mersenneforum.org 4 Points in a Plane
 Register FAQ Search Today's Posts Mark Forums Read

 2007-04-14, 07:09 #1 Kees     Dec 2005 C316 Posts 4 Points in a Plane Choose 4 points at random in a plane. What is the probability that the convex hull of these 4 points is a triangle ?
 2007-04-14, 08:23 #2 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 23×727 Posts Is the plane infinite? What is the type of random distribution?
 2007-04-14, 08:51 #3 davieddy     "Lucan" Dec 2006 England 6,451 Posts Is this the same as asking whether they can constitute the vertices of a quadrilateral whose interior angles are all < 180 degrees?
 2007-04-14, 08:51 #4 Kees     Dec 2005 110000112 Posts the plane is infinite, the distribution is that every point in the plane is as likely to be selected
2007-04-14, 08:56   #5
Kees

Dec 2005

3038 Posts

Quote:
 Is this the same as asking whether they can constitute the vertices of a quadrilateral whose interior angles are all < 180 degrees?
Not sure what you mean exactly, but 4 points in a plane have either a triangle or a quadrilateral as convex hull.

 2007-04-14, 09:57 #6 davieddy     "Lucan" Dec 2006 England 6,451 Posts I say it's evens. Take three points at random. Produce the three sides of the triangle in both directions. This divides the plane into seven regions: the triangle itself, 3 unbounded regions adjoining an edge of the triangle, and 3 unbounded regions with one vertex. The two types of infinite region each span 180 degrees. The fourth point has equal probability of lying in each type of infinite region. (The hull is a triangle iff the fourth point lies in a "vertex" type region or the triangle itself) (I shudder to think what Silverstone will make of this ) David Last fiddled with by davieddy on 2007-04-14 at 10:04
 2007-04-14, 10:03 #7 Kees     Dec 2005 3·5·13 Posts My solution is essentially the same and I too fear that it is not very rigourous
 2007-04-14, 10:46 #8 davieddy     "Lucan" Dec 2006 England 6,451 Posts If we consider a sphere rather than a plane, does this clarify or obfuscate the problem? David
2007-04-14, 11:19   #9
davieddy

"Lucan"
Dec 2006
England

6,451 Posts

Quote:
 Originally Posted by Kees My solution is essentially the same and I too fear that it is not very rigourous
I suspect that Retina may have been anticipating these problems.

2007-04-14, 14:57   #10
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

581610 Posts

Quote:
 Originally Posted by Kees the plane is infinite, the distribution is that every point in the plane is as likely to be selected
In that case, I think the probability approaches zero (P --> 0).

2007-04-14, 21:33   #11
davieddy

"Lucan"
Dec 2006
England

6,451 Posts

Quote:
 Originally Posted by retina In that case, I think the probability approaches zero (P --> 0).
In the context of my analysis, I think you are referring
to the probability of the fourth point lying inside the triangle
formed by the other three.

 Similar Threads Thread Thread Starter Forum Replies Last Post bhelmes Math 0 2018-03-04 18:50 devarajkandadai Math 19 2012-12-15 14:15 Unregistered Information & Answers 14 2011-09-27 05:34 petrw1 PrimeNet 8 2007-08-19 19:27 Mystwalker Prime Sierpinski Project 6 2006-01-03 23:32

All times are UTC. The time now is 14:25.

Tue Oct 27 14:25:06 UTC 2020 up 47 days, 11:36, 0 users, load averages: 3.17, 2.99, 2.98