Welcome to the forum.
I've been hoping that someone would post an answer to this and I could learn too. No such luck
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 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.
Bob
]]>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 .
]]>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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