When the origin is shifted to (h,k), the relation between old (x,y) and new (X,Y) coordinates is: x = X + h, y = Y + k. Equivalently, X = x - h, Y = y - k.

Purpose: To simplify the equation of a curve by moving the origin to a special point (center of a conic, vertex of a parabola).

For the general conic ax^2+2hxy+by^2+2gx+2fy+c=0, translation to the center (found by solving dF/dx=0, dF/dy=0 simultaneously) removes the linear terms 2gx and 2fy, simplifying the equation.

The center of the conic (no xy-term) ax^2+by^2+2gx+2fy+c=0 is at (-g/a, -f/b).