tangencies

Tangencies describe a contact of geometric objects at a point where they share a common tangent line. For a plane curve, a tangent line is the line that best approximates the curve near that point; two curves are tangent at a common point when they meet there and have the same tangent line (the intersection has multiplicity at least two).

In the case of circles, tangency means the circles touch at exactly one point. Externally tangent circles have centers separated by the sum of their radii; internally tangent circles have separation equal to the difference of their radii. If the circles intersect at two points, they are not tangent; if they coincide, every point is a point of tangency.

Analytically, tangency can be detected by derivatives. If a curve y = f(x) is differentiable at x0, its tangent slope is f′(x0). A line with that slope through (x0, f(x0)) is the tangent. For curves given parametrically, the velocity vector at a common point is parallel to the tangent line.

Higher-order contact occurs when curves not only share a tangent line but match curvature at the point. The osculating circle is the circle that has second-order contact with a curve, matching position, slope, and curvature there. The notion extends to higher dimensions via tangent planes or tangent spaces, which describe directions of immediate motion along a surface or manifold.

In algebraic geometry, tangency is described by intersection multiplicity greater than 1: a line is tangent to a curve if it meets it with contact of at least second order at the point.