Defect (geometry)

Defect (geometry)

In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.

Classically the defect arises in two ways:

  • the defect of a vertex of a polyhedron;
  • the defect of a hyperbolic triangle;

and the excess arises in one way:

  • the excess of a spherical triangle.

In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).

In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem.

Read more about Defect (geometry):  Defect of A Vertex, Examples, Descartes' Theorem, A Potential Error

Famous quotes containing the word defect:

    The child realizes to every man his own earliest remembrance, and so supplies a defect in our education, or enables us to live over the unconscious history with a sympathy so tender as to be almost personal experience.
    Ralph Waldo Emerson (1803–1882)