Osculating Circles
Colored pencils on black paper, scanned.
Traditional descriptive geometry construction.
The osculating circle to a curve matches its slope and its curvature. For an ellipse (or conic section in general), you can construct it exactly.
Here, I let a hypothetical particle move on a meandering path. I zoom in on one squiggle - which I happen to match one half of an ellipse (yellow) exactly.
Turning the ellipse so that it lies flat in the drawing plane, I can construct the osculating circle (red)… which appears as an ellipse viewed from any other direction.
I draw every ellipse freehand, but guided by its osculating circles. So, also the ellipses representing the "true" osculating circle have their own osculating circles.
I am trying to indicate the three directions (or the three planes) that characterize this path every instant: The tangent to the curve, the principal normal (pointing from the position of the particle to the center of the osculating circle), and the direction perpendicular to both - the "instantaneous plane" of this motion. The three of them are connected by the equations I added.
