Φ = 0.3141592653589793 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
θ = 0.5934119456780721 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
φ = 0.4363323129985824 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
P: {-sin(θ), cos(θ), 0, -cos(θ) sin(φ), -sin(θ) sin(φ), cos(φ), cos(θ) cos(φ), sin(θ) cos(φ), sin(φ)}
i_3: {1, 0, 0}
j_3: {0, 1, 0}
k_3: {0, 0, 1}
O_3: {0, 0, 0}
O: (Element[O_3, 1], Element[O_3, 2])
i = Vector(O, (Element[i_3, 1], Element[i_3, 2]))
j = Vector(O, (Element[j_3, 1], Element[j_3, 2]))
U: (Element[P, 1] Element[i_3, 1] + Element[P, 2] Element[i_3, 2] + Element[P, 3] Element[i_3, 3], Element[P, 4] Element[i_3, 1] + Element[P, 5] Element[i_3, 2] + Element[P, 6] Element[i_3, 3])
V: (Element[P, 1] Element[j_3, 1] + Element[P, 2] Element[j_3, 2] + Element[P, 3] Element[j_3, 3], Element[P, 4] Element[j_3, 1] + Element[P, 5] Element[j_3, 2] + Element[P, 6] Element[j_3, 3])
W: (Element[P, 1] Element[k_3, 1] + Element[P, 2] Element[k_3, 2] + Element[P, 3] Element[j_3, 3], Element[P, 4] Element[k_3, 1] + Element[P, 5] Element[k_3, 2] + Element[P, 6] Element[k_3, 3])
H = (1, 0)
G = (-1, 0)
u = Vector(O, U)
v = Vector(O, V)
w = Vector(O, W)
R_Φ: {cos(Φ), -sin(Φ), 0, sin(Φ), cos(Φ), 0, 0, 0, 1}
Mtest_0: O + cos(Φ) u + sin(Φ) v
M_0: O + Element[R_Φ, 1] Element[i_3, 1] u + Element[R_Φ, 4] Element[i_3, 1] v + Element[R_Φ, 7] Element[i_3, 1] w
N_0: O + Element[R_Φ, 2] Element[j_3, 2] u + Element[R_Φ, 5] Element[j_3, 2] v + Element[R_Φ, 8] Element[j_3, 2] w
Ψ = 0.6632251157578453 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
Mtest: O + cos(Φ) cos(Ψ) u + sin(Φ) cos(Ψ) v + sin(Ψ) w
Coordsph: {cos(Φ) cos(Ψ), sin(Φ) cos(Ψ), sin(Ψ)}
M: (Element[P, 1] Element[Coordsph, 1] + Element[P, 2] Element[Coordsph, 2] + Element[P, 3] Element[Coordsph, 3], Element[P, 4] Element[Coordsph, 1] + Element[P, 5] Element[Coordsph, 2] + Element[P, 6] Element[Coordsph, 3])
coefΓ_0: {1, 0, 1 / sin(φ)^(2)}
Γ_0: Element[coefΓ_0, 1] x^(2) + Element[coefΓ_0, 2] x y + Element[coefΓ_0, 3] y^(2) = 1
den_Φ: cos(Φ) * (Element[P, 1] Element[P, 6] - Element[P, 4] Element[P, 3]) + sin(Φ) * (Element[P, 2] Element[P, 6] - Element[P, 5] Element[P, 3])
coefMer_Φ: {(Element[P, 6]^(2) + (cos(Φ) Element[P, 4] + sin(Φ) Element[P, 5])^(2)) / den_Φ^(2), -2 * (Element[P, 3] Element[P, 6] + (cos(Φ) Element[P, 4] + sin(Φ) Element[P, 5]) * (cos(Φ) Element[P, 1] + sin(Φ) Element[P, 2])) / den_Φ^(2), (Element[P, 3]^(2) + (cos(Φ) Element[P, 1] + sin(Φ) Element[P, 2])^(2)) / den_Φ^(2)}
Mer_{Phi}: Element[coefMer_Φ, 1] x^(2) + Element[coefMer_Φ, 2] x y + Element[coefMer_Φ, 3] y^(2) = 1
coefMer_0: {(Element[P, 6]^(2) + Element[P, 4]^(2)) / (Element[P, 1] Element[P, 6] - Element[P, 4] Element[P, 3])^(2), -2 * (Element[P, 3] Element[P, 6] + Element[P, 4] Element[P, 1]) / (Element[P, 1] Element[P, 6] - Element[P, 4] Element[P, 3])^(2), (Element[P, 3]^(2) + Element[P, 1]^(2)) / (Element[P, 1] Element[P, 6] - Element[P, 4] Element[P, 3])^(2)}
Mer_0: Element[coefMer_0, 1] x^(2) + Element[coefMer_0, 2] x y + Element[coefMer_0, 3] y^(2) = 1
coefMer_{90}: {(Element[P, 6]^(2) + Element[P, 5]^(2)) / (Element[P, 2] Element[P, 6] - Element[P, 5] Element[P, 3])^(2), -2 * (Element[P, 3] Element[P, 6] + Element[P, 5] Element[P, 2]) / (Element[P, 2] Element[P, 6] - Element[P, 5] Element[P, 3])^(2), (Element[P, 3]^(2) + Element[P, 2]^(2)) / (Element[P, 2] Element[P, 6] - Element[P, 5] Element[P, 3])^(2)}
Mer_{90}: Element[coefMer_{90}, 1] x^(2) + Element[coefMer_{90}, 2] x y + Element[coefMer_{90}, 3] y^(2) = 1
I: (sin(θ) sin(φ) / sqrt(sin(θ)^(2) sin(φ)^(2) + cos(θ)^(2)), cos(θ) / sqrt(sin(θ)^(2) sin(φ)^(2) + cos(θ)^(2)))
J: (-sin(θ) sin(φ) / sqrt(sin(θ)^(2) sin(φ)^(2) + cos(θ)^(2)), -cos(θ) / sqrt(sin(θ)^(2) sin(φ)^(2) + cos(θ)^(2)))
K: (-cos(θ) sin(φ) / sqrt(cos(θ)^(2) sin(φ)^(2) + sin(θ)^(2)), sin(θ) / sqrt(cos(θ)^(2) sin(φ)^(2) + sin(θ)^(2)))
L: (cos(θ) sin(φ) / sqrt(cos(θ)^(2) sin(φ)^(2) + sin(θ)^(2)), -sin(θ) / sqrt(cos(θ)^(2) sin(φ)^(2) + sin(θ)^(2)))
horiz_1 = Translate(xAxis, Vector[11 / 2 j])
horiz_2 = Translate(xAxis, Vector[7 / 2 j])
ΓGHdev_0 = Arc(Γ_0, G, H)
ΓGHder_0 = Arc(Γ_0, G, H)
ΓHGdev_0 = Arc(Γ_0, H, G)
ΓHGder_0 = Arc(Γ_0, H, G)
GH = Segment(G, H)
MerIJdev = Arc(Mer_0, I, J)
MerJIdev = Arc(Mer_0, J, I)
MerIJder = Arc(Mer_0, I, J)
MerJIder = Arc(Mer_0, J, I)
IJ = Segment(I, J)
MerKLdev = Arc(Mer_{90}, K, L)
θ90a270: 90 ° < θ ∧ θ < 270 °
φ90a270: 90 ° < φ ∧ φ < 270 °
MerLKdev = Arc(Mer_{90}, L, K)
MerKLder = Arc(Mer_{90}, K, L)
MerLKder = Arc(Mer_{90}, L, K)
KL = Segment(K, L)