My current equation is: (x(t), y(t)) = (1.53cos(t), 0.99sin(t)+0.42cos(2t))
From my understanding, this is an explicitly-defined parametric equation. (See this screenshot for a visualized version of my equation: (x(t), y(t)) = (1.53cos(t), 0.99sin(t)+0.42cos(2t)) Plotted)
However, I would like to modify my equation to mimick a specific shape: (See this screenshot: Heart or bean-shaped equation)
Can anyone please provide advice or guidance on how to modify my current equation to become more similar to the shape that I am aiming for? It does not have to be identical. It just needs to be similar to close enough. Thank you for the understanding.
First, notice that your target is not centered in the plane at all similarly to how your current parametric equation is centered, so you should think about what kind of extra coefficient(s) you would need to add to your function to compensate for that.
The next step is to sample your target to a list of numeric points, each point being a pair of (x,y) values. (For example, if the darker line in the center of the bean is at $x=0$, I estimate that (0, -1.7) is on the bean curve --- it is at the center of the indentation.)
The third step is called "fitting", see Wikipedia. For this you need to think about how your problem translates into a system of equations based on your target points.
Then solve that system as well as you can. Ordinarily in such a problem, you will end up with more equations than you have variables --- this is called an "overdetermined system". In such a case, you will usually want to minimize some kind of sum of squares which represent your errors, and a powerful tool to minimize is looking at the partial derivatives. Or alternatively, find software which does this for you.