![]() ![]() But these ellipses can be transformed into any other ellipses using rotations and translations. This describes only ellipses centered at the origin and having axes along the coordinate axes. We also know that the standard equation for an ellipse is x 2 / a 2 + y 2 / b 2 = 1, which is quadratic. Every Ellipse is Given by a Quadratic EquationĪt 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. Consequently, the final simplified version of the equation will once again be a quadratic in x and y. That produces terms that are multiples of x 2, y 2, x y, x, y, as well as constants. We can only obtain linear or quadratic combinations of x and y, and constants. It is enough to observe what sorts of terms occur. However, it is not necessary to actually carry this step out. Therefore, we substitute p x + q y for x and r x + s y for y in the original equation to deriveĪ( p x + q y) 2 + B( p x + q y)( r x + s y) + C( r x + s y) 2 + D( p x + q y) + E( r x + s y) + F = 0.Īs in the example, the next step is to multiply out all of the binomial factors and collect like terms. If ( x, y) is a point of the new curve, transformed from ( p x + q y, r x + s y), then this latter point satisfies the original equation. Now we are assuming that the original curve has an equation of the formĪ x 2 + B x y + C y 2 + D x + E y + F = 0. The point ( x, y) on the blue ellipse comes from the point ( p x + q y, r x + s y) on the red circle.įigure 1. The situation is illustrated in Figure 1, which shows a red circle transformed into a blue ellipse. All that is of interest is how they affect the equation of the transformed curve. Here it is not important to know the values of these numbers, nor how they depend on the transformation. In particular, when this inverse transformation is applied to the point ( x, y), the result is something of the form ( p x + q y, r x + s y), where p, q, r, and s are fixed real numbers depending on the original linear transformation. A nonsingular linear transformation always has an inverse transformation that is also linear. ![]() That is really the same as applying to the point ( x, y) the inverse of the transformation. 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. 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. Expand the squared factors, and collect like terms, to derive This can be expressed in the standard form for a quadratic polynomial in x and y. The resulting equation isĪ point ( x, y) is on the transformed curve if and only if this equation holds. 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. 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). Following the method above, we consider a point not of the original curve, but rather of the new curve. 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). Let the original curve be the ellipse x 2 / 4 + y 2 / 9 = 1. 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. The coordinates of that original point must satisfy the original equation. We begin with a point of the new curve, ( x, y), and ask what point of the original curve it came from. There is a general method for answering this question. If a transformation is applied to every point of the curve, a new curve is produced. Suppose a curve is defined by an equation in x and y. The Most Marvelous Theorem in Mathematics, Dan Kalman Equation of a Transformed Ellipse ![]() ![]() The Journal of Online Mathematics and Its Applications, Volume 8 (2008) ![]()
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