# Finding the normal of a simplex facet in n-dimensions

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. ## 1 Answer

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.

• That is just what I was looking for, thank you so much! – Michael Murray May 21 '20 at 11:25
• You are welcome! – Reinhard Meier May 21 '20 at 11:28