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Line Segment Range

Range

A number of points on a line segment. The term was first used by Desargues (Cremona 1960, p. x). If the points A, B, C, ... lie on a line segment with the coordinates of the points such that A<B<C, they are said to form a range, denoted {ABC...}. Let AB denote the signed distance B-A. Then the range {ABC} satisfies the relation

AB+BC+CA=0.
(1)

The range {ABCD} satisfies

BC·AD+CA·BD+AB·CD=0
(2)

and

BC·AD^2+CA·BD^2+AB·CD^2+BC·CA·AB=0,
(3)

the latter of which holds even when D is not on the line ABC (Lachlan 1893).

Graustein (1930) and Woods (1961) use the term "range" to refer to the totality of points on a straight line, making it the dual of a pencil.

See also

Axis, Homographic, Line, Line Segment, Pencil, Pencil Section, Perspectivity, Range

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References

Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960.Durell, C. V. "Concurrency and Collinearity." Ch. 4 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 37-39, 1928.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 40, 1930.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 14-15, 1893.Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 8, 1961.

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Line Segment Range

Cite this as:

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

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