dünaamiline

Dünaamiline is a term used in theoretical geometry to denote a line whose defining parameters vary with a secondary parameter, typically time, making the line dynamic rather than static. It is used to describe the motion of a line in Euclidean or projective space.

In 2D, a dünaamiline can be represented as the set of points (x, y) satisfying a(t)x + b(t)y + c(t) = 0, where a(t), b(t), and c(t) are real functions. Under certain normalizations, it is convenient to parameterize by slope m(t) and intercept b(t). In homogeneous coordinates, the line is given by l(t) = (l1(t), l2(t), l3(t)) with l(t)·X = 0. The instantaneous line exists for each fixed t, and the family forms a trajectory in the space of lines.

Properties include the envelope—the locus of points touched by the moving line as t varies—and the velocity of the line, described by derivatives of a(t), b(t), and c(t). If the line moves with constant speed and rotation, it can represent a rigid motion of the line; otherwise, deformation occurs in the dual space, reflecting changes in orientation and position.

Applications of dünaamiline appear in kinematics and robotics for modeling path-like constraints with moving obstacles, in computer graphics for animating dynamic guides or trajectories, and in optics or material science where a line-like reference evolves over time.

Historically, the concept appears in theoretical discussions and computational methods as a descriptive tool rather than a standardized object, with terminology and formalisms varying among authors. See also: dynamical geometry, moving line, envelope, dual space.