ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 70)

Appendix C: Intersection point of two lines
defined by the end points
Line
L
1
is defined by its end points
P
1
(X
1
,Y
1
)
and
P
2
(X
2
,Y
2
).
Line
L
2
is defined by end
points
P
3
(X
3
,Y
3
)
and
P
4
(X
4
,Y
4
).
Note:
∆X
1
= X
2
– X
1
, ∆Y
1
= Y
2
– Y
1
, ∆X
2
= X
4
– X
3
, ∆Y
2
= Y
4
– Y
3
,
k
1
= ∆Y
1
/∆X
1
, q
1
= ∆X
1
/∆Y
1
, k
2
= ∆Y
2
/∆X
2
, q
2
= ∆X
2
/∆Y
2
Calculation of the two lines intersection point
(X,Y)
depends on the each line position
relative to coordinate axes. The following cases are possible.
a) X
1
≠ X
2
, Y
1
≠ Y
2
, X
3
≠ X
4
, Y
3
≠ Y
4
To define intersection point, use
(X – X
1
)∆Y
1
= (Y – Y
1
)∆X
1
(a3.1)
(X – X
3
)∆Y
2
= (Y – Y
3
)∆X
2
(a3.2)
or
X∆Y
1
– X
1
∆Y
1
= Y∆X
1
– Y
1
∆X
1
(a3.3)
X∆Y
2
– X
3
∆Y
2
= Y∆X
2
– Y
3
∆X
2
(a3.4)
From (a3.3) the results are
X = X
1
+ Yq
1
– Y
1
q
1
(a3.5)
and substituting into (a3.4) the results are
(X
1
+ Yq
1
– Y
1
q
1
)∆Y
2
– X
3
∆Y
2
= Y∆X
2
– Y
3
∆X
2
(a3.6)
that produces
Y = [(q
1
Y
1
+ X
3
– X
1
)∆Y
2
–Y
3
∆X
2
]/(q
1
∆Y
2
– ∆X
2
) (a3.7)
and X can be calculated by (a3.5).
b) X
1
≠ X
2
, Y
1
≠ Y
2
, X
3
= X
4
, Y
3
≠ Y
4
To find an intersection point use
(X – X
1
)/∆X
1
= (Y – Y
1
)/∆Y
1
(a3.8)
X = X
3
(a3.9)
Maestro Software Manual
MAN-MLT (Ver. 2.0)
C-1