- Nov 8, 2022
Suppose P is a point on the circle ⊙C. When is the sum of distances from P to two edges of ∠O extremal? It turns out to be related to angle bisectors (the intersections of ⊙C and the bisector of ∠O or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of ⊙C and edges of ∠O) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting ⊙C at P lies, translate the region to make it center at C, and see whether P is inside the translated region.
- Nov 6, 2022
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…
- Jun 13, 2020
Joukowsky transformation of a circle centered at (1,1) of radius 1 is a curve resembling a heart.
- May 31, 2020
We may generalize Euler–Lagrange equation to higher dimensional optimization problems: find a function defined inside a region to extremize a functional defined as an integral over that region, with the constraint that the value of the function is fixed on the boundary of the region.
- Mar 3, 2020
This article gives the formula for the normal vectors of a surface defined by a scalar field on Rn. The normal vector of the graph of the function y=f(x) at (x0,f(x0)) is (∇f(x0),−1). 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.
- Jan 25, 2020
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.