Curve of constant slope

A curve the tangent to which forms a constant angle with a fixed direction. A helical line is an example of this. The ratio between the torsion and the curvature of a curve of constant slope is constant. The spherical indicatrix of the tangents of a curve of constant slope is a circle. If $\mathbf{r} = \mathbf{r}(s)$ is the natural parametrization of a curve of constant slope, then $(\mathbf{r}'', \mathbf{r}''', \mathbf{r}'''') = 0$ (see [2]). The evolutes of a plane curve $\gamma$ are curves of constant slope, the tangents of which are inclined towards the plane of the curve $\gamma$ at a constant angle (see [1]). For every curve of constant slope there is a comoving cone connected with its accompanying trihedron, the vertex of which lies on the curve, while the generators describe developable surfaces.

References

[1]	W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , 1 , Springer (1921)
[2]	A.R. Forsyth, "Lectures on the differential geometry of curves and surfaces" , Cambridge (1912)
[3]	P.E. Appell, Arch. Math. Phys. , 64 : 1 (1879) pp. 19–23

Comments

For the various concepts defined for curves in three-dimensional space mentioned above, such as trihedron, cf. Differential geometry.