(41) Intersection points between circles

[rawi]

Hi Albert,

I tried it and you are totally right. Thanks for pointing that out to me.

Here is the improved method description:

Let (x1, y1) be the coordinates of midpoint of circle 1 and (x2, y2) be the coordinates of midpoint of circle 2 and r1 resp. r2 be the radii.
Let z be the distance between the midpoints: z=((x1-x2)²+(y1-y2)²)^0.5
The angle alpha between the line between the midpoints and the parallel to the x-axis through (x1, y1) using the sinus-function is:

alpha = arcsin((y2-y1)/z)
Beta can be computed by using the Law of Cosine:

beta = arccos((z²+r1²-r2²)/(2*z*r1))
We must take into account, whether x2<x1 or x2>=x1.
We set sg = 1 if x2>=x1 and sg1 = -1 if x2<x1
Then the coordinates of the two intersection points are:

sx1 = x1 + r1*(cos(alpha+beta))*sg
sy1 = y1 + r1*(sin(alpha+beta))
sx2 = x1 + r1*(cos(alpha-beta))*sg
sy2 = y1 + r1*(sin(alpha-beta))

And here is the improved code, which is shorter and easier than the previous one:

Code:

01 LBL „CIP“
02 CF 01
03 “CIRCLE1?”
04 PROMPT        Input of midpoint and radius of circle 1
05 STO 03
06 RDN
07 STO 02
08 RDN
09 STO 01
10 “CIRCLE2?”
11 PROMPT        Input of midpoint and radius of circle 2
12 STO 06
13 RDN
14 STO 05
15 RCL 02
16 -
17 X²
18 X<>Y
19 STO 04
20 RCL 01 
21 X>Y?
22 SF 01
23 -
24 X²
25 +
26 SQRT
27 STO 07        Distance between midpoints
28 RCL 05
29 RCL 02
30 -
31 X<>Y
32 /
33 ASIN
34 STO 08        Angle between line between midpoints and x-axis
35 RCL 07
36 X²
37 RCL 03
38 X²
39 +
40 RCL 06
41 X²
42 -
43 2
44 /
45 RCL 07
46 /
47 RCL 03
48 /
49 ACOS
55 STO 09
51 +
52 COS
53 LASTX
54 SIN
55 RCL 03
56 *
57 RCL 02
58 +
59 STO 12
60 X<>Y
61 RCL 03
62 *
63 FS? 01
64 CHS
65 RCL 01 
66 +
67 STO 11
68 STOP        X-register: x-coordinate, Y register: y-coordinate of first intersection point
69 RCL 08
70 RCL 09
71 -
72 COS
73 LASTX
74 SIN
75 RCL 03
76 *
77 RCL 02
78 +
79 STO 14
80 X<>Y
81 RCL 03
82 *
83 FS? 01
84 CHS
85 RCL 01
86 +
87 STO 13
88 GTO 02
89 LBL 01
90 “NO SOLUTION”
91 AON
92 STOP
93 AOFF
94 LBL 02
95 CF 01
96 END