Osculating curve
![A curve C containing a point P where the radius of curvature equals r, together with the tangent line and the osculating circle touching C at P](/Images/godic/202501/31/Osculating_circle.svg4835.png")
In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if F is a family of smooth curves, C is a smooth curve (not in general belonging to F), and p is a point on C, then an osculating curve from F at p is a curve from F that passes through p and has as many of its derivatives at p equal to the derivatives of C as possible.