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October 17th, 2017, 06:21 PM  #1 
Senior Member Joined: Jul 2015 From: Florida Posts: 116 Thanks: 2 Math Focus: noneuclidean geometry  Does this identity belong with the circle functions or with the conic sections?
$$\alpha={\cot}^{1 }(\cos\upsilon\tan{\sin}^{1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon}))$$ The equation describes the relationship between 3 dihedral angles in 3D. It is expressed as: $$\alpha=f(\lambda)$$ To understand the angles, it’s easiest to construct a unit sphere with North and South poles and an axis that contains both poles. If a small circle is placed on the sphere such that it passes through the North pole, then $$\alpha$$ is the slope of the tangent to a point on the small circle (relative to a line of longitude) and $$\lambda$$ is the dihedral angle between the North pole and the tangent point. In order to define a family of functions, $$\upsilon$$ is a third variable that expresses the dihedral angle between the North pole and the center of the small circle. With no spherical excess $$\upsilon = 0$$ the equation produces a sine curve. With maximum spherical excess $$\upsilon = \frac{\pi}{2}$$ the equation produces a hyperbola. Even though the identity can produce either a sine or a hyperbola, it doesn’t seem to belong to either class of functions. 
October 18th, 2017, 02:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,380 Thanks: 542 
Your question is confusing. I don't see an identity.

October 18th, 2017, 04:44 PM  #3 
Senior Member Joined: Jul 2015 From: Florida Posts: 116 Thanks: 2 Math Focus: noneuclidean geometry  I'm sort of confused, hence a confusing question. I think it is a sin vs. cos type of thing: $$\sin{\upsilon = \frac{\sin\frac{\lambda}{2 }}{\sin\frac{\phi}{2 }}}$$ $$\cos\upsilon = \frac{\cot\alpha}{\tan\frac{\phi}{2 }}$$ is a parametrization of the equation. I thought that since $$\cos^2(\upsilon)+\sin^2(\upsilon)=1$$ that this would mean that $$(\frac{\sin\frac{\lambda}{2 }}{\sin\frac{\phi}{2 }})^2+(\frac{\cot\alpha}{\tan\frac{\phi}{2 }})^2 = 1$$ I know I'm doing something wrong, I just don't know what it is. 
October 18th, 2017, 08:01 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,642 Thanks: 960 Math Focus: Elementary mathematics and beyond 
What does $\phi$ represent?

October 18th, 2017, 09:02 PM  #5 
Senior Member Joined: Jul 2015 From: Florida Posts: 116 Thanks: 2 Math Focus: noneuclidean geometry 
It's the small circle arc length from the North pole to the tangent point. $$0\leq\phi\leq\pi$$ Last edited by steveupson; October 18th, 2017 at 09:28 PM. 
October 19th, 2017, 04:55 AM  #6 
Senior Member Joined: Jul 2015 From: Florida Posts: 116 Thanks: 2 Math Focus: noneuclidean geometry 
Using the range and domain given, only a quarter of the full wave is present, as shown in the attachment. If the limits are changed to $$0\leq\upsilon\leq\pi$$ and $$0\leq\phi\leq2\pi$$ then this should produce the full wave function.
Last edited by steveupson; October 19th, 2017 at 05:27 AM. 
October 25th, 2017, 10:08 AM  #7  
Senior Member Joined: Jul 2015 From: Florida Posts: 116 Thanks: 2 Math Focus: noneuclidean geometry  Quote:
$$\left(\begin{array}{c}{\cos\frac{\phi}{2 }}\:{\sin\frac{\lambda}{2 }}\end{array}\right)^2+\left(\begin{array}{c}{\cot \frac{\phi}{2 }}\:{\cot\alpha}\end{array}\right)^2 = 1$$  

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