tangenciales

Tangenciales is the plural form of tangencial, used in Spanish to refer to objects or concepts that touch a curve or surface at a single point without crossing it. In geometry and related fields, these are commonly lines or planes that have the same direction as the first-order approximation of the figure at the point of contact.

In plane geometry, a line is tangencial to a smooth curve at a point if it intersects the curve there and shares the tangent direction to the curve at that point. For a differentiable function y = f(x), the tangent line to the curve at x0 is y = f(x0) + f′(x0)(x − x0). For a circle, the tangent at a point is perpendicular to the radius drawn to that point.

In three dimensions, the concept extends to tangent planes. A plane is tangencial to a smooth surface at a point if it touches the surface at that point and approximates it to first order. The tangent plane provides a locally linear approximation of the surface near the point of tangency.

Tangenciales are contrasted with secants, which intersect a curve at two or more points, and with normals, which are perpendicular to the tangent at the contact point. More advanced topics include higher-order contact or osculation, where the curve and the tangent line agree beyond the first order.

Applications appear across disciplines, including physics, computer graphics, and engineering, wherever local linear approximations of curves or surfaces are useful. See also tangent, tangential, and osculation.