[SOLVED] Question in geometry

Started by nashwaan, October 04, 2011, 07:43:44 PM

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nashwaan

Suppose we have two circles sharing same center O. Circle 1 is defined by r1 and circle 2 is defined by r2. Lets say we pick an arbitrary point A that lies on circumference of circle 1. If we strech this line OA, it will cross circle 2 at point B. If we define another line at point A with a specific angle phi (in respect to line AB), this new line will cross circle 2 at point E. Now, Line OB and line OE are defining angle theta. How can we find value of theta?

If the above doesn't make much sense, then looking at the below figure only: If {r1, r2, phi} are all known values, how can we calculate theta?


Thanks,
Yousuf.
Give me six hours to chop down a tree and I will spend the first four sharpening the axe — Abraham Lincoln

JuneTheSecond

Though my answer may include many mistakes.
Best Regards,

Junichi Yoda
http://june.minibird.jp/

wapperdude

I haven't followed thru all of JuneTheSecond's math, but, he's usually correct.  However, most trig / geo reference books should contain formulas since you know two sides & an angle of a triangle.  The sides would be OA & OE (the two radii), and the angle, OAE, which is 180 - phi.

HTH
Wapperdude
Visio 2019 Pro

wapperdude

A quick check of Math Table & Formula Handbook provides the necessary equations, see attached pdf.
Visio 2019 Pro

JuneTheSecond

#4
Found my mistake in the last lines.
Best Regards,

Junichi Yoda
http://june.minibird.jp/

nashwaan

#5
Ok, the problem is solved using the following equation (following wapperdude advice):



theta = phi–ASIN(r1/r2*SIN(PI()–phi))

Note, I have verified this equation with numbers and it gave me correct results.
Thanks alot  wapperdude

Yousuf.
Give me six hours to chop down a tree and I will spend the first four sharpening the axe — Abraham Lincoln

nashwaan

#6
JuneTheSecond,

I appreciate your efforts trying to help me.. but unfortunately i wasn't able to follow your method:

  • In step (7), I think there is a mistake (see figure below)
  • I don't understand what did you do exactly to go from step (17) to step (18)


theta = ASIN(COS(phi)*(r1/r2*SIN(phi)+SQRT(1-(r1/r2)^2*SIN(phi)^2)))

I have tested your last equation (both cases + and -) with numbers but it did not give me correct results.

Anyway, i got the correct equation as i showed in my previous reply (following wapperdude method).

Again, i really appreaciate all the efforts and and time you spent on this.

Thanks,
Yousuf.
Give me six hours to chop down a tree and I will spend the first four sharpening the axe — Abraham Lincoln