Curves of constant slope (generalized helices)

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Problem 3.1.4 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

(a)(b)(c)

Let $\gamma : (a,b) \to \mathbb {R}^{3}$ be smooth with $\left|\gamma '\right| = 1$ and curvature $k$ and torsion $\tau$ , both nonvanishing. Denote the Frenet frame by $\{ T, N, B\}$ . Assume that there exists a unit vector $a \in \mathbb {R}^{3}$ with

T \cdot a = \text{constant} = \cos \alpha .

(a)

Show that a circular helix is an example of such a curve.

(b)

Show that $N \cdot a = 0$ .

(c)

Show that $k/\tau = \text{constant} = \pm \tan \alpha$ .

Problem 2305 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Prove that for the moving trihedron of the curve

x = e^{t}\sin t, \qquad y = e^{t}\cos t, \qquad z = e^{t}

each of its edges makes a constant angle with $Oz$ .

Problem 1.24.133 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

(1)(2)(3)

A curve is called a curve of constant slope if its tangents form a constant angle with some straight line.

Prove that a twice continuously differentiable curve with nonzero curvature is a curve of constant slope if and only if one of the following conditions holds:

(1)

all the principal normals are parallel to some plane;

(2)

all the rectifying planes are parallel to some line;

(3)

in the case of a three-times continuously differentiable curve, there exists a constant $c > 0$ such that $\varkappa (s) = c k(s)$ ; moreover $\left|c\right| = \operatorname {ctg} \alpha$ , where $\alpha$ is the angle referred to in the definition of a curve of constant slope.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.