O1 is a circle with diameter AOB, of radius r.
C is a point on AB between O and A, and the length of AC is s.
O2 is a second circle, radius t, centred on C, such that s < t < r, so that some part of O2 will always fall outside O1.
D is the common area of O1 and O2.
If r, s, and t are all integer values less than 100 units, say of centimetres, when is D closest to an integer value?
For example, if r=97, s=2, and t=78, then D = 8185.006 cm^2, a near miss. Is there a neater solution than this?
5 bestdiff=1
6 kill "objintgr.txt":open "objintgr.txt" for output as #2
10 For r = 3 To 100
20 For t = 2 to r-1
30 For s = 1 to t-1
40 cosa = (r * r + (r - s) * (r - s) - t * t) / (2 * r * (r - s))
50 sina = Sqr(1 - cosa * cosa)
60 a = Atn(sina / cosa)
70 hgt = r * sina
80 base = r * cosa
90 d1 = r * r * a - base * hgt
100
110 cosc = (t * t + (r - s) * (r - s) - r * r) / (2 * t * (r - s))
120 sinc = Sqr(1 - cosc * cosc)
121 if cosc=0 then
122 :c=#pi/2
123 :else
130 :c = Atn(sinc / cosc)
140 If c < 0 Then c = c + #pi
150 hgt = t * sinc
160 base = t * cosc
170 d2 = t * t * c - base * hgt
180
190 d = d1 + d2
191 dround=int(d+.5)
192 if abs(d-dround) <= bestdiff then
193 :print r,t,s,d
194 :print #2, r,t,s,d
195 :bestdiff=abs(d-dround)
210 next
220 next
230 next
240 close #2
to find successively better approximations to integers:
r t s D
3 2 1 9.3204778955757026782
5 2 1 9.6991563660426482314
5 3 1 18.2246946609861792419
5 3 2 24.082570710619696239
7 5 4 72.0498673837428549039
8 3 1 18.9787625988235752569
9 6 1 60.0117226015716722812
10 7 6 146.0034085891044524847
11 9 6 209.9991389657466712032
22 18 6 625.0007644328349099103
23 9 4 187.9996978940880101749
24 19 4 613.999794044358302884
32 11 5 285.0001721145300130949
35 34 25 3049.0000917415865148564
36 14 1 310.0000559545887104217
45 13 1 275.000030503785316484
45 33 24 2989.0000155083033543965
73 22 2 798.9999990508283636297
verified in VB:
3 2 1 9.3204778955757
5 2 1 9.69915636604265
5 3 1 18.2246946609862
5 3 2 24.0825707106197
7 5 4 72.0498673837429
8 3 1 18.9787625988236
9 6 1 60.0117226015717
10 7 6 146.003408589104
11 9 6 209.999138965747
22 18 6 625.000764432835
23 9 4 187.999697894088
24 19 4 613.999794044358
32 11 5 285.00017211453
35 34 25 3049.00009174159
36 14 1 310.000055954589
45 13 1 275.000030503785
45 33 24 2989.0000155083
73 22 2 798.999999050828
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
pi = Atn(1) * 4
bestdiff = 1
For r = 3 To 100
For t = 2 To r - 1
For s = 1 To t - 1
DoEvents
cosa = (r * r + (r - s) * (r - s) - t * t) / (2 * r * (r - s))
sina = Sqr(1 - cosa * cosa)
a = Atn(sina / cosa)
hgt = r * sina
base = r * cosa
d1 = r * r * a - base * hgt
cosc = (t * t + (r - s) * (r - s) - r * r) / (2 * t * (r - s))
sinc = Sqr(1 - cosc * cosc)
If cosc = 0 Then
c = pi / 2
Else
c = Atn(sinc / cosc)
End If
If c < 0 Then c = c + pi
If c < 0 Then c = c + pi
hgt = t * sinc
base = t * cosc
d2 = t * t * c - base * hgt
d = d1 + d2
dround = Int(d + 0.5)
If Abs(d - dround) <= bestdiff Then
Text1.Text = Text1.Text & r & Str(t) & Str(s) & " " & d & crlf
bestdiff = Abs(d - dround)
End If
Next
Next
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2016-06-14 15:43:05 |
computer solution
