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#1 2018-12-04 21:10:55

Spicca
Member
Registered: 2018-12-04
Posts: 1

Probability function

Pick 4 random points on a sphere. what Probability function has the volume of tetrahedron with these points?

I've seen the question of average volume from somewhere and I am interested in the probabilities of the volume. (unfortunately I couldn't understand the answer...)

I'm sorry if my english is bad. If you don't understand anything please let me know. Thank you!

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#2 2018-12-10 16:45:29

mr.wong
Member
Registered: 2015-12-01
Posts: 252

Re: Probability function

Hi  Spicca , 

I  really  cannot  solve  this  problem . Or  we  may  consider  a  similar  but  simpler  one  of  2- dimensional  ,  perhaps  we  can  get  some  hint .

From  a  circle  with  radius  1  unit , 3  points  are  picked  randomly  on  its  circumference . Find  the  expected  area  of  the  triangle  so  formed .

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#3 2018-12-22 19:01:27

mr.wong
Member
Registered: 2015-12-01
Posts: 252

Re: Probability function

For  the  problem  of  2-dimensional  , let  O  be  the  centre  of  the  circle  . Take  a  point  A  on  the  circumference  ,  and  let  AD  be  the  diameter  with  length  2  unit . If  we  take  2  more  points  B  and  C  randomly  on  the  circumference  to  form  a  triangle ,  the  chance  that  B  and  C  lie  on  the  same  side  of  AD  is  1/2 ,  while  the  chance  that  B  and  C  lie  on  the  opposite  sides  of  AD  is  also  1/2 . We  shall  consider  the  1st  case  firstly . Let  x  be  the  length  of  AB , and  let  y  be  the  length  of  AC . ( For  simplicity  let  x ≤ y ) Then  both  x  and  y  varies  from  0  to  2  units .
We  know  that  the  area  of  Δ ABC  =  Δ ABO  +  Δ OBC  -  Δ ACO .  But  how  to  find  the  areas  of  the  last  3   Δ s  ?

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#4 2018-12-23 03:07:09

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,507

Re: Probability function

hi Spicca

Welcome to the forum.

I've been hoping that someone would post an answer to this and I could learn too.  No such luck sad

So I'm having a go myself.  Please note: I've never done this before, so what follows may be rubbish.  Please comment …. ask for clarification  … tell me why it's wrong etc.  Maybe between us, we can arrive at the correct answer.  And oh yes … your English is good.

So let's work with 3D coordinates with the x-y plane horizontal and z going straight up.  Further, let's make the sphere have unit radius (cannot see any harm in that) and centred on the origin, O.

If P is one point of the tetrahedron, then we can specify its position using spherical coordinates theta and phi as shown here: https://en.wikipedia.org/wiki/Spherical … ate_system  And let P' be the point in the x-y plane below P.

Now, what would be helpful is to have a formula for the volume of a tetrahedron in terms of theta and phi but I cannot find one.  Plenty of internet pages giving the formula in vector terms such as https://math.stackexchange.com/question … ot-product

and as a determinant https://stackoverflow.com/questions/986 … n-4-points

I could also expand either into a large algebraic formula but it would take ages to enter all the LaTex so you'll have to ask nicely smile if you want this.

phi can take any random value from 0 to 2pi and theta any from 0 to pi. 

So you can construct your function with 8 variables and there it is.  Hhhmmmm. dizzy

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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