Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form

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Theorem

Let $K$ be an ellipse aligned in a cartesian plane in reduced form.


Let:

the major axis of $K$ have length $2 a$
the minor axis of $K$ have length $2 b$.


The equation of $K$ in parametric form is:

\(\ds x\) \(=\) \(\ds a \cos \theta\)
\(\ds y\) \(=\) \(\ds b \sin \theta\)

where $\theta$ is the eccentric angle of the point $P = \tuple {x, y}$ with respect to $K$.


Proof

Let the point $\tuple {x, y}$ satisfy the equations:

\(\ds x\) \(=\) \(\ds a \cos \theta\)
\(\ds y\) \(=\) \(\ds b \sin \theta\)


Then:

\(\ds \frac {x^2} {a^2} + \frac {y^2} {b^2}\) \(=\) \(\ds \frac {\paren {a \cos \theta}^2} {a^2} + \frac {\paren {b \sin \theta}^2} {b^2}\)
\(\ds \) \(=\) \(\ds \frac {a^2} {a^2} \cos^2 \theta + \frac {b^2} {b^2} \sin^2 \theta\)
\(\ds \) \(=\) \(\ds \cos^2 \theta + \sin^2 \theta\)
\(\ds \) \(=\) \(\ds 1\) Sum of Squares of Sine and Cosine

$\blacksquare$


Sources

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