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Bull's eye! (Posted on 2003-11-22) Difficulty: 3 of 5
Two points have polar coordinates as follows: θ=130,r=.35 (point A) and θ=70,r=.6 (point B). There is a surrounding circle, r=1, that acts as a mirror, and you wish to send a light ray from point A to point B by bouncing it once off the circle. What two alternative directions could you send it in (use an angular measure paralleling the θ coordinate it would have if directed from the origin)?

No Solution Yet Submitted by Antonio    
Rating: 3.6000 (5 votes)

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Solution solution (solved numerically) | Comment 5 of 16 |
I'm ashamed to admit that I have forgotten much of my vector calculus as well as much of my polar coordinate education.

This shouldn't be such a difficult problem for me.

Anyway, I set up a bunch of relationships and then solved it numerically (that basically means I set up a bunch of constraints, and let some variable, theta, change until the constraints settled into an appropriate range... usually zero).

That being said, this is a very unsatisfactory post for me.
_______________________________________

Assuming that I did all my calculations correctly, and I look forward to someone verifying this work... preferably analytically.

The numerical results I found were to aim for the points on the circle where from the origin's perspective the radian measure would be 83.43201955 degrees, and -85.75353466 degrees.
These correspond to aiming at points:
(0.11438199,0.993436843) and (0.074046968,-.997254755)

By the way, this makes a reflection angle of 18.50560563 degrees, and 9.049321949 degrees respectively. (The reflection angle, as Popstar Dave already mentioned, is measured as the angle between the ray of light and line segment along the radius of the circle at the point of reflection)
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But that's not what Antonio asked.

Antonio asked what direction it should point from point A's perspective.

To aim at the first point, A should point at 64.92641392 degrees (off the horizontal, so aimed up and to the right.)
(This is roughly the angle a '/' makes. :-)

To aim at the second point, A should point at -76.70421272 degrees (off the horizontal which means down and to the right... mostly down... :-)

Please advise if this isn't clear. And apologies for not providing an analytic solution.
  Posted by SilverKnight on 2003-11-23 04:47:38
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