To find the evolute of the given ellipse equation:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$

we can proceed as follows:

Step-by-step process:

1. Parametric Form of the Ellipse: The ellipse can be parameterized as: $x = a \cos \theta, \quad y = b \sin \theta,$ where $\theta$ is the parameter.

2. First and Second Derivatives of Parametric Equations: For the parameterized coordinates, we calculate the first derivatives with respect to $\theta$: $\frac{dx}{d\theta} = -a \sin \theta, \quad \frac{dy}{d\theta} = b \cos \theta.$

   The second derivatives are: $\frac{d^2x}{d\theta^2} = -a \cos \theta, \quad \frac{d^2y}{d\theta^2} = -b \sin \theta.$

3. Radius of Curvature ($R$): The formula for the radius of curvature $R$ in parametric form is: $R = \frac{\left( \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 \right)^{3/2}}{\left| \frac{dx}{d\theta} \frac{d^2y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2x}{d\theta^2} \right|}.$

   Plugging in the values of the first and second derivatives, we get: $R = \frac{\left( a^2 \sin^2 \theta + b^2 \cos^2 \theta \right)^{3/2}}{ab}.$

4. Center of Curvature Coordinates: The coordinates of the center of curvature $(x_c, y_c)$ for an ellipse are given by: $x_c = x - \frac{(b^2/a) \cos^3 \theta}{(a^2 \sin^2 \theta + b^2 \cos^2 \theta)^{1/2}},$ $y_c = y - \frac{(a^2/b) \sin^3 \theta}{(a^2 \sin^2 \theta + b^2 \cos^2 \theta)^{1/2}}.$

   Thus, the parametric equations for the evolute of the ellipse describe the path traced by the center of curvature as the point moves along the ellipse.

Would you like further details or have any questions on this solution? Here are some related questions to explore:

1. What is the significance of an evolute in geometry?
2. How do the parameters $a$ and $b$ affect the shape of the evolute?
3. Can we find the evolute for a circle using a similar method?
4. What is the geometric interpretation of the radius of curvature?
5. How does the curvature vary along different points of an ellipse?

Tip: The evolute is the envelope of normals to the curve, meaning it's a curve where all the centers of curvature lie.