Archive of posts in category “math”
Normal vectors of a scalar field
This article gives the formula for the normal vectors of a surface defined by a scalar field on $\mathbb R^n$. The normal vector of the graph of the function $y=f\!\left(\mathbf x\right)$ at $\left(\mathbf x_0,f\!\left(\mathbf x_0\right)\right)$ is $\left(\nabla f\!\left(\mathbf x_0\right),-1\right)$. This also provides us a way to recover a scalar field from the normal vectors of its graph: normalizing the vectors so that the last component is $-1$, and then integrate the rest components.
Hyperellipsoids in barycentric coordinates
In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.
- Categories: math
- Tags: linear algebra, long paper