How to draw "non-function" graph: f(x) = g(y)

Question

I want to know is there any fast way to draw a graph of a "non-function" curve. For example

x^2+3x = y^3-4y+1

I know for normal function, like y=x^2, we can iterate x and calculate y, then draw the points. But for non-function curve, it will take a lot of times to iterate x, then solve function of y (using Newton method or alike). So please suggest me the correct way to draw them.

Thanks & Regards.

Answer 1

I am afraid there is no "generic" way except for the method you describe yourself: iterate over one variable and solve for the other.

Complications

Note that you have to be careful to find all solutions, not just a solution. This is a major stumbling block in creating a working general algorithm.

Another stumbling block is the singularity points: when f'(x)=0, you will want to solve for y and, vice versa, when g'(y)=0, you will want to solve for x. What if both are 0 at the same time? You will need to do some paper-and-pencil analysis.

Special Cases

There are some problem-specific simplifications though.

In your specific case the equation for x is quadratic, so a well known simple closed formula exists. This means that iterating over y and solving for x is easier. (The equation for y is cubic, so a less well known and much more complicated formula exist too).

Another way is to find a parametric representation of your curve (e.g., x^2+y^2=1 is equivalent to x=cos(t); y=sin(t); 0<=t<2*pi).