area has the right amount of probability: . choose z uniformly in [-1,+1] . choose phi uniformly in [0,2pi] . set x=sqrt(1-z^2) cos phi and y=sqrt(1-z^2) sin phi It’s a very nice exercise. Basically the way the circumference of the circle of latitude decreases as z moves away from zero is exactly compensated for by the way the area of a bit of surface of height dz increases as it tilts from vertical to horizontal. Try it! - Cris
On Jul 25, 2019, at 12:10 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Is it really true that the z coordinate is uniformly distributed?
A really common question is, "What's the right way to generate pseudo random points on the surface of a sphere?" The most common answer, and the only one I know offhand, is, "Generate three Gaussian coordinate and normalize.". Why have I never heard, "Generate a uniform z coordinate and a uniform azimuth."?
On Thu, Jul 25, 2019, 1:47 PM Cris Moore <moore@santafe.edu> wrote:
This reminds me of the lovely fact (mentioned before on this forum) that a random point on a sphere of radius r has a _uniformly_ random z-coordinate from +r to –r. (This is true only in 3 dimensions.)
- Cris
On Jul 25, 2019, at 8:30 AM, Veit Elser <ve10@cornell.edu> wrote:
Jim,
Strogatz did quite a bit of research on this for his recent book (pre-calculus ideas). I would first look there.
-Veit
On Jul 25, 2019, at 7:25 AM, James Propp <jamespropp@gmail.com> wrote:
One can prove that the expected distance from a random point on the surface of a sphere to the equatorial plane is half the radius. Assuming we could rephrase this claim in a form that Archimedes would recognize, how would he have proved it?
As an example of the kind of proof I would like to see, consider the proposition that the expected distance from a random point in a disk to the boundary of the disk is 1/3 of the radius. One can prove this using the formula for the volume of a cone. (I came up with this myself but I’m sure others have too.)
Further examples of the kind of proof I have in mind are Archimedes’ determination of the surface area and volume of the sphere.
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Cris Moore moore@santafe.edu The disorders and miseries which result gradually incline the minds of men to seek security and repose in the absolute power of an individual; and sooner or later the chief of some prevailing faction, more able or more fortunate than his competitors, turns this disposition to the purposes of his own elevation, on the ruins of public liberty. — George Washington