Cayley-Menger determinants

These determinants give the volume of a tetrahedron from its edges.
They are generalized to any dimension : in two dimensions we get the area of a triangle from its sides, resulting into the Heron formula.
In 3 dimensions, we get the Piero della Francesca formula for the tetrahedron.
In > 3 dimensions, it is more weird... Generally speaking, in dimension n :

V² = (-1)n+1/(2n(n!)²) × |Ď|

With Ď being a (n+2)×(n+2) matrix from matrix D by prepending a firt line (0,1,1,1...) and a first column (0,1,1,1,...)T
The matrix D being the matrix {d²ij} of the squares of distances from vertex i to vertex j.
Of course dij = dji and dii = 0.

In 2 dimensions, we get the area of a triangle :

            | 0   1    1    1   |
S² = -1/16  | 1   0   d²1213 |
            | 1  d²21   0   d²23 |
            | 1  d²3132   0  |
By naming simpler a,b,c the three sides :
            | 0  1   1   1  |
S² = -1/16  | 1  0   a²  b² |
            | 1  a²  0   c² |
            | 1  b²  c²  0  |
After developing, we get :

16 S² = 2a²b² + 2a²c² + 2b²c² - a4 - b4 - c4

Factoring into 16S² = (a + b + c)(a + b - c)(a - b + c)(-a + b + c) doesn't seem obvious...
Just checking this identity knowing the result is not a problem !
The traditional form of the Heron formula :

S = p(p - c)(p - b)(p - a)

is obtained by dividing by 16, with (a + b - c)/2 = ((a + b + c) - 2c)/2 = (a + b + c)/2 - c = p - c etc...

Piero della Francesca Formule

In 3 dimensions, the Cayley-Menger determinant is :
            | 0   1    1    1     1   |
            | 1   0   d²121314 |
V² = 1/288  | 1  d²21   0   d²2324 |
            | 1  d²3132   0   d²34 |
            | 1  d²414243  0   |
Naming a,b,c the three edges of a face and x,y,z the opposite edges a↔x, b↔y, c↔z
            | 0  1   1   1   1  |
            | 1  0   a²  b²  z² |
V² = 1/288  | 1  a²  0   c²  y² |
            | 1  b²  c²  0   x² |
            | 1  z²  y²  x²  0  |
Note that just giving opposite edges does not suffice : exchanging a and x, or {a,b,c} and {x,y,z}, we get a different tetrahedron in which x,y,z build a face, and a,b,c share a same vertex, with a different volume in general !

Raw developing of this determinant gives the Piero della Francesca formula :


144 V²  = - a²b²c² - a²y²z² - b²x²z² - c²x²y² 
          + a²c²z² + b²c²z² + a²b²y² + b²c²y²
          + b²y²z² + c²y²z² + a²b²x² + a²c²x² 
          + a²x²z² + c²x²z² + a²x²y² + b²x²y²
          - c²c²z² - c²z²z² - b²b²y² - b²y²y² - a²a²x² - a²x²x²

Note : We have
- the 4 faces with a "-" sign
- the 12 other combinations of three edges, with a "+" sign
- the 6 combinations u4v² with u and v opposite edges, with a "-" sign

There is no complete factoring of this formula.
There are several partial factoring or equivallent formulas :

144V²   = 4x²y²z² + (y²+z²-a²)(z²+x²-b²)(x²+y²-c²)
          - x²(y²+z²-a²)² - y²(z²+x²-b²)² - z²(x²+y²-c²)² 

144 V²  = a²x²[-(a²+x²)+(b²+y²)+(c²+z²)] 
          + b²y²[(a²+x²)-(b²+y²)+(c²+z²)] 
          + c²z²[(a²+x²)+(b²+y²)-(c²+z²)]
          - (a²+x²)(b²+y²)(c²+z²)/2  - (a²-x²)(b²-y²)(c²-z²)/2
That last one grouping two opposite edges in each term.
Note the only assymetry is in the last term with "a²-x²" factors, changing sign when exchanging a and x.
This proves the two "dual" tetrahedrons really have different volumes, unless this term is 0, that is when two opposite edges are equal.

 

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