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# Line Segment Range

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

 (1)

The range satisfies

 (2)

and

 (3)

the latter of which holds even when is not on the line (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.

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.

## Referenced on Wolfram|Alpha

Line Segment Range

## Cite this as:

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