Equation of a Transformed Ellipse

Suppose a curve is defined by an equation in x and y.  If a transformation is applied to every point of the curve, a new curve is produced.  What is the equation for the new curve?

There is a general method for answering this question.  We begin with a point of the new curve, (x, y), and ask what point of the original curve it came from.  The coordinates of that original point must satisfy the original equation.  So if the coordinates of the starting point are expressed in terms of x and y, and substituted in the original equation, we will obtain the new equation.

Here is an example to illustrate.  Let the original curve be the ellipse x² / 4 + y² / 9 = 1.  And suppose that the given transformation is a translation by the fixed vector (−2,7), so that every point (x, y) is carried to the shifted point (x − 2, y + 7). Following the method above, we consider a point not of the original curve, but rather of the new curve.  Call this point (x, y).  It is the result of transforming the point (x + 2, y − 7), because that is the point that goes to (x, y) when it is translated by (−2, 7).  Now the original point satisfies the equation of the original ellipse, so substitute x + 2 for x and y − 7 for y in the original equation.  The resulting equation is

A point (x, y) is on the transformed curve if and only if this equation holds.  This can be expressed in the standard form for a quadratic polynomial in x and y.  Expand the squared factors, and collect like terms, to derive

Generalizing from this example, we can see that for any nonsingular linear transformation, if the starting equation is quadratic in x and y, so is the resulting equation.  Note that the first step of the general algorithm is to begin with a point (x, y) on the new curve, and ask what point this came from on the original curve.  That is really the same as applying to the point (x, y) the inverse of the transformation.  A nonsingular linear transformation always has an inverse transformation that is also linear.  In particular, when this inverse transformation is applied to the point (x, y), the result is something of the form (px + qy, rx + sy), where p, q, r, and s are fixed real numbers depending on the original linear transformation.  Here it is not important to know the values of these numbers, nor how they depend on the transformation.  All that is of interest is how they affect the equation of the transformed curve.  The situation is illustrated in Figure 1, which shows a red circle transformed into a blue ellipse.  The point (x, y) on the blue ellipse comes from the point (px + qy, rx + sy) on the red circle.

Figure 1.  Circle transformed into an ellipse.

Now we are assuming that the original curve has an equation of the form

Ax² + Bxy + Cy² + Dx + Ey + F = 0.

If (x, y) is a point of the new curve, transformed from (px + qy, rx + sy), then this latter point satisfies the original equation.  Therefore, we substitute px + qy for x and rx + sy for y in the original equation to derive

A(px + qy)² + B(px + qy)(rx + sy) + C(rx + sy)² + D(px + qy) + E(rx + sy) + F = 0.

As in the example, the next step is to multiply out all of the binomial factors and collect like terms.  However, it is not necessary to actually carry this step out.  It is enough to observe what sorts of terms occur.  We can only obtain linear or quadratic combinations of x and y, and constants.  That produces terms that are multiples of x², y², xy, x, y, as well as constants. Consequently, the final simplified version of the equation will once again be a quadratic in x and y.

Every Ellipse is Given by a Quadratic Equation

At this point, we know that any curve described by a quadratic equation is transformed by a linear transformation to a new curve described by a different quadratic equation.  We also know that the standard equation for an ellipse is x² / a² + y² / b² = 1, which is quadratic.  This describes only ellipses centered at the origin and having axes along the coordinate axes.  But these ellipses can be transformed into any other ellipses using rotations and translations.  Rotations are linear transformations.  As just shown, since the standard equation of an ellipse is quadratic, so is the equation of a rotated ellipse centered at the origin.  We have also seen that translating by a curve by a fixed vector (h, k) has the effect of replacing x by x − h and y by y − k in the equation of the curve.  This operation also changes a quadratic equation into another quadratic equation.  Thus, every ellipse is described by a quadratic equation.

In conclusion, this page has shown both that linear (and affine) transformations take curves with quadratic equations to curves with quadratic equations, and also that all ellipses are given by quadratic equations.  Combined, these show that linear and affine transformations of ellipses always have quadratic equations.