Cone

One point on the sphere

({{Cos[θ], -Sin[θ], 0}, {Sin[θ], Cos[θ], 0}, {0, 0, 1}}) . ({{Cos[φ], 0, -Sin[φ]}, {0, 1, 0}, {Sin[φ], 0, Cos[φ]}}) . ({{1}, {0}, {0}}) [[1]]

{Cos[θ] Cos[φ], Cos[φ] Sin[θ], Sin[φ]}

({{Cos[θ], -Sin[θ], 0}, {Sin[θ], Cos[θ], 0}, {0, 0, 1}}) . ({{Cos[φ], 0, -Sin[φ]}, {0, 1, 0}, {Sin[φ], 0, Cos[φ]}}) . ({{x}, {y}, {z}}) [[1]]

{x Cos[θ] Cos[φ] - y Sin[θ] - z Cos[θ] Sin[φ], y Cos[θ] + x Cos[φ] Sin[θ] - z Sin[θ] Sin[φ], z Cos[φ] + x Sin[φ]}

({{Cos[θ], -Sin[θ], 0}, {Sin[θ], Cos[θ], 0}, {0, 0, 1}}) . ({{Cos[φ], 0, -Sin[φ]}, {0, 1, 0}, {Sin[φ], 0, Cos[φ]}}) . ({{d}, {0}, {0}}) [[1]]

{d Cos[θ] Cos[φ], d Cos[φ] Sin[θ], d Sin[φ]}

Formula and solve

Ph : φ

Th : θ

Jh : φ

1

{d Cos[θ] Cos[φ], d Cos[φ] Sin[θ], d Sin[φ]} = {x, y, z}

d = (x - x0)/ Cos[θ] Cos[φ]

d = (y - y0)/Cos[φ] Sin[θ]

d = (z - z0)/Sin[φ]

S = Simplify[Solve[{(x - x0)/ Cos[θ] Cos[φ] == (y - y0)/Cos[φ] Sin[θ] == (z - z0)/Sin[φ], z == h}, {x, y, z}][[1]]] ;

{x, y, z}/.S

{x0 + (h - z0) Cos[θ] Csc[φ] Sec[φ], y0 + (h - z0) Cot[φ] Csc[θ], h}

2

d = (x - x0)/(-Cos[t] Sin[θ] Tan[φ] + Cos[θ] (Cos[φ] - Sin[t] Sin[φ] Tan[φ]))

d = (y - y0)/(Cos[φ] Sin[θ] + (Cos[t] Cos[θ] - Sin[t] Sin[θ] Sin[φ]) Tan[φ])

d = (z - z0)/(Sin[φ] + Cos[φ] Sin[t] Tan[φ])

{x//InputForm, y//InputForm, z//InputForm}/.S[[1]]

3

{x0 + d (Cos[θ] Cos[φ] - Sin[θ] Tan[φ]), y0 + d (Cos[φ] Sin[θ] + Cos[θ] Tan[φ]), z0 + d Sin[φ]}

Simplify[Transpose[({{Cos[Th], -Sin[Th], 0}, {Sin[Th], Cos[Th], 0}, {0, 0, 1}}) . ({{Cos[Jh], 0, -Sin[Jh]}, {0, 1, 0}, {Sin[Jh], 0, Cos[Jh]}}) . ({{D}, {0}, {D Tan[Ph]}})][[1]]] + {x0, y0, z0}

{x0 + D Cos[Jh + Ph] Cos[Th] Sec[Ph], y0 + D Cos[Jh + Ph] Sec[Ph] Sin[Th], z0 + D Sec[Ph] Sin[Jh + Ph]}