# GraphicMaths

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 →

## Ellipses

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 →