In transportation engineering, the design of horizontal curves on highways is directly tied to the radius of curvature. When a vehicle travels along a curve, it experiences centrifugal force pushing it outward. The friction between the tires and the road, along with the road's superelevation (banking), resists this force. Engineers use the formula $R_{min} = \frac{V^2}{127(e + f)}$ (where $V$ is design speed, $e$ is superelevation, and $f$ is the friction factor) to determine the absolute minimum radius of curvature a road can have to safely support vehicles traveling at the speed limit. By analyzing the curvature of the proposed centerline geometry, engineers verify that the radius at every point exceeds $R_{min}$.

In structural analysis, when a beam is subjected to bending moments, its longitudinal axis deforms into a curve called the elastic curve. The fundamental differential equation of the elastic curve is given by $\frac{1}{R} = \frac{M}{EI}$, where $R$ is the radius of curvature, $M$ is the internal bending moment, $E$ is the modulus of elasticity, and $I$ is the moment of inertia. For very small deflections typical in civil structures, the slope $y'$ is nearly zero, so the radius of curvature formula simplifies to $\frac{1}{R} \approx y''$. Thus, the curvature of a loaded beam is directly proportional to the bending moment at that section, and engineers integrate the curvature twice to determine the exact vertical deflection of the beam.

Find the radius of curvature of the parabola $y = x^2$ at the origin $(0, 0)$ and at the point $(1, 1)$.

Find the radius of curvature of the curve $y = \ln x$ at the point $(1, 0)$.

Determine the curvature $K$ of the function $y = \sin x$ at $x = \frac{\pi}{2}$.

Find the center of curvature for the parabola $y = x^2$ at the point $(1, 1)$.

Find the center of curvature for $y = \frac{1}{x}$ at the point $(1, 1)$.

Find the curvature of the circle defined by the parametric equations $x = r \cos t$ and $y = r \sin t$ for any value of $t$.

Find the curvature of the cycloid given by $x = t - \sin t$ and $y = 1 - \cos t$ at $t = \pi$.

Find the curvature of the cardioid defined by the polar equation $r = 1 + \cos\theta$ at $\theta = 0$.

Find the curvature of the Archimedean spiral $r = \theta$ for any angle $\theta > 0$.