Figure 3.1  Mathematica Model

The animation in Fig. 3.1 above is an illustration of a mathematical model that was produced by Hans Milton in Mathematica.  The model captures two smooth functions that are interdependent in normal, orthogonal, Euclidean 3-space.  The illustration shows an animation of the visual components of this relationship.

Also in the animation are two sets of changing numbers; one set at the top of the image adjacent to the slider, and another set in the small rectangle directly above the model.   These two sets of numbers are ordered pairs of a function that expresses the model.

The animation includes two fixed planes (tan=$\mathbb{T}$ and magenta=$\mathbb{M}$), three moving planes (yellow=$\mathbb{Y}$, blue=$\mathbb{B}$, and green=$\mathbb{G}$), a sphere, a circle, and an intersection point for the three moving planes that traces a path along the circle.

The upper set of numbers (E in the animation) is the angle between the yellow plane and the tan plane.   Originally, this was referred to as the elevation angle (hence the E variable) but this nomenclature has been changed in order to make this angle the lambda angle, $\lambda$.  The other set of numbers represents the angle between the green and blue planes.  These planes are the longitude and tangent planes, respectively, and the angle between them is the alpha angle, $\alpha$.

The function $\alpha=f(\lambda)$ produces the set of ordered pairs in the animation.  This set of ordered pairs is for a circle having a diameter equal to 45º of latitude.  The model that illustrates the function can have a circle of any size relative to the sphere.

Note that the sphere utilized in the model is only there for discussion purposes.  The sphere is utilized in order to better explain the model and to orient the model components and to solve for the dihedral angles of interest.  This is a subtle but extremely important observation.  The orientation of the components (their gross position in space) is what is of interest, and the spherical surface is only utilized in order to help visualize the arrangement.

In order to express the function, an additional angle is defined as part of the derivation.  This additional angle represents the arc length of the travel of the intersecting point along the circle.  We call this the phi angle, $\phi$.  The derivation then reduces to solving an isosceles spherical triangle.   The solution of the triangle yields two simultaneous equations:

$\cot\alpha = \cos\upsilon\tan\frac{\phi}{2 }\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\:(1)$

$\sin\frac{\lambda}{2 } = \sin\frac{\phi}{2 }\sin{\upsilon}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\:\:\:(2)$

The function $\alpha=f(\lambda)$ is:

$\alpha={\cot}^{-1 }(\cos\upsilon\tan{\sin}^{-1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon}))\quad\quad\quad\quad\:\:(3)$

It is a smooth function that approaches a sine curve when $\upsilon\to0$, and that approaches a hyperbola when $\upsilon\to\frac{\pi}{2}$.

Equations (1) and (2) can be rewritten in the form of:

$\sin{\upsilon}= \frac{\sin\frac{\lambda}{2}}{\cos\frac{\phi}{2}}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\:\:\:\:(4)$

$\cos{\upsilon}= \frac{\cot\alpha}{\cot\frac{\phi}{2}}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\:\:\:\:(5)$

and since,

$\sin^2 \upsilon+\cos^2\upsilon=1$

then we should get a new spherical trigonometry identity:

$(\cos\frac{\phi}{2}\,\sin\frac{\lambda}{2})^2 +(\cot\frac{\phi}{2}\,\cos\alpha)^2=1\quad\quad\:\:\:(6)$

Special thanks and credit go to tashirosgt for developing the method used for expressing and solving the model.

The model was produced using Mathematica, which has a downloadable viewer/player.   In addition to the .gif animation above, there is also another viewable version of the model in the form of a .cdf file.   The file requires a free viewer/player that can be downloaded from the Mathematica website.  Viewing the model this way allows the image to be stopped and rotated in 3D, and allows manual positioning of the slider.  This provides for a much better visual examination of the angles of interest ($\alpha$ and $\lambda$.)