Parametric equations of various shapes

[Raghav Gupta]

How we can know the parametric equation for any curve?
Is there some trick?
Like for parabola $y^2 =4a x$
It has general coordinates$(at^2 , 2at)$
It will satisfy the equation but how in first place we know it?
Also we can have $(a/t^2, -2a/t)$, how?

 

[Raghav Gupta]

Can any ideas, discussion or some help be given?

 

[HallsofIvy]

There exist an infinite number of different parametric equations for a curve. The simplest way to get parametric equations for $y= ax^2$ is to use x itself as parameter: If $x= t$ then $y= at^2$. That can, in fact, be done for any function- if $y= f(x)$ then $x= t$, $y= f(t)$ are parametric equations.

For non-function curves, we have to be a little more creative. For example, the relation, $x^2+ y^2= a^2$ describe a circle with center at (0, 0) and radius a. We know that $cos^2(t)+ sin^2(t)= 1$ so $a^2 cos^2(t)+ a^2 sin^2(t)= a^2$ so we can take $x= a cos(t)$, $y= a sin(t)$ as parametric equations. Of course, $x= a sin(t)$, $y= a cos(t)$ would work as well.

For a more general circle, $(x- x_0)^2+ (y-y_0)^2= a^2$, still with radius a but now with center at $(x_0, y_0)$, with the same analysis as before, we have $x- x_0= a cos(t)$, $y- y_0= a sin(t)$ so $x= a cos(t)+ x_0$, $y= a sin(t)+ y_0$ are parametric equations.

We can think of $\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$, the relation describing an ellipse with axes, along the x and y axes of lengths a and b, respectively, as $\left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2= 1$ and see that we can take $\frac{x}{a}= cos(t)$, $\frac{y}{b}= sin(t)$ or $x= a cos(t)$, $y= b sin(t)$ as parametric equations for that ellipse.

 

[Raghav Gupta]

Got it, thanks.