One speaks also of curves and geometric objects having " k "-th order contact at a point : this is also called " osculation " ( i . e . kissing ), generalising the property of being tangent . ( Here the derivatives are considered with respect to arc length . ) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact ( same tangent angle and curvature ), etc.
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