Volume of n-simplex | n維單純形的體積

An is an n-simplex which is supported by n vectors (a1,a2,…,an) of same origin.
Vn is the signed volume of An.

After we apply Gram–Schmidt orthogonalization to (a1,a2,…,an), we will get (a1,a2’,…,an’), the orthogonal basis of n-dimensional vector space.
Transpose of a1 =(h1,0,…,0)
Transpose of a2’=(0,h2,…,0)
…
Transpose of an’=(0,0,…,hn)

① V1=h1
② t = h2-h

V2=∫h20V1(th2)dt

V3=∫h30V2(th3)2dt

...

Vn=∫hn0Vn−1(thn)n−1dt

hn,Vn−1 can be + or -.
t/hn should always be (and it will be) + to make Vn−1 and Vn−1(thn)n−1 the same sign.

Vn=Vn−1hn−1ntnn∣∣∣hn0=hnnVn−1=hnnhn−1n−1Vn−2=...=hnhn−1...h2n!V1=1n!hnhn−1...h1

=1n!det(a1,a2′,...,an′)=1n!det(a1,a2,...,an)

a′2=a2−c21⋅a1
a′3=a3−c31⋅a1−c32⋅a2
a′4=a4−c41⋅a1−c42⋅a2−c43⋅a3
…
a′n=an−cn1⋅a1−cn2⋅a2−...−cnn−1⋅an−1

=det(a1,a2′,...,an′)
=det(a1,a2,a3′,...,an′)
=det(a1,a2,a3,...,an′)=...
=det(a1,a2,...,an−1,an′)
=det(a1,a2,...,an)

Reference
- Simplex Volumes and … http://mathpages.com/home/kmath664/kmath664.htm
- Volume of n-d parallelepiped http://blog.csdn.net/u010476094/article/details/44746143