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Line-Line Angle

For two lines in the plane with endpoints (x_1,x_2) and (x_3,x_4), the angle between them is given by

costheta=((x_2-x_1)·(x_4-x_3))/(|x_2-x_1||x_4-x_3|).
(1)

The angle theta between two lines in the plane specified in trilinear coordinates by

lalpha+mbeta+ngamma=0
(2)
l^'alpha+m^'beta+n^'gamma=0
(3)

is given by

tantheta=y/x,
(4)

where

x=ll^'+mm^'+nn^'-(mn^'+m^'n)cosA-(nl^'+n^'l)cosB-(lm^'+l^'m)cosC
(5)
y=(mn^'-m^'n)sinA+(nl^'-n^'l)sinB+(lm^'-l^'m)sinC
(6)

(Kimberling 1998, p. 31).

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