styczne

Styczne is a mathematical term used in Polish to denote tangents, that is lines or curves that touch another curve at a point without crossing it locally. In common usage, the phrase prosta styczna or linia styczna refers to a tangent line to a given curve, while the plural styczne can denote multiple tangent lines at different points.

In geometry and calculus, a tangent line to a differentiable function y = f(x) at a point (x0, f(x0)) has slope f′(x0). The equation of the tangent line is y = f′(x0)(x − x0) + f(x0). The concept extends to higher dimensions, where the tangent plane to a surface at a point is the best linear approximation to the surface near that point.

Two curves are called tangent if they share a common tangent at a point of contact, meaning they touch there with a certain order of contact. The most common case is simple tangency (first-order contact), where the curves meet but do not cross to first order. Higher-order tangency describes a higher degree of contact.

In the context of circles and other curves, the tangent line at a point is perpendicular to the radius drawn to the point of contact. Tangency plays a central role in analytic geometry, differential geometry, and applications such as physics and computer graphics, providing a local linear approximation and a means to study curvature and contact relations. See also tangency, tangent line, tangent plane, derivative.