Differential Geometry

Fundamental Theorem of Space Curves Visualization

In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size) completely determined by its curvature and torsion.

Given curvature \kappa and torsion \tau, the Frenet equation relates the tangent, normal, and binormal vector of the curve at each position.

    \[ \begin{cases} T'_{\alpha} & =\kappa_{\alpha}N_{\alpha}\\ N'_{\alpha} & =-\tau T_{\alpha}-\kappa_{\alpha}B_{\alpha}\\ B'_{\alpha} & =\tau N_{\alpha} \end{cases} \]

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Linear Programming

Simplex Algorithm

I have coded recently the simplex algorithm. It is an algorithm that solves the optimizations of a problem of linear programming. It gives a function to minimize and constrains the variables with an inequality form.

It can be proven that the simplex algorithm solves this problem in Non-Polynomial time.

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Pathfinding Algorithms

We usually want to know the shortest/fastest route between two points. This is a well-known problem in graph theory. There are several algorithms that solve this problem: A*, Dijkstra, etc.

I have implemented in C++ those algorithms and a graph of the roads in California.

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Connect 4 AI Solver

Recurrent algorithms C++ and Python

I have been working on recurrent algorithms. They’re a bunch of examples in which you can implement these algorithms. However, we usually see that they are efficient.

Connect 4 is a simple game that can be easily programmed and solved to win every time. I have made in Python an AI that solves and wins. It is not programmed in C++ because I wanted a GUI.

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Dynamic Systems

N-Order Linear Systems

I have done a project for my math class where we analyze the linear systems. We have classified the first and second-order linear systems and give a general formula for an n-linear system. We have used algebra notation and we have encountered with the companion matrix power to the n problem.

The n-linear systems are used widely in finance for loans, interests, etc.

PDF download: N-LinearDynamicSystem

GitHub repository: DynamicSystem