# Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature

So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve that is: Smooth, Monotone, and has the same curvature as a helix $(\cos(t),\sin(t),t)$.
Anyway this type of curve should exist but I cannot construct it..

Alternativly, I was thinking could we reconstruct the curve from its torion and curvature functions; since they determine a unique curve (up to rigid motion) in Euclidean space.
If so, the curve would have to satisfy $k(s)=1/\sqrt 2$ and $t(s)=1-2s$.

Many thanks in advance! :)

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## 1 Answer

Any torus knot works. That is, any torus knot embedded with constant slope.

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