Retrieving "Canonical Equation" from the archives

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1. Axis Of The Parabola

   Linked via "canonical equation"

   Orientation along the $y$-axis
   If the parabola opens vertically, its canonical equation centered at the origin$(0, 0)$ is given by:
   $$x^2 = 4py$$
   In this configuration, the Axis of the Parabola is the $y$-axis, defined by the equation $x=0$. The focus is located at $(0, p)$ and the directrix is the line $y = -p$.

2. Ellipsoid Of Revolution

   Linked via "canonical equation"

   Canonical Equation and Definitions
   When centered at the origin/), the canonical equation for an ellipsoid of revolution in Cartesian coordinates $(x, y, z)$ depends on whether the generating ellipse is rotated about its major axis or minor axis.
   If the ellipse is rotated about its minor axis (producing an oblate spheroid [($\text{oblate spheroid}$)], wider than it is tall, like the Earth), the equation is: