# A proof of the fundamental theorem of curves in space and its applications

## TL;DR

This paper establishes a necessary and sufficient condition for the existence of space curves with specified curvature and torsion, solving a nonlinear differential equation, and explores applications to general and slant helices.

## Contribution

It provides a new fundamental condition linking curvature and torsion to space curve existence, with solutions to associated differential equations and applications to specific helix types.

## Key findings

- Derived a necessary and sufficient condition for space curve existence.
- Solved a nonlinear second-order differential equation related to curvature and torsion.
- Applied results to general helices and slant helices.

## Abstract

We give a necessary and suficente condition for the existence of a space curve with curvature $\kappa$ and torsion $\tau$ finding a solution of a nonlinear differential equation of second order and some applications are given for the general helices and slant helices.