Visualising maths

Cartesian equation of ellipse

The Cartesian equation of an ellipse is: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ Understanding the equation Starting from the parametric equations: $$ x = a \cos{t}\\ y = b \sin{t} $$ These can be rearranged to give: $$ \cos{t} = \frac{x}{a}\\ \sin{t} = \frac{y}{b} $$ We also know that: $$ \cos^2{t} + \sin^2{t} = 1 $$ This follows from Pythagoras. Substituting for $\cos{t}$ and $\sin{t}$ gives Read more →


An ellipse is a stretched circle. The longest diameter, $PQ$, is called the major axis. The shortest diameter, $RS$, is called the minor axis. The major and minor axes are always perpendicular. An ellipse can be defined by the parameters $a$ and $b$, where the major axis has length $2a$ and the minor axis has length $2b$ (or vice versa). An ellipse centred on the origin with its major and minor axes aligned with the x and y axes looks like this: Read more →

Parametric equation of ellipse

The Parametric equation of an ellipse is: $$ x = a \cos{t}\\ y = b \sin{t} $$ Understanding the equations We know that the equations for a point on the unit circle is: $$ x = \cos{t}\\ y = \sin{t} $$ Multiplying the $x$ formula by $a$ stretches the shape in the x direction, so that is the required width (crossing the x axis at $x=\pm a$: Read more →