Translation of Axes

Translation of Axes

Translation of axes means shifting the origin to a new point while keeping the direction of both axes unchanged — the new $x'$-axis is parallel to the old $x$-axis, and the new $y'$-axis is parallel to the old $y$-axis. Nothing about the curve or the point in the plane changes; only the frame we use to name its coordinates changes.

Suppose the origin $O$ is shifted to a new point $O'(h,k)$, measured in the old coordinate system. Let a point $P$ have old coordinates $(x,y)$ referred to $O$, and new coordinates $(x',y')$ referred to $O'$. Since the new axes are parallel to the old ones, the position of $P$ relative to $O'$ is simply its position relative to $O$ minus the shift $(h,k)$. This gives the fundamental translation formulas:

Equivalently, solving for the new coordinates in terms of the old ones,

These two forms are used in different directions: if we are given an equation in the old variables $(x,y)$ and want the equation in the new variables $(x',y')$, we substitute $x=x'+h$ and $y=y'+k$ into the given equation and simplify. If instead we are given the new coordinates of a point and want its old coordinates, we use $x=x'+h,\,y=y'+k$ directly; if we are given the old coordinates and want the new ones, we use $x'=x-h,\,y'=y-k$.

Translation is especially useful for simplifying the equation of a curve such as a circle, parabola, ellipse or hyperbola whose axis is parallel to a coordinate axis but whose centre or vertex is not at the origin. By translating the origin to the centre or vertex, the linear terms in the equation disappear and the equation reduces to its standard form. Translation, however, cannot remove a genuine $xy$ (cross) term from a second-degree equation — that requires a rotation of axes, taken up next.