Order of Contact Between 2 Curves and the Concept of Osculation

1. Two Curves intersecting at a Point \(P\) are said to have:
   1. \(0\)th-Order Contact if the Curves have a Simple Crossing that is Not Tangent.
   2. \(1\)st-Order Contact if the Curves have a Same Tangent .
   3. \(2\)nd-Order Contact if the Curves have a Same Curvature.
   4. \(3\)rd-Order Contact if the Curves have the Same First Derivative of Curvature with Respect to Arc Length.
   5. \(4\)th-Order Contact if the Curves have the Same Second Derivative of Curvature with Respect to Arc Length.
   6. \(\vdots\)
   7. \(K\)th-Order Contact if the Curves have the Same \(K-2\) th Derivative of Curvature with Respect to Arc Length.
2. Two Curves are said to be Osculating at any given Point if they have a Minimum of \(2\)nd-Order Contact at that Point.