ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 72)

Appendix D: Circle – line intersection
points
The line is defined by its end points (X
1
,Y
1
) and (X
2
,Y
2
). The circle is defined by its radius R and
center coordinates (X
o
, Y
o
). Consider the general case X
1
≠ X
2
and Y
1
≠ Y
2
. In this case, calculate
the intersection points using
(X – X
1
)/(X
2
– X
1
) = (Y – Y
1
)/(Y
2
– Y
1
) (a4.1)
(X – X
o
) 2 + (Y – Y
o
) 2 = R 2 (a4.2)
Note that k = (Y
2
– Y
1
)/(X
2
– X
1
)
and
C
1
= Y
1
– kX
1
equation (a4.1) can be written in
the following form
Y = kX + (Y
1
– kX
1
) = kX+ C
1
(a4.3)
Substituting (a4.3) into (a4.2) results in
(X – X
o
) 2 + (kX + C
1
– Y
o
) 2 – R 2 = 0 (a4.4)
Simplifying (a4.4) results in the following equation
C
3
X 2 + C
4
X + C
5
= 0 (a4.5)
where
C
2
= C
1
– Y
o
, C
3
= 1 + k 2 ,
C
4
= 2*(–X
o
+ kC
2
), C
5
= (X
o
) 2 + (C
2
) 2 – R 2
Note that
d = (C
4
) 2 – 4C
3
C
5
and for intersection point
X
coordinates the results are
X
1
= (–C
4
+ d 1/2 )/(2C
3
) (a4.6)
X
2
= (–C
4
– d 1/2 )/(2C
3
) (a4.7)
Respective
Y
coordinates can be calculated by (a4.3) as
Y
1
= kX
1
+ C
1
(a4.8)
Y
2
= kX
2
+ C
1
(a4.9)
Consider the case
X
1
= X
2
. In this case line equation
X = X
1
and substituting into (a4.2) the results are
(Y – Y
o
) 2 = R 2 – (X
1
– X
o
) 2 (a4.10)
and for Y the results are
Y 1= Y
o
+ [R 2 – (X
1
– X
o
) 2 ] 1/2 (a4.11)
Y 2= Y
o
– [R 2 – (X
1
– X
o
) 2 ] 1/2 (a4.12)
Maestro Software Manual
MAN-MLT (Ver. 2.0)
D-1