Helix (source code)
= Helix
{wiki}
= Helices
{synonym}
A general <helix> is a regular <space curve> whose <unit tangent vector> makes a constant <angle> with a fixed axis direction $u$. In <arc length> parameter $s$, this means $\gamma'(s)\cdot u$ is constant. Equivalently, the axial <coordinate> grows at a constant rate \b[per unit <arc length>].
The familiar circular <helix> has constant distance from its axis:
$$
\gamma(t)=(R\cos t,R\sin t,at),\qquad R>0.
$$
Its speed is $\sqrt{R^2+a^2}$, and its tangent's axial component is $a/\sqrt{R^2+a^2}$. Its signed <pitch of a helix> is $2\pi a$. A general <helix> need not have constant radius or constant axial displacement per turn: the <logarithmic conical helix> has a varying <radius> while retaining a constant tangent-axis <angle>.
Back to article page
