A hyperbola is a type of conic section. A hyperbola is a set of points where the distances between the line called the directrix and point called the focus remain in a constant ratio (this ratio is the hyperbola's eccentricity). A hyperbola is visually similar to a parabola, but with two mirrored sides. The formula for a hyperbola (assuming they are orientated facing left and right rather than up and down) is
The reciprocal function which shows one variable is inversely proportional to another also describes a hyperbola but oriented differently. It can be shown that this equation is in fact a hyperbola rotated 45 degrees about the origin.
Start with the Cartesian form
Rewrite in a parametric/vector form.
Pre-multiply by rotation matrix representing a 45 degree anti-clockwise rotation about the origin from the x-axis (we are making a new equation).
Take out the factor of half square root 2
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