Expression 15: theta Subscript, 0 , Baseline equals arc sine left parenthesis, tan left parenthesis, alpha , right parenthesis tan left parenthesis, beta , right parenthesis , right parenthesisθ0=arcsintanαtanβ
equals=
0.2 9 2 9 5 0 0 3 0 8 1 90.292950030819
15
Hidden Label: "e" Subscript, 1 , Baseline with "t" equals StartFraction, theta Subscript, 0 , Baseline Over 2 pi , EndFractione1witht=θ02π
Hidden Label: "e" Subscript, 1 , Baseline with "t" equals StartFraction, negative theta Subscript, 0 , Baseline minus pi Over 2 pi , EndFractione1witht=−θ0−π2π
Expression 18: left brace, tan left parenthesis, alpha , right parenthesis tan left parenthesis, beta , right parenthesis less than 1 : left parenthesis, "e" Subscript, 1 , Baseline for StartFraction, negative theta Subscript, 0 , Baseline minus pi Over 2 pi , EndFraction less than or equal to "t" less than or equal to StartFraction, theta Subscript, 0 , Baseline Over 2 pi , EndFraction , right parenthesis , left parenthesis, "e" Subscript, 1 , Baseline for 0 less than or equal to "t" less than or equal to 1 , right parenthesis , right brace for beta equals left bracket, negative 1 , negative 8 ninths ...1 , right bracket StartFraction, pi Over 2 , EndFractiontanαtanβ<1:e1for−θ0−π2π≤t≤θ02π,e1for0≤t≤1forβ=−1,−89...1π2
18
longitude
19
Expression 20: varphi Subscript, 0 , Baseline equals 1 half arc cosine left parenthesis, StartFraction, tan squared left parenthesis, alpha , right parenthesis minus sin squared left parenthesis, gamma , right parenthesis Over tan squared left parenthesis, alpha , right parenthesis plus sin squared left parenthesis, gamma , right parenthesis , EndFraction , right parenthesisφ0=12arccostan2α−sin2γtan2α+sin2γ
equals=
0.8 2 5 4 2 1 5 9 6 9 1 90.825421596919
20
stays on top/bottom; rotates smoothly with 0 < γ < π
21
Hidden Label: "T" equals "e" Subscript, 2 , Baseline with "t" equals left brace, gamma greater than or equal to 0 : StartFraction, varphi Subscript, 0 , Baseline Over 2 pi , EndFraction , StartFraction, negative varphi Subscript, 0 , Baseline plus pi Over 2 pi , EndFraction , right braceT=e2witht=γ≥0:φ02π,−φ0+π2π
Expression 25: left brace, gamma greater than or equal to 0 : left parenthesis, "e" Subscript, 2 , Baseline for StartFraction, varphi Subscript, 0 , Baseline Over 2 pi , EndFraction less than or equal to "t" less than or equal to StartFraction, varphi Subscript, 0 , Baseline plus pi Over 2 pi , EndFraction , right parenthesis , left parenthesis, "e" Subscript, 2 , Baseline for StartFraction, negative varphi Subscript, 0 , Baseline Over 2 pi , EndFraction less than or equal to "t" less than or equal to StartFraction, pi minus varphi Subscript, 0 , Baseline Over 2 pi , EndFraction , right parenthesis , right brace for gamma equals left bracket, negative 1 , negative 8 ninths ...1 , right bracket StartFraction, pi Over 2 , EndFractionγ≥0:e2forφ02π≤t≤φ0+π2π,e2for−φ02π≤t≤π−φ02πforγ=−1,−89...1π2
25
stays on left/right; rotates smoothly with -π/2 < γ < π/2
26
Hidden Label: "R" equals left parenthesis, cos left parenthesis, gamma , right parenthesis cos left parenthesis, varphi Subscript, 0 , Baseline , right parenthesis , StartFraction, gamma Over StartAbsoluteValue, gamma , EndAbsoluteValue , EndFraction cos left parenthesis, alpha , right parenthesis sin left parenthesis, varphi Subscript, 0 , Baseline , right parenthesis plus sin left parenthesis, alpha , right parenthesis sin left parenthesis, gamma , right parenthesis cos left parenthesis, varphi Subscript, 0 , Baseline , right parenthesis , right parenthesisR=cosγcosφ0,γγcosαsinφ0+sinαsinγcosφ0
