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From: trainor@ucla-cs.UUCP
Newsgroups: net.math
Subject: Re: volume of a tetrahedron
Message-ID: <5847@ucla-cs.ARPA>
Date: Tue, 4-Jun-85 20:30:36 EDT
Article-I.D.: ucla-cs.5847
Posted: Tue Jun  4 20:30:36 1985
Date-Received: Sat, 8-Jun-85 05:38:56 EDT
References: <758@gloria.UUCP>
Reply-To: trainor@ucla-cs.UUCP (Douglas J. Trainor)
Followup-To: trainor@ucla-cs.UUCP (Douglas J. Trainor)
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Organization: UCLA Computer Science Department
Lines: 50

More about volumes of tetrahedrons...

>>Yeah, but it is easier to remember that the determinant
>>of the deformation matrix changes volumes.

>Not if you never knew it in the first place.  What is the
>deformation matrix?  How does this give volumes?  Is there
>a fairly simple way to show it?

I can never remember formulas...  If you have some object A
which encloses a volume V, and a 4x4 deformation matrix M
which transforms A, then the volume of the new object is just

    V' = V * |det(M)|

The absolute value is for reflections and the like which have
negative determinants.  For this particular problem, all you
need to find is M, which transforms the standard tetrahedron to
whatever tetrahedron you want.  For the tetrahedron P,Q,R,S:

         1  |   [P-S 0]|    1  |   [P 1]|
    V = --- |det[Q-S 0]| = --- |det[Q 1]|
         6  |   [R-S 0]|    6  |   [R 1]|
            |   [ S  1]|       |   [S 1]|


From and old UCLA Math 169 exam:

Q:  Find the volume of the tetrahedron with vertices
    [1 1 1], [2 4 8], [3 9 27], [5 25 125]

A:       1  |   [1  1   1 1]|
    V = --- |det[2  4   8 1]|
         6  |   [3  9  27 1]|
            |   [5 25 125 1]|

         1  |   [1 1  1   1]|
      = --- |det[1 2  4   8]|
         6  |   [1 3  9  27]|
            |   [1 5 25 125]|

         1
      = --- (2-1)(3-1)(5-1)(3-2)(5-2)(5-3)
         6

      =  8

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