The center of curvature and the equation of the circle of curvature at a given point $(x, y)$ on a curve can be found using the following formulas:

1. Formula for the Center of Curvature:

The center of curvature at a point $(x_0, y_0)$ on a curve is given by:

$\left( x_c, y_c \right) = \left( x_0 - \frac{y'_0 (1 + (y'_0)^2)}{y''_0}, y_0 + \frac{(1 + (y'_0)^2)}{y''_0} \right)$

Where:

• $y'_0 = \frac{dy}{dx} \bigg|_{x = x_0}$ is the slope of the tangent line at the point $(x_0, y_0)$,
• $y''_0 = \frac{d^2y}{dx^2} \bigg|_{x = x_0}$ is the curvature or second derivative at the point.

2. Radius of Curvature:

The radius of curvature $R$ at the point $(x_0, y_0)$ is given by:

$R = \frac{\left(1 + \left(y'_0\right)^2\right)^{3/2}}{|y''_0|}$

3. Equation of the Circle of Curvature:

The equation of the circle of curvature (also called the osculating circle) at the point $(x_0, y_0)$ is:

$(x - x_c)^2 + (y - y_c)^2 = R^2$

Where:

• $(x_c, y_c)$ is the center of curvature as derived above,
• $R$ is the radius of curvature.

Summary:

• Center of curvature: $\left( x_c, y_c \right)$
• Radius of curvature: $R$
• Equation of the circle of curvature: $(x - x_c)^2 + (y - y_c)^2 = R^2$

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Do you need further details or have any questions on this topic?

Related Questions:

1. What is the geometric interpretation of the radius of curvature?
2. How do you calculate the radius of curvature in parametric coordinates?
3. How does the center of curvature relate to the normal to the curve?
4. Can you apply this method to a specific curve like $y = x^2$?
5. How does the curvature change along different points of a curve?

Tip: The smaller the radius of curvature, the sharper the curve at that point.