Line Geometry
(1) Line equation:
y =
x +
Two points on line:
x
_{1}
:
y
_{1}
:
x
_{2}
:
y
_{2}
:
Angle (α):
Intercepts: (x
_{i}
,0):
(0,y
_{i}
):
Midpoint:
x
_{d}
:
y
_{d}
:
Line length:
Point in plane:
x
_{p}
:
y
_{p}
:
The point is on the line
Distance from point (x
_{p}
, y
_{p}
) to line (1):
Equation of line passing through (x
_{p}
, y
_{p}
)
and is perpendicular to line (1):
Intersection coordinates:
x:
y:
Degree
Radian
Line basic definition
▲
The most general equation of a line is of the form:
(1)
where A, B and C are any real number and A and B are not both zero. If B ≠ 0 then we can divide
equation (1) by B to obtain the form:
(2)
This is the equation of a line whose slope is:
and
Equation (2) can be normalized to the general form:
(3)
the slope of the line (m) is defined in terms of the inclination
(4)
Note: if the angle α is greater then 90 degree then the slope is negative.
α (0 - 90)
degree
:positive slope
α (90 - 180)
degree
:negative slope
Necessary condition for two lines to be perpendicular to each other is that their slopes fulfills the
condition:
m
_{1}
m
_{2}
= − 1
(5)
In order to find the intersection point of two lines we have to solve the system of linear equations
representing the lines.
A x + B y = −C
D x + E y = −F
solution by matrix:
If the
determinant
:
then intersection point exists.
Basic line of the form
y = ax + b or Ax + By + C = 0
▲
Slope (m) of the line
m = a
y
_{intercept}
(y
_{i}
)
y
_{i}
= b
y
_{i}
= −mx
_{i}
y
_{i}
= y
_{1}
− mx
_{1}
x
_{intercept}
(x
_{i}
)
tan θ
tan θ = m
Line angle (θ)
from x axis (range 0 ≤ θ < π)
θ = arctan(m)
Slope (M) of a line perpendicular to a given slope (m)
Line midpoint
Point (x, y) which divides the line connecting two points (x
_{1}
, y
_{1}
) and (x
_{2}
, y
_{2}
) in the ratio p:q
Point (x, y) which divides the line connecting two points (x
_{1}
, y
_{1}
) and (x
_{2}
, y
_{2}
) externally at a ratio p:q
Note: the (x,y) point is in the direction from point 1 to point 2,to get the other side extension change the point 1 with point 2 and vice versa
A point (x, y) which is located at a distance d from a point (x
_{1}
, y
_{1}
) on the line
Angle θ between two lines:
Angle between two lines given by
Ax + By + C = 0
Dx + Ey + F = 0
Lines equations
▲
Equation of a line passing through a point (x
_{1}
, y
_{1}
)
y = mx + (y
_{1}
− mx
_{1}
)
y − y
_{1}
= m(x − x
_{1}
)
Equation of a line passing through two points (x
_{1}
, y
_{1}
), (x
_{2}
, y
_{2}
)
Equation of a line perpendicular to a given slope m and passing through a point (x
_{p}
, y
_{p}
)
Equation of a line perpendicular to a line which is defined by two points (x
_{1}
, y
_{1}
) and (x
_{2}
, y
_{2}
) and passing through the point (x
_{p}
, y
_{p}
)
Equation of a line passing through the intercepts x
_{i}
, y
_{i}
x
_{i}
y = −y
_{i}
x + x
_{i}
y
_{i}
Equation of a line passing through the point (x
_{p}
,y
_{p}
) and parallel to a line which is defined by two points (x
_{1}
, y
_{1}
) and (x
_{2}
, y
_{2}
)
Equation of a line parallel to the line Ax + By + C = 0 and at a distance d from it.
Equation of the midline between the lines Ax + By + C = 0
Dx + Ey + F = 0
Equation of a line perpendicular to the line Ax + By + C = 0
Bx − Ay + C = 0
Equation of a horizontal line
y = b
Equation of a vertical line
x = a
Lines distances y = ax + b Ax + By + C = 0
▲
Distance between two points (D)
Distance between intercepts x
_{i}
and y
_{i}
Distance from a line to the origin
Distance from a line given by two points (x
_{1}
,y
_{1}
),(x
_{2}
, y
_{2}
) to the origin
Distance from a line to the point
(x
_{p}
, y
_{p}
)
Distance from a line given by two points (x
_{1}
, y
_{1}
) , (x
_{2}
, y
_{2}
) to the point (x
_{p}
, y
_{p}
)
Distance between two parallel lines
y = ax + b
y = cx + d
or
Ax + By + C = 0
Dx + Ey + F = 0