Assuming everything in 2D, if there is a circle with centre at origin, with radius R , we can write its cartesian equation as : x^2 + y^2 = R^2 .
It's vector form in cartesian coordinates will be : $$R\cos(\theta)\hat{i} + R\sin(\theta)\hat{j}$$ ;where R is the radius & $$\theta$$ is the parameter.
It's polar equation will be : r=R.
But what will be it's vector form in 2D polar coordinates?
Also, I have two basic fundamental doubts : a) In 3D if the number of parameters is 2 , then will the equation always represent a surface or are there any conditions?
b) If we reduce the dimensions of the coordinates then will the shape represented by the equation changes? And will the shape of the object always remain the same in different systems of coordinates of same dimensions?
The first question: I understand that you are looking for the vectorial equation in polar. That is:
$$\vec{r}=R\hat{r}$$ , where $\hat{r}$ is the unit vector in the direction that goes from the origin to a point in the circle.
Whit respect to a) Normally, under some dependence assumptions quite general: Yes. But in some circumstances the dependence with the two parameters can collapse or get related, so that the equation could end up representing a figure whose dimensionality is less than the number of parameters (in your case 2) . In these singular cases you could get a line (dim=1) or a point (dim=0) or even a fractal figure (with non integer dimensionality between 0 and 2)
With respect to b) I don't understand exactly what you mean.