tasopinnat

Tasopinnat are flat, two‑dimensional surfaces embedded in three‑dimensional space. In geometry, a tasopinta is a subset of space that lies entirely in a single plane, extending in two independent directions. In mathematical terms, it can be described by a linear equation ax + by + cz + d = 0 with at least one of a, b, c nonzero, where the vector (a, b, c) is normal to the plane. A parametrization is also common: r(u, v) = p0 + u v1 + v v2, where p0 is a point on the plane and v1, v2 are non-collinear direction vectors lying in the plane.

Tasopinnat have several defining properties. They are flat surfaces with zero curvature (Gaussian curvature 0). Any line contained in a tasopinta is straight, and the Euclidean metric on the plane agrees with the ambient space restricted to the surface. In many contexts, tasopinnat are treated as affine subspaces of three‑space.

In practical use, tasopinnat appear as the faces of polyhedra, as elements in computer graphics and computer‑aided design, and as simplifying surfaces in architecture, engineering, and geographic information systems. They are distinguished from curved surfaces such as spheres or cylinders by their lack of curvature and their straight‑line simplicity.

Common examples include the horizontal plane z = 0 (the xy‑plane) and any plane given by equations such as x + y + z = 1. While mathematical tasopinnat are infinite in extent, real‑world surfaces labeled as tasopinnat are often finite patches bounded by edges in drawings or models.

See also: Plane (geometry), Surface, Polyhedron.

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