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Pencil

Pencil

The set of all lines through a point. The term was first used by Desargues (Cremona 1960, p. x). The six angles of any pencils of four rays O{ABCD} are connected by the relation

sinBOCsinAOD+sinCOAsinBOD+sinAOBsinCOD=0

and the lengths satisfy

BC·AD+CA·BD+AB·CD=0

(Lachlan 1893).

Woods (1961) uses the term pencil as a synonym for line segment range, and Altshiller-Court (1979, p. 12) uses the term to mean sheaf of planes.

See also

Near-Pencil, Pencil Section, Perspectivity, Radial Line, Range, Sheaf of Planes, Star

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References

Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979.Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960.Lachlan, R. "Relations Connecting the Angles of a Pencil." §29 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 16-18, 1893.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 36, 1930.Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 8 and 11-12, 1961.

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Pencil

Cite this as:

Weisstein, Eric W. "Pencil." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pencil.html

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