To determine the radius of curvature for a curve, we typically use the formula based on the geometry of the curve at a specific point. The radius of curvature tells us how sharply the curve bends at that point. Here's how you can find it:

1. For a function $y = f(x)$ (Cartesian coordinates):

The radius of curvature $R$ at a point on the curve $y = f(x)$ is given by:

$R = \frac{ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{3/2} }{ \left| \frac{d^2y}{dx^2} \right| }$

Where:

• $\frac{dy}{dx}$ is the first derivative (the slope of the curve),
• $\frac{d^2y}{dx^2}$ is the second derivative (which tells us how the slope is changing, i.e., the curvature).

2. For a parametric curve:

If the curve is given parametrically as $x = x(t)$ and $y = y(t)$, the radius of curvature is:

$R = \frac{ \left( \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 \right)^{3/2} }{ \left| \frac{d^2x}{dt^2} \cdot \frac{dy}{dt} - \frac{d^2y}{dt^2} \cdot \frac{dx}{dt} \right| }$

Where $\frac{dx}{dt}$, $\frac{dy}{dt}$, $\frac{d^2x}{dt^2}$, and $\frac{d^2y}{dt^2}$ are the first and second derivatives of $x(t)$ and $y(t)$, respectively.

3. For polar coordinates:

If the curve is given in polar coordinates $r = r(\theta)$, the radius of curvature can be calculated using:

$R = \frac{ \left( 1 + \left( \frac{dr}{d\theta} \right)^2 \right)^{3/2} }{ \left| r + \theta \frac{d^2r}{d\theta^2} - \left( \frac{dr}{d\theta} \right)^2 \right| }$

Where $\frac{dr}{d\theta}$ and $\frac{d^2r}{d\theta^2}$ are the first and second derivatives of $r$ with respect to $\theta$.

Summary of Steps:

1. Find the first and second derivatives of the function (depending on how the curve is given: Cartesian, parametric, or polar).
2. Plug those derivatives into the appropriate formula.
3. Simplify the expression to get the radius of curvature.

Would you like a detailed example with a specific function or curve?