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- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.05%3A_Conic_SectionsA plane section of a cone is either an ellipse, a parabola or a hyperbola, depending on whether the angle that the plane makes with the base of the cone is less than, equal to or greater than the angl...A plane section of a cone is either an ellipse, a parabola or a hyperbola, depending on whether the angle that the plane makes with the base of the cone is less than, equal to or greater than the angle that the generator of the cone makes with its base. However, given the definitions of the ellipse, parabola and hyperbola that we have given, proof is required that they are in fact conic sections.
- https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/06%3A_General_Planar_Motion/6.03%3A_Motion_Under_the_Action_of_a_Central_ForceA central force is a force that points along the (positive or negative) radial direction ˆr, and whose magnitude depends only on the distance r to the origin - so \(\boldsymbol{F...A central force is a force that points along the (positive or negative) radial direction ˆr, and whose magnitude depends only on the distance r to the origin - so F(r)=F(r)ˆr.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_(Fowler)/14%3A_Mathematics_for_Orbits/14.01%3A_Preliminaries-_Conic_SectionsTaking the cone to be x2+y2=z2, and substituting the z in that equation from the planar equation →r⋅→p=p, where →p is the vector perpendicular to the pl...Taking the cone to be x2+y2=z2, and substituting the z in that equation from the planar equation →r⋅→p=p, where →p is the vector perpendicular to the plane from the origin to the plane, gives a quadratic equation in x,y. Draw two lines: one from P to the point F where the small sphere touches, the other up the cone, aiming for the vertex, but stopping at the point of intersection with the circle C.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_SectionsA particle moving under the influence of an inverse square force moves in an orbit that is a conic section; that is to say an ellipse, a parabola or a hyperbola. We shall prove this from dynamical pri...A particle moving under the influence of an inverse square force moves in an orbit that is a conic section; that is to say an ellipse, a parabola or a hyperbola. We shall prove this from dynamical principles in a later chapter. In this chapter we review the geometry of the conic sections. We start off, however, with a brief review (eight equation-packed pages) of the geometry of the straight line.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.06%3A_The_General_Conic_SectionThus if x is first replaced with x+ˉg/ˉc and y with y+ˉf/ˉc, and then the new x is replaced with xcosθ−ysinθ and the new y with \(x \sin θ...Thus if x is first replaced with x+ˉg/ˉc and y with y+ˉf/ˉc, and then the new x is replaced with xcosθ−ysinθ and the new y with xsinθ+ycosθ, the Equation will take the familiar form of a conic section with its major or transverse axis coincident with the x axis and its centre at the origin.
