📂Geometry

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.

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• $T$ 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.