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- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/56%3A_Geodesy/56.04%3A_Vincentys_Formul-_IntroductionOne set solves the direct problem: given one point on the Earth's surface (latitude and longitude), a direction, and a distance, these equations find the latitude and longitude of the ending point. Th...One set solves the direct problem: given one point on the Earth's surface (latitude and longitude), a direction, and a distance, these equations find the latitude and longitude of the ending point. The other set solves the inverse problem: given two points on the Earth's surface (latitudes and longitudes), these equations find the distance between the two points, as well as the direction from one point the other.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/57%3A__Celestial_Mechanics/57.01%3A_Keplers_LawsEach of the planets orbits the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. Here (r,θ) are the plane polar coordinates of the planet, a is the semi-major a...Each of the planets orbits the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. Here (r,θ) are the plane polar coordinates of the planet, a is the semi-major axis of the orbit, e is the eccentricity of the orbit, and ω is the argument of perihelion. The square of the period of the orbit is proportional to the cube of the semi-major axis.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/25%3A_Celestial_Mechanics/25.06%3A_Keplers_LawsUsing analytic geometry in the limit of small Δθ , the sum of the areas of the triangles in Figure 25.9 is given by The period of revolution T of a planet about the sun is related to the semi-major ax...Using analytic geometry in the limit of small Δθ , the sum of the areas of the triangles in Figure 25.9 is given by The period of revolution T of a planet about the sun is related to the semi-major axis a of the ellipse by T2=ka3 where k is the same for all planets. Using Equation (25.2.1) for reduced mass, the square of the period of the orbit is proportional to the semi-major axis cubed,
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/56%3A_Geodesy/56.06%3A_Vincentys_Formul-_Inverse_Problem& \Delta \sigma=B \sin \sigma\left\{\cos \left(2 \sigma_{m}\right)+\frac{1}{4} B\left[\cos \sigma\left(-1+2 \cos ^{2}\left(2 \sigma_{m}\right)\right)-\frac{1}{6} B \cos \left(2 \sigma_{m}\right)\left(...& \Delta \sigma=B \sin \sigma\left\{\cos \left(2 \sigma_{m}\right)+\frac{1}{4} B\left[\cos \sigma\left(-1+2 \cos ^{2}\left(2 \sigma_{m}\right)\right)-\frac{1}{6} B \cos \left(2 \sigma_{m}\right)\left(-3+4 \sin ^{2} \sigma\right)\left(-3+4 \cos ^{2}\left(2 \sigma_{m}\right)\right)\right]\right\} \\ \notag\\ & \alpha_{1}=\arctan \left(\frac{\cos U_{2} \sin \lambda}{\cos U_{1} \sin U_{2}-\sin U_{1} \cos U_{2} \cos \lambda}\right)\\ \notag\\
