Graphing the Function

December 24, 2016 By admin in Background No Comments

Figure 4.1 $\alpha=f(\lambda)$

This graph was generated by Strange, using data from the Mathematica model written by Hans Milton.

Variables

March 15, 2017 By admin in Formula No Comments

C = circle made by intersection of the xy-plane and unit sphere S

C_O = (0,0,0) = origin = center of circle C

r = $\sqrt{1-{s_z}^2}$ = radius of circle C

(r,0,0) = intersection of positive x-axis with unit sphere S = S_N (north pole)
(-r,0,0) = intersection of negative x-axis with unit sphere S
(0,r,0) = intersection of positive y-axis with unit sphere S
(0,-r,0) = intersection of negative y-axis with unit sphere S

z = length from origin to center of unit sphere

S = unit sphere

S_O = (0,0, s_z ) = center of unit sphere S, where -1 < s_z < 1

S_N = (r,0,0) = north pole of unit sphere S

S_C = (0,0,1-z) = projection of line segment between S_O and C_O to the surface of sphere S

N = cone having circle C as its base and sphere center S_O as its apex

P = (r cos $\phi$ , r sin $\phi$ ,0) = tangent point on circle C where planes $\mathbb{B}$ , $\mathbb{G}$ , and $\mathbb{Y}$ intersect

h = arc length of circle C from P to S_N
h = ${2r}\sin\frac{\phi}{2}$ (from analyzing a chord in circle C)
h = ${2}\sin\frac{\lambda}{2}$ (from analyzing isosceles triangle ∆S_NS_OP, which has two sides of length 1)

$\mathbb{T}$ = plane containing point S_O and also the equator of sphere S

$\mathbb{M}$ = plane containing point S_O and also the tangent (in the xy-plane) to circle C at point S_N

$\mathbb{B}$ = plane containing point S_O and also the tangent (in the xy-plane) to circle C at point P

$\mathbb{G}$ = plane containing point S_O and point P and point S_N

$\mathbb{Y}$ = plane containing point S_O and point P and plane $\mathbb{Y}$ is perpendicular to plane $\mathbb{G}$

a = great circle arc λ
b = great circle arc υ
c = great circle arc ∠S_CS_OP

D = midpoint of side a

$\alpha$ = ∠ $\mathbb{BG}$

$\phi$ = arc length in degrees from (r,0,0) to point P = ∠S_NC_OP = dihedral ∠S_N-S_OC_O-P

$\lambda$ = ∠PS_OS_N

$\upsilon$ = angle between the axis of cone N and the cardinal axis of sphere S = ∠S_NS_OC_O.

Alpha as a Function of Phi

March 15, 2017 By admin in Formula, Gallery No Comments

Figure 4. Dihedral Angle α

Referring to Fig. 4, since b and c both have endpoints on circle C, Δabc is an isosceles triangle with dihedral angle $\phi$ between the two equal sides. If we construct a great circle arc from point S_C to midpoint D of side a, we will bisect Δabc into two congruent right spherical triangles.

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Lambda as a Function of Phi

March 15, 2017 By admin in Formula, Gallery No Comments

Figure 3. Isosceles Spherical Triangle ∆abc

Point S_C is the intersection of the z-axis with sphere S such that ∠S_NS_OC_O = $\upsilon$ = ∠S_NS_OS_C. There is a spherical triangle ∆abc that, when solved, will express the relationship between $\alpha$ , $\phi$ , $\lambda$ , and $\upsilon$ such that:

Side a is great circle arc of length $\lambda$
Side b is great circle arc of length $\upsilon$ , or ∠S_CS_OS_N
Side c is great circle arc of length $\upsilon$ , or ∠S_CS_OP

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The Problem

March 15, 2017 By admin in Formula No Comments

The objective is to define a family of functions which are based on different values of $\upsilon$ , and which express α as a function of $\lambda$ . The approach will be to find $\alpha$ and $\lambda$ , each as a function of $\phi$ . One of the methods used here will be to solve a spherical isosceles triangle using spherical trigonometry.

At the top right corner of the animation there is a set of numbers representing $\lambda$ that change between 0° and 90°, while below them, in the rectangular box, there is another set of numbers representing $\alpha$ that also change between 0° and 90°. The mathematical model used for the animation defines a specific member of a family of functions. The z-coordinate for the center of sphere S defines which member of the family we are analyzing. The model shows $\alpha$ as a function of $\lambda$ with angle $\upsilon$ equal to 45°, or z = $\frac{1}{\sqrt{2}}$ .

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The Angles

March 15, 2017 By admin in Formula No Comments

Plane $\mathbb{G}$ can be considered the longitude plane because the great circle made by its intersection with sphere S is always a line of longitude. Plane $\mathbb{B}$ can be considered the tangent plane since it always contains a line tangent to circle C at point P. The dihedral angle between planes $\mathbb{G}$ and $\mathbb{B}$ is angle $\alpha$ , the angle of interest.

Plane $\mathbb{Y}$ can be considered the elevation plane and it stays perpendicular to plane $\mathbb{G}$ and passes through sphere center S_O and point P. The smallest angle between the cardinal axis of the sphere and plane $\mathbb{G}$ is angle $\lambda$ , or ∠S_NS_OP.

Point P can be defined as an arc length along circle C equal to angle $\phi$ , or ∠S_NC_OP.

The angle between the axis of cone N and the cardinal axis of sphere S is angle $\upsilon$ , or ∠S_NS_OC_O.

The Planes

March 15, 2017 By admin in Formula No Comments

The two planes that remain stationary in the animation each contain S_O and a line tangent to circle C. For tan-colored plane $\mathbb{T}$ this tangent is at (-r,0,0) and for magenta-colored plane $\mathbb{M}$ it is at S_N which is also (r,0,0). Because these two planes contain the center of the sphere, which need not be in the xy-plane, these stationary planes are not parallel to, or perpendicular to, the xy-plane except for when r is at either limit r = 0, or r = 1.

The point on C where 3 moving planes intersect is P. Blue-colored plane $\mathbb{B}$ contains S_O and a line tangent to the circle at point P. The animation starts with P at (r,0,0) and then moves P along a 180º arc of circle C.

Green-colored plane $\mathbb{G}$ contains the point P and pivots about an axis as P moves, but this axis is not one of the coordinate axes. It is an axis defined by a line containing the line segment from S_O to S_N, which is the main or cardinal axis of sphere S.

Yellow-colored plane $\mathbb{Y}$ (or light green) contains points S_O and P, and is perpendicular to $\mathbb{G}$ .

The Coordinate System

March 14, 2017 By admin in Formula, Gallery No Comments

Figure 2. Coordinate System

In the model as illustrated in Fig. 2, small circle C has a circumference that is 45º of latitude. We will be using the conventional terminology where a circle on the surface of a sphere that is made by the intersection of the sphere with a plane passing through the sphere center is called a great circle. All other circles, where the intersecting plane does not pass through the sphere center S_O, are called small circles.

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SB Left – 2 Column