Canonical equation

By choosing a convenient reference frame, with the foci symmetric with respect to the origin on the $x$-axis, the condition of the definition translates into a particularly simple equation.

Property — Canonical equation

If $F_1(-c;0)$ and $F_2(c;0)$ with $c>0$ and $2a>2c$, the ellipse has canonical equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad \text{con}\quad b^2 = a^2 - c^2.$ The vertices on the $x$-axis are $V_1(-a;0)$ and $V_2(a;0)$; the vertices on the $y$-axis are $W_1(0;-b)$ and $W_2(0;b)$. The two numbers $a$ and $b$ are called semi-axes: $a$ is the semi-major axis (if $a > b$), $b$ is the semi-minor axis.

The relation $b^2 = a^2 - c^2$ links together the three fundamental parameters of the ellipse: the semi-major axis $a$, the semi-minor axis $b$ and the focal semi-distance $c$. Note that it is entirely analogous to Pythagoras’ theorem, with $a$ as the hypotenuse: the points $(c;0)$, $(0;b)$ and the origin do in fact form a right triangle with hypotenuse $a$.

Links

Topics: Ellipse
Concepts: Ellipse · Canonical equation · Foci · Semi-axes · Vertices
Methods: Canonical ellipse
Skills: Analytic geometry · Using formulae