I am attempting to find a generalised formula for the normal of a simplex facet in n-dimensions.
For example if I had the 2 dimensional simplex formed by the vertices ABC below. Then I want to find the normal for the facet formed by the vertices BC. The reason I want the normal is because I'm trying to find the hyperplane that intersects vertices BC, and is equal to 1 at vertex A.
Is there a nice way of doing this?
One way I thought of would be to find D. and then use the formulas: $$W = \frac{A-D}{||A-D||_2}$$ $$b = -D\cdot{W}$$ for the plane: $$xW + b = y$$ However, I don't know how to find D for an n-dimensional simplex facet either.
I'm not very knowledgeable in linear algebra or geometry, so any help would be great.
Let $x_0,\ldots,x_n$ be the vertices of the $n$-dimensional simplex.
You can find all normals of the facets using the inverse of the following matrix: $$ \begin{pmatrix} \;\; & x_0^T & \;\; & 1 \\ \;\; & x_1^T & \;\; & 1 \\ \;\; & \vdots & \;\; & \vdots \\ \;\; & x_n^T & \;\; & 1 \\ \end{pmatrix} $$ Note that this matrix is invertible unless the simplex is degenerate. The determinant of this matrix is $\pm n!$ times the volume of the simplex.
Let's compose the inverse of the matrix of vectors $y_0,\ldots,y_n \in\mathbb{R}^n$ and numbers $c_0,\ldots,c_n\in\mathbb{R}$ as follows: $$ \begin{pmatrix} \;\; & x_0^T & \;\; & 1 \\ \;\; & x_1^T & \;\; & 1 \\ \;\; & \vdots & \;\; & \vdots \\ \;\; & x_n^T & \;\; & 1 \\ \end{pmatrix} \begin{pmatrix} \; & & & \; \\ y_0 & y_1 & \cdots & y_n \\ \; & & & \; \\ -c_0 & -c_1 & \cdots & -c_n \end{pmatrix} = \begin{pmatrix} 1 & & & 0 \\ & 1 & & \\ & & \ddots & \\ 0 & & & 1 \end{pmatrix} $$ Then you get $\langle x_i\, ,\,y_i\rangle = 1+c_i$ and $\langle x_i\, ,\,y_k\rangle = c_k,\;i\neq k.$ This means $$ i\neq k,\,j\neq k\; \Rightarrow \; \langle x_i-x_j\, , \,y_k \rangle =\langle x_i\, , \,y_k \rangle - \langle x_j\, , \,y_k \rangle = c_k - c_k = 0 $$ which shows that $y_k$ is perpendicular to the facet opposite to $x_k.$
Note that the vectors $y_k$ are not necessarily unit vectors. You might have to scale them.