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joshmccraney

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Attached is a sketch of geometry in question, sketch.pdf. I am trying to determine the circle's radius ##R## given half corner angle ##\beta##, contact angle ##\alpha##, and displaced vertex ##h##. What I find from law of sines is ##R = h \sin\beta / \cos\alpha##.

However, when I plot this I do not always get a desired answer. See case.pdf. Here I set ##\alpha=60^\circ## yet obviously in this case ##\alpha > 90^\circ##. In this case ##\alpha=120^\circ##.

Any help here is greatly appreciated. I know law of sines has 0,1,2 cases, though I can't see how that is applicable here.

Edit: For what it's worth, it is clear in the limit case ##h\to0\implies\alpha\to 90^\circ##. Just an observation.

However, when I plot this I do not always get a desired answer. See case.pdf. Here I set ##\alpha=60^\circ## yet obviously in this case ##\alpha > 90^\circ##. In this case ##\alpha=120^\circ##.

Any help here is greatly appreciated. I know law of sines has 0,1,2 cases, though I can't see how that is applicable here.

Edit: For what it's worth, it is clear in the limit case ##h\to0\implies\alpha\to 90^\circ##. Just an observation.

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