1.6.1 The Osculating Circle for Space Curves
Recall we defined the osculating circle to a plane curve to be a circle of best fit to the curve. We did this by making the curvature of the osculating circle equal to the curvature of the curve at the point of tangency. Nearly everything we did for plane curves can be transferred nicely to space curves - curvature has the same definition (length of where is understood to be an arclength parameterization). The same formulas for curvature that were introduced earlier still apply.
However when we consider the osculating circle to a space curve we are confronted with a choice. At each point along the curve there are infinitely many planes containing that point in which we could draw the osculating circle - which one should we choose?
The GeoGebra applet below illustrates the choice of plane - the osculating circle is drawn in the plane defined by and (recall the unit normal is the unit tangent to the unit tangent). As you've seen already, these two vectors are always orthogonal. This plane is called the osculating plane.
Write a parameterization for the osculating circle to a space curve in terms of the curvature , unit tangent , and unit normal vectors.