Depends on what you mean by "center point", I'd say. If your curve does not intersect itself (i.e. if it forms a simple polygon), then one usually refers to its centroid (aka center of mass) as its center point. According to this Wikipedia page, you can compute the centroid of a simple polygon as follows:

1. /*
2. Assume x[] and y[] are the vertices of your polygon, and that
3. the polygon is not closed, i.e. x[0] != x[x.length - 1] or
   
4. y[0] != y[y.length - 1]. Also assume that x and y have the same
5. number of elements.
6. */
7. 
   
8. float centerX = 0;
9. float centerY = 0;
10. float area = 0;
    
11. for (int i = 0; i < x.length - 1; i++) {
12.   float f = x[i] * y[i + 1] - x[i + 1] * y[i];
13.   area += f;
    
14.   centerX += (x[i] + x[i + 1]) * f;
    
15.   centerY += (y[i] + y[i + 1]) * f;
16. }
17. area /= 2;
    
18. centerX /= 6 * area;
19. centerY /= 6 * area;

You may have to some adjustments depending on whether the vertices are in clockwise or counterclockwise order, see the Wikipedia page cited above.

Is this what you were looking for?