-
8/13/2019 Distance Between a Point and a Line
1/6
Point-Line Distance--2-Dimensional
The equation of a line in slope-intercept form is given by
(1
)
so the line has slope . Now consider the distance from a point
to the line. Points
onthe line have the vector coordinates
(
)
Therefore! the vector
(")
isparallelto the line! and the vector
(#
)
isperpendicularto it. Now! avectorfrom the point to the line is
given by
($
)
Pro%ecting onto !
(&)
(')
()
()
http://mathworld.wolfram.com/Slope.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Parallel.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Perpendicular.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Slope.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Parallel.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Perpendicular.htmlhttp://mathworld.wolfram.com/Vector.html
-
8/13/2019 Distance Between a Point and a Line
2/6
(1*
)
(11
)
+f the line is specified by two points and ! then
avectorperpendicularto
the line is given by
(1
)
,et be a vector from the point to the first point on the
line
(1"
)
then the distance from to the line is again given by pro%ecting
onto ! giving
(1#
)
s it must! this formula corresponds to the distance in the
three-dimensional case
(1$
)
with all vectors having ero -components! and can be written in
the slightly more concise
form
(1&
)
where denotes a determinant.
The distance between a point with e/act trilinear coordinates
and a line
is
(1'
)
http://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Perpendicular.htmlhttp://mathworld.wolfram.com/Perpendicular.htmlhttp://mathworld.wolfram.com/Determinant.htmlhttp://mathworld.wolfram.com/Determinant.htmlhttp://mathworld.wolfram.com/ExactTrilinearCoordinates.htmlhttp://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/Perpendicular.htmlhttp://mathworld.wolfram.com/Determinant.htmlhttp://mathworld.wolfram.com/ExactTrilinearCoordinates.html
-
8/13/2019 Distance Between a Point and a Line
3/6
Point-Line Distance--3-Dimensional
,et a line in three dimensions be specified by two points
and
lying on it! so a vector along the line is given by
(1
)
The squared distance between a point on the line with parameter
and a point is
therefore
(
)
To minimie the distance! set and solve for to obtain
("
)
where denotes the dot product. The minimum distance can then be
found by plugging bac0
into () to obtain
sing the vector quadruple product
('
http://mathworld.wolfram.com/DotProduct.htmlhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html#eqn2http://mathworld.wolfram.com/VectorQuadrupleProduct.htmlhttp://mathworld.wolfram.com/notebooks/SolidGeometry/Point-LineDistance3-Dimensional.nbhttp://mathworld.wolfram.com/DotProduct.htmlhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html#eqn2http://mathworld.wolfram.com/VectorQuadrupleProduct.html
-
8/13/2019 Distance Between a Point and a Line
4/6
)
where denotes the cross productthen gives
(
)
and ta0ing the square root results in the beautiful formula
()
(1*
)
(11
)
2ere! the numeratoris simply twice theareaof the triangle formed
by points ! ! and !
and the denominatoris the length of one of the bases of the
triangle! which follows since!
from the usual triangle areaformula! .
344 ,356 7ollinear!,ine!Point! Point-,ine
8istance---8imensional!Triangle rea
http://mathworld.wolfram.com/CrossProduct.htmlhttp://mathworld.wolfram.com/Numerator.htmlhttp://mathworld.wolfram.com/Numerator.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/Denominator.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/Collinear.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Point.htmlhttp://mathworld.wolfram.com/Point.htmlhttp://mathworld.wolfram.com/Point-LineDistance2-Dimensional.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/CrossProduct.htmlhttp://mathworld.wolfram.com/Numerator.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/Denominator.htmlhttp://mathworld.wolfram.com/TriangleArea.htmlhttp://mathworld.wolfram.com/Collinear.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Point.htmlhttp://mathworld.wolfram.com/Point-LineDistance2-Dimensional.htmlhttp://mathworld.wolfram.com/TriangleArea.html
-
8/13/2019 Distance Between a Point and a Line
5/6
Collinear
Three or more points ! ! ! ...! are said to be collinear if they
lie on a single straight line
. line on which points lie! especially if it is related to a
geometric figure such as a triangle!
is sometimes called an a/is.
Two points are trivially collinear since two points determine a
line.
Three points for ! ! " are collinear iffthe ratios of distances
satisfy
(1
)
slightly more tractable condition is obtained by noting that the
areaof a triangledetermined
by three points will be ero iffthey are collinear (including the
degenerate cases of two or all
three points being concurrent)! i.e.!
(
)
or! in e/panded form!
("
)
This can also be written in vector form as
(#
)
where is the sum of components! ! and .
The condition for three points ! ! and to be collinear can also
be e/pressed as the
statement that the distance between any one point and the line
determined by the other two is
ero. +n three dimensions! this means setting in thepoint-line
distance
($
)
giving simply
http://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Triangle.htmlhttp://mathworld.wolfram.com/Axis.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Area.htmlhttp://mathworld.wolfram.com/Triangle.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.htmlhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.htmlhttp://mathworld.wolfram.com/classroom/Collinear.htmlhttp://mathworld.wolfram.com/notebooks/PlaneGeometry/Collinear.nbhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Triangle.htmlhttp://mathworld.wolfram.com/Axis.htmlhttp://mathworld.wolfram.com/Line.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Area.htmlhttp://mathworld.wolfram.com/Triangle.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html
-
8/13/2019 Distance Between a Point and a Line
6/6
(&
)
where denotes the cross product.
3ince three points are collinear if for some constant ! it
follows thatcollinear points in three dimensions satisfy
('
)
(
)
by the rules of determinant arithmetic. 9hile this is a
necessarycondition for collinearity! it is
not sufficient. (+f any single point is ta0en as the origin! the
determinant will clearly be ero.nother countere/ample is provided
by the noncollinear points ! !
! for which but .)
Three points ! ! and in trilinear coordinatesare collinear if
the
determinant
(
)
(:imberling 1! p. ).
,et points ! ! and lie! one each! on the sides of a triangle or
their e/tensions!
and reflect these points about the midpoints of the triangle
sides to obtain ! ! and . Then
! ! and are collineariff ! ! and are (2onsberger 1$).
344 ,356 /is!7oncyclic! 7onfiguration!8irected ngle! 8ro-;arny
Theorem!
