Spirals - Archimedean, logarithmic, hyperbolic and clothoid

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Problem 1053 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.2 "Derivatives of Inverse Functions. Derivatives of Functions Given Parametrically. Derivatives of Implicit Functions"

Find the derivative $y'_{x}$ of the following implicitly defined function: $\arctg \dfrac {y}{x}=\ln \sqrt{x^{2}+y^{2}}$ (logarithmic spiral).

Problem 1605 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.14 "Tangency of Curves. Circle of Curvature. Evolute"

Find the radius of curvature of the curve given by the following polar equation: the Archimedean spiral $r=a\varphi$ .

Problem 1606 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.14 "Tangency of Curves. Circle of Curvature. Evolute"

Find the radius of curvature of the curve given by the following polar equation: the logarithmic spiral $r=ae^{m\varphi }$ .

Problem 1.14.28 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.14.2 "Problems"

Find the angle between the tangent and the polar radius of the point of tangency for the following curves:

(1)

the spiral of Archimedes $r = a \varphi$ ;

(2)

the hyperbolic spiral $r = a / \varphi$ ;

(3)

the logarithmic spiral $r = a e^{b \varphi }$ ;

(4)

the cardioid $r = a(\cos \varphi +1)$ ;

(5)

the arc of the lemniscate of Bernoulli $r^{2} = a^{2} \cos 2 \varphi , 0 \leqslant \varphi < \pi / 4$ .

Problem 1.24.92 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.24.2 "Problems"

Find the condition on the parameter $a$ of the logarithmic spiral $\rho = c a^{\varphi }$ under which its evolute coincides with the spiral itself.

Problem 1.24.94 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.24.2 "Problems"

Prove that for points of the Archimedean spiral $\rho = a \varphi$ , as $\varphi \rightarrow +\infty$ , the difference between the length of the radius vector and the radius of curvature tends to zero.

Problem 1.24.95 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.24.2 "Problems"

Prove that under the conditions of Problem 94 the center of curvature moves along a curve tending to coincide with the circle $\rho = 1$ .

Problem 1.24.101 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.24.2 "Problems"

Prove that the centers of curvature at points of the Archimedean spiral $\rho = a \varphi$ lying on one ray are located on an ellipse whose semi-axes do not depend on the choice of ray.

Problem 1.7.102 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.7.2 "Problems"

(1)(2)(3)(4)

Plot the graph of the function in polar coordinates:

(1)

$r = 1 / \varphi$ (hyperbolic spiral);

(2)

$r = e^{\varphi }$ (logarithmic spiral);

(3)

$r = 8 \sin (\varphi -\pi / 3)$ ;

(4)

$r = 1 /(1-\sin \varphi )$ .

Problem 2.7.30 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

(1)(2)(3)

Find the area of the sector $\varphi_{1} \leqslant \varphi \leqslant \varphi_{2}, \varphi_{2}-\varphi_{1} \leqslant 2 \pi$ , bounded by the curve:

(1)

$r = a \varphi /(2 \pi )$ (Archimedean spiral);

(2)

$r \varphi = a$ (hyperbolic spiral);

(3)

$r = R e^{k \varphi }, \quad k > 0$ (logarithmic spiral).

Problem 2.7.31 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the area of the figure bounded by the $n$ -th turn of the Archimedean spiral

r = a \varphi /(2 \pi ), \quad 2 \pi (n-1) \leqslant \varphi \leqslant 2 \pi n, \quad n \in \mathbb {N}

and a segment of the polar axis.

Problem 2.7.32 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

(1)(2)(3)

Find the area of the figure bounded by a segment of the polar axis and two turns of the spiral corresponding to the values of the polar angle $\varphi \in [2 \pi n ; 2 \pi (n+1)]$ and $\varphi \in [2 \pi (n+1) ; 2 \pi (n+2)]$ :

(1)

$r = \frac{a}{2 \pi } \varphi$

(2)

$r \varphi = a$ ;

(3)

$r = R e^{k \varphi }, \quad k > 0$ .

Problem 2.7.79 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

(1)(2)

Find the length of the curve:

(1)

$x = \int_{0}^{t} \cos \varphi^{2} d \varphi , \quad y = \int_{0}^{t} \sin \varphi^{2} d \varphi , \quad 0 \leqslant t \leqslant t_{0}$ (clothoid);

(2)

$x = \int_{1}^{t} \frac{\sin \varphi }{\varphi } d \varphi , \quad y = \int_{1}^{t} \frac{\cos \varphi }{\varphi } d \varphi , \quad 1 \leqslant t \leqslant t_{0}$ .

Problem 2.7.82 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the length of the arc of the curve:

(1)

$r = a \sin \varphi$

(2)

$r = a e^{k \varphi }, \quad \varphi_{1} \leqslant \varphi \leqslant \varphi_{2}$ (logarithmic spiral);

(3)

$r = a(1-\cos \varphi )$ (cardioid);

(4)

$r = 2(1+\cos \varphi ), \quad r \leqslant 1$ ;

(5)

$r = a(1-\sin \varphi ), \quad -\pi / 2 \leqslant \varphi \leqslant -\pi / 6$ ;

(6)

$r = a \operatorname {th}(\varphi / 2), \quad 0 \leqslant \varphi \leqslant \varphi_{0}$ .

Problem 2.7.83 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Prove that the lengths of the arcs of successive turns of the logarithmic spiral

r = a e^{k \varphi }, \quad 2 \pi n \leqslant \varphi \leqslant 2 \pi (n+1)

form a geometric progression, and find its common ratio.

Problem 2.7.84 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Let $s(\alpha )$ be the length of the arc of the logarithmic spiral

r = a e^{k \varphi }, \quad k > 0, \quad \alpha \leqslant \varphi \leqslant 0.

Find $\lim_{\alpha \rightarrow -\infty } s(\alpha )$ .

Problem 2.7.89 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the length of the arc of the curve:

(1)

$r = a \varphi , \quad 0 \leqslant \varphi \leqslant \varphi_{0}$

(2)

$r = a \varphi^{2}, \quad 0 \leqslant \varphi \leqslant 4$ ;

(3)

$r = a \varphi^{4}, \quad 0 \leqslant \varphi \leqslant 3$ ;

(4)

$r = a \varphi^{3}, \quad 0 \leqslant \varphi \leqslant 4$ .

(5)

Problem 226 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 2.2 "Deriving Equations of Curves in the Plane"

(Archimedean spiral.) A ray rotates about a point $O$ with constant angular velocity $\omega$ . A point $M$ moves along it with constant speed $v$ . Set up the equation, in polar coordinates, of the curve traced by the point $M$ , if at the initial moment of motion the ray coincides with the positive direction of the abscissa axis, and the point $M$ with the origin $O$ . Sketch the resulting curve.

Problem 227 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 2.2 "Deriving Equations of Curves in the Plane"

Prove that in the preceding problem the Archimedean spiral is tangent to the rotating ray at the initial moment of time.

Problem 228 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 2.2 "Deriving Equations of Curves in the Plane"

Prove that the Archimedean spiral is not an algebraic curve.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

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

[ККЧ+03b] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.

[Сми16] Смирнов Ю.М. Сборник задач по аналитической геометрии и линейной алгебре. МЦНМО, 2016.