释义 |
LineEuclid defined a line as a ``breadthless length,'' and a straight line as a line which ``lies evenly with thepoints on itself'' (Kline 1956, Dunham 1990). Lines are intrinsically 1-dimensional objects, but may be embedded in higher dimensionalSpaces. An infinite line passing through points and is denoted . A Line Segment terminating at these points is denoted . A line is sometimes called a Straight Line or, more archaically, a Right Line (Casey 1893), to emphasize thatit has no curves anywhere along its length.
Consider first lines in a 2-D Plane. The line with x-Intercept and y-Intercept is given bythe intercept form
 | (1) |
The line through with Slope is given by the point-slope form
 | (2) |
The line with -intercept and slope is given by the slope-intercept form
 | (3) |
The line through and is given by the two point form
 | (4) |
Other forms are
 | (5) |
 | (6) |
 | (7) |
A line in 2-D can also be represented as a Vector. The Vector along the line
 | (8) |
is given by
 | (9) |
where . Similarly, Vectors of the form
 | (10) |
are Perpendicular to the line. Three points lie on a line if
 | (11) |
The Angle between lines
is
 | (14) |
The line joining points with Trilinear Coordinates and is the set of point satisfying
 | (15) |
 | (16) |
Three lines Concur if their Trilinear Coordinates satisfy
in which case the point is
 | (20) |
or if the Coefficients of the lines
satisfy
 | (24) |
Two lines Concur if their Trilinear Coordinates satisfy
 | (25) |
The line through is the direction and the line through in direction intersect Iff
 | (26) |
The line through a point Parallel to
 | (27) |
is
 | (28) |
The lines
are Parallel if
 | (31) |
for all , and Perpendicular if | |  | (32) | for all (Sommerville 1924). The line through a point Perpendicular to(32) is given by
 | (33) |
In 3-D Space, the line passing through the point and Parallel to the Nonzero Vector
 | (34) |
has parametric equations
 | (35) |
See also Asymptote, Brocard Line, Collinear, Concur, Critical Line, Desargues'Theorem, Erdös-Anning Theorem, Line Segment, Ordinary Line, Pencil,Point, Point-Line Distance--2-D, Point-Line Distance--3-D, Plane, Range (Line Segment), Ray, Solomon's Seal Lines, Steiner Set,Steiner's Theorem, Sylvester's Line Problem References
Casey, J. ``The Right Line.'' Ch. 2 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 30-95, 1893. Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 32, 1990. Kline, M. ``The Straight Line.'' Sci. Amer. 156, 105-114, Mar. 1956. MacTutor History of Mathematics Archive. ``Straight Line.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Straight.html. Sommerville, D. M. Y. Analytical Conics. London: G. Bell, p. 186, 1924. Spanier, J. and Oldham, K. B. ``The Linear Function and Its Reciprocal.'' Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 53-62, 1987. |