Radius of curvature of plane curves

Starting from the definition of curvature, find the length of an arc of $1^{\circ }$ of the Earth's meridian at the pole and at the equator. The meridian may be taken as an ellipse with semi-axes $a=6378$ km and $b=a-21$ km (the polar semi-axis).

Find the radius of curvature of the curve $3ay^{2}=2x^{3}$.

Find the radius of curvature of the curve $y=x^{3}$ at the point $(1,1)$.

Find the radius of curvature of the curve $x=t^{2}$, $y=t^{3}$ at the point $(1,1)$.

Find the radius of curvature of the curve $y=a\ch \dfrac {x}{\alpha }$ (the catenary) at the point $(0,a)$.

Find the radius of curvature of the curve $y=a\ln \cos \dfrac {x}{a}$.

Find the radius of curvature of the curve $x^{n}+y^{n}=2a^{n}$ at the point $(a,a)$.

Find the radius of curvature of the curve $x=a\cos^{4}t$, $y=a\sin^{4}t$ at the point $(a,0)$.

Find the radius of curvature of the curve $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$.

Find the radius of curvature of the curve $x=a(t-\sin t)$, $y=a(1-\cos t)$.

Find the radius of curvature of the curve $r=a(1+\cos \varphi )$.

Find the radius of curvature of the curve $r^{2}=a^{2}\cos 2\varphi$, at $\varphi =0$.

Find the radius of curvature of the curve

Find the radius of curvature of the curve

Find the radius of curvature of the curve

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

Prove that for the curves $r^{n}=a^{n}\cos n\varphi$, the polar normal is $n+1$ times the radius of curvature.

Prove that for the curves $r^{n}=a^{n}\sin n\varphi$, the part of the radius vector lying inside the circle of curvature equals $\dfrac {2r}{n+1}$.

Prove the equality $R=\dfrac {r\, dr}{dp}$, where $R$ is the radius of curvature, $r$ is the radius vector, and $p$ is the perpendicular from the pole to the tangent.

The coordinates of the points of a curve can be expressed in terms of the arc length of the curve: $x=x(s)$, $y=y(s)$. Denoting by $\Delta s$ a small increment of $s$, and by $\Delta l$ the corresponding chord, prove the equality

where $R$ is the radius of curvature of the curve.

The point of tangency is taken as the origin, the tangent as the axis $Ox$, and the axis $Oy$ is directed from the point of tangency toward the center of curvature. Prove that in a neighborhood of the point of tangency, i.e.for small values of $x$, the following formulas hold:

where $R$ is the radius of curvature at the point $(0,0)$, $y_{1}$ is the ordinate of points of the curve, and $y_{2}$ is the ordinate of points of the circle of curvature.