1.4.5 The Osculating Circle
The unit normal vector points "in the direction of curvature". This means that at a point along an image curve if you were to build a circle whose curvature is equal to the unit normal would point towards the center of the circle. This circle is called the osculating circle. If a tangent line is a line of best fit, you can think of the osculating circle as the circle of best fit.
The GeoGebra applet below allows you to type in different paths and view the osculating circle attached to a point as it moves around the path.
You have all the tools necessary to write a parameterization of the osculating circle. Do so.