Parametric & Polar Forms
| Shape | Cartesian Equation | Parametric / Polar Equation | Key Characteristics |
|---|---|---|---|
| Line / Segment | $$ y = mx+b $$ | Parametric: $$ x = a+bt, y = c+dt $$ Polar: $$ \theta = c $$ | The parametric form defines a line segment if \(t\) is restricted. |
| Circle | $$ x^2 + y^2 = R^2 $$ | Parametric: $$ x=R\cos t, y=R\sin t $$ Polar: $$ r = R $$ Polar: $$ r = a\cos\theta $$ | Standard parametric form traces counter-clockwise. In polar, \(r=a\cos\theta\) is on the x-axis. |
| Ellipse | $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ | Parametric: $$ x=a\cos t, y=b\sin t $$ | A stretched circle where horizontal radius is \(a\) and vertical radius is \(b\). |
| Parabola | $$ y = Ax^2+Bx+C $$ | Parametric: $$ x = t, y = At^2+Bt+C $$ Parametric (Trig): $$ x=\tan t, y=\sec^2 t $$ | Often formed by one linear and one quadratic parametric equation. |
| Cardioid | (Complex) | Polar: $$ r = a \pm a\sin\theta $$ Polar: $$ r = a \pm a\cos\theta $$ | Heart-shaped curve. Key is that the coefficients are equal (e.g., \(r=2+2\cos\theta\)). |
| Rose Curve | (Complex) | Polar: $$ r=a\cos(n\theta) $$ Polar: $$ r=a\sin(n\theta) $$ | If \(n\) is odd, there are \(n\) petals. If \(n\) is even, there are \(2n\) petals. |