Archive of posts in category “math”

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• Nov 8, 2022

  The point on the circle farthest to two lines

  Suppose $P$ is a point on the circle $\odot C$. When is the sum of distances from $P$ to two edges of $\angle O$ extremal? It turns out to be related to angle bisectors (the intersections of $\odot C$ and the bisector of $\angle O$ or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of $\odot C$ and edges of $\angle O$) 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 $\odot C$ at $P$ lies, translate the region to make it center at $C$, and see whether $P$ is inside the translated region.

  • Categories: math
  • Tags: elementary geometry, trigonometry, from zhihu
• Nov 6, 2022

  Solving linear homogeneous ODE with constant coefficients

  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…

  • Categories: math
  • Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu
• Jun 13, 2020

  Drawing a heart using Joukowsky transformation

  Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.

  • Categories: math
  • Tags: complex
• May 31, 2020

  Generalization of Euler–Lagrange equation

  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.

  • Categories: math
  • Tags: calculus
• Mar 3, 2020

  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.

  • Categories: math
  • Tags: calculus, vector analysis
• Jan 25, 2020

  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

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