In non parametric form, $y^2=4ax$

In parametric form, $x=tan^2\phi$

$y=\pm 2\sqrt {atan\phi}$

where $0\le \phi \le \frac{\pi}{2}$

But, the curve does not give maximum inscribed area, which is required to define the best parametric representation.

Hence, it is not a adopted.

In other parametric representation; giving maximum inscribed area under the curve and hence defining the best parametric representation.

$x=a\theta ^2$

$y=2a\theta$

where, $0\le \theta \le \infty$, sweeps out the entire upper limit of the parabola.

Choose a minimum or maximum value of $\theta$ for parabola.

If range of x is limited, then $\theta _{min}=\frac{x_{min}}{a}$ and $\theta _{max}=\frac{x_{max}}{a}$

If range of y is limited, then $\theta _{min}=\frac{y_{min}}{2a}$ and $\theta _{max}=\frac{y_{max}}{2a}$

Incremental generation of Parabola:-

Let $\delta \theta=$ fixed increment in $\theta$

$\theta _{i+1}=\theta _i+\delta \theta$

$\therefore x_{i+1}=a(\theta _i+\delta \theta)^2=a\theta _i^2+2a\theta _i\delta \theta+a(\delta \theta)^2$

$y_{i+1}=2a(\theta _i+\delta \theta)=2a\theta _i+2a\delta \theta$

$\therefore x_{i+1}=x_i+y_i\delta \theta +a(\delta \theta)^2$

$y_{i+1}=y_i+2a\delta \theta$

Consists of 3 additions; and multiplications in inner loop