ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 67)

Appendix A: Projection of a point on a
line defined by the end points
The line
L
is defined by its end points
P
1
(X
1
,Y
1
)
and
P
2
(X
2
,Y
2
). Drop a perpendicular
from some point
(X
o
,Y
o
)
on the line
L
and find coordinates
(X
pr
, Y
pr
)
of the projection
point and the length of the perpendicular h .
a) X
1
= X
2
X
pr
= X
1
Y
pr
= Y
o
h = |X
o
– X
1
|
b) Y
2
= Y
1
X
pr
= X
o
Y
pr
= Y
1
h= |Y
o
– Y
1
|
c) X
1
≠ X
2
and Y
2
≠ Y
1
Line equation
(X– X
1
)/(X
2
– X
1
) = (Y – Y
1
)/(Y
2
– Y
1
)
(a1.1)
That can be transformed into
(Y – Y
1
)∆X = (X – X
1
)∆Y
or
Y = (Y
1
dX + X∆Y – X
1
∆Y)/∆X = Y
1
+ k(X – X
1
)
(a1.2)
Where
k = ∆Y/∆X = (Y
2
– Y
1
)/(X
2
– X
1
)
An equation of the perpendicular dropped from (Xo,Yo) on L can be written in a form
(Y - Y
o
) = (-1/k)(X – X
o
) = q(X – X
o
)
(a1.3)
or
Y = Y
o
+ q(X – X
o
) (a1.4)
Substituting (a1.4) into (a1.2) the results are
Yo + q(X – Xo) = Y
1
+ k(X – X
1
)
(a1.5)
That produces for X coordinate
X = (Y
o
– Y
1
+ kX
1
– qX
o
)/(k – q)
(a1.6)
Maestro Software Manual
MAN-MLT(Ver. 2.0)
A-1