Tangenter

Tangenter is a term used in some mathematical and computational contexts to denote a class of objects that generalize the notion of a tangent to a curve or manifold. In its broad sense, a tangenter is intended to capture the first-order behavior of a map at a point along a designated direction, functioning as a directional linear approximation that may incorporate constraints beyond a simple tangent line.

Definition and variants: A tangenter at a point p on a function f can be described by a direction vector d and a linear map L that best approximates the change of f in the direction of d near p, with an error term that becomes small for small displacements. The tangenter may be represented as a pair (d, L) or as a projection of f’s differential along d. In higher dimensions, the tangenter can be extended as a subspace of directional derivatives consistent with tangency, or as a constructed object that combines directionality with a constrained linear approximation.

Relation to related concepts: Tangenter is distinguished from the conventional tangent space by emphasizing a chosen direction and potential nonlinearity of the ambient space. It resembles ideas such as directional derivatives, derivative along a curve, or the tangent bundle, but it is not a universally standardized mathematical object with a single agreed-upon definition. In computer graphics and design software, “tangenter” is sometimes used as a vendor-specific label for tools that generate tangent-based interpolations or directional constraints on curves and surfaces.

History and usage: The term has appeared in niche publications and online discussions as a provisional or vendor-specific label. There is no formal, universally accepted definition. Readers using the term should refer to the specific source’s definition for precise meaning.