# Graphing the Function

Figure 4.1

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

A New Perspective

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Figure 4.1

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

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

C_{O} = (0,0,0) = origin = center of circle C

r = = 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, r sin,0) = tangent point on circle C where planes , , and intersect

h = arc length of circle C from P to S_{N}

h = (from analyzing a chord in circle C)

h = (from analyzing isosceles triangle ∆S_{N}S_{O}P, which has two sides of length 1)

= plane containing point S_{O} and also the equator of sphere S

= plane containing point S_{O} and also the tangent (in the xy-plane) to circle C at point S_{N}

= plane containing point S_{O} and also the tangent (in the xy-plane) to circle C at point P

= plane containing point S_{O} and point P and point S_{N}

= plane containing point S_{O} and point P and plane is perpendicular to plane

a = great circle arc λ

b = great circle arc υ

c = great circle arc ∠S_{C}S_{O}P

D = midpoint of side a

= ∠

= arc length in degrees from (r,0,0) to point P = ∠S_{N}C_{O}P = dihedral ∠S_{N}-S_{O}C_{O}-P

= ∠PS_{O}S_{N}

= angle between the axis of cone N and the cardinal axis of sphere S = ∠S_{N}S_{O}C_{O}.

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**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 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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**Figure 3. Isosceles Spherical Triangle ∆abc**

Point S_{C} is the intersection of the z-axis with sphere S such that ∠S_{N}S_{O}C_{O} = = ∠S_{N}S_{O}S_{C}. There is a spherical triangle ∆abc that, when solved, will express the relationship between , , , and such that:

Side a is great circle arc of length

Side b is great circle arc of length , or ∠S_{C}S_{O}S_{N}

Side c is great circle arc of length , or ∠S_{C}S_{O}P

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The objective is to define a family of functions which are based on different values of , and which express α as a function of . The approach will be to find and , each as a function of . 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 that change between 0° and 90°, while below them, in the rectangular box, there is another set of numbers representing 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 as a function of with angle equal to 45°, or z = .

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

Plane can be considered the elevation plane and it stays perpendicular to plane and passes through sphere center S_{O} and point P. The smallest angle between the cardinal axis of the sphere and plane is angle , or ∠S_{N}S_{O}P.

Point P can be defined as an arc length along circle C equal to angle , or ∠S_{N}C_{O}P.

The angle between the axis of cone N and the cardinal axis of sphere S is angle , or ∠S_{N}S_{O}C_{O}.

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The two planes that remain stationary in the animation each contain S_{O} and a line tangent to circle C. For tan-colored plane this tangent is at (-r,0,0) and for magenta-colored plane 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 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 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 (or light green) contains points S_{O} and P, and is perpendicular to .

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**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.