Definition of a General Spiral
Definition 1
A helix is defined as a regular curve $\alpha$ that, for some fixed unit vector $\mathbf{u}$, satisfies $\left< T, \mathbf{u} \right>$ as a constant, and $\mathbf{u}$ is called the axis.
- $T$ is a tangent.
- $\left< \cdot , \cdot \right>$ is an inner product.
Description
According to the definition, since $\mathbb{R}^{3}$ always satisfies $\left< T, \mathbf{u} \right> = 0$ when $\mathbf{u} = B$, all regular curves lying in a plane are helices; however, this is merely a definition and, in reality, these are not explored as ‘helices’ per se.
The equivalence condition is known by Lancret’s theorem.
Millman. (1977). Elements of Differential Geometry: p32. ↩︎