V² = (-1)^{n+1}/(2^{n}(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 {d²_{ij}} of the squares of distances from vertex i to vertex j.
Of course d_{ij} = d_{ji} and d_{ii} = 0.
In 2 dimensions, we get the area of a triangle :
| 0 1 1 1 | S² = -1/16 | 1 0 d²_{12} d²_{13} | | 1 d²_{21} 0 d²_{23} | | 1 d²_{31} d²_{32} 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 - a^{4} - b^{4} - c^{4} |
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...
| 0 1 1 1 1 _{ }| | 1 0 d²_{12} d²_{13} d²_{14} | V² = 1/288 | 1 d²_{21} 0 d²_{23} d²_{24} | | 1 d²_{31} d²_{32} 0 d²_{34} | | 1 d²_{41} d²_{42} d²_{43} 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 u^{4}v² 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)/2That last one grouping two opposite edges in each term.