1. The formula follows by verifying it for the osculating circle . 2. This is the osculating circle to the curve. 3. The "'center of curvature "'is the center of the osculating circle . 4. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point. 5. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. 6. The curve " C " may be obtained as the envelope of the one-parameter family of its osculating circles . 7. Each constraint can be a point, angle, or curvature ( which is the reciprocal of the radius of an osculating circle ). 8. If " P " is a vertex then " C " and its osculating circle have contact of order at least four. 9. Given any curve and a point on it, there is a unique circle or line which most closely approximates the curve near, the osculating circle at. 10. :I'm not well-versed on the mathematics involved, but you might be able to construct the parabola by treating the log as an osculating circle .
