To complete the analysis of the general equation of an ellipse, note that translating a curve by a fixed vector ( h, k) simply has the effect of replacing x by x − h and y by y − k in the equation for that curve (see Equation of a Transformed Ellipse). We have seen that a rotated ellipse, centered at the origin, is always given by an equation of the form A x 2 + B x y + C y 2 = 1, where A and C are positive, and B 2 − 4 A C < 0. And this conclusion is easily verified by computing B 2 − 4 A C = −4 / ( a b) 2 using the coefficients in equation (1). Since A t 2 + B t + C therefore has no real roots, we conclude that B 2 − 4 A C < 0. Meanwhile t takes on all real values as the point ( x, y) traces the ellipse between the two x intercepts. This shows that A t 2 + B t + C can never equal 0. This is valid for any point on the ellipse, except the x intercepts where y = 0. If we factor out y 2, we obtain ( A t 2 + B t + C) = 1 / y 2, where t = x / y is the reciprocal of the slope from the origin to the point ( x, y). In the form A x 2 + B x y + C y 2 = 1, we recognize a generic quadratic equation. In this way we see that the equation for a rotated ellipse, centered at the origin is a quadratic with a nonzero x y term. Which is in the form A x 2 + B x y + C y 2 = 1, with A and C positive. Applying the methods of Equation of a Transformed Ellipse now leads to the following equation for a standard ellipse which has been rotated through an angle α.Įxpanding the binomial squares and collecting like terms gives (1) The inverse operation can be obtained by rotating through 2π − α, and hence carries ( x, y) to ( x cos α + y sin α, y cos α − x sin α). Rotation counterclockwise about the origin through an angle α carries ( x, y) to ( x cos α − ysin α, ycos α+ x sin α) (derived here). In developing a general equation for ellipses, we will use rotate then translate. This shows that every ellipse can be obtained from one in the standard position by either a rotation followed by a translation, or a translation followed by a rotation. But this is the same as first rotating x, and then translating by R v. The result will be R( x + v) = R x + R v, because R is linear. To see this, let R represent a rotation, and consider what happens to a point x = ( x, y) if we first translate by vector v, and then apply R. It is a matter of choice whether we rotate and then translate, or the opposite. Accordingly, we can find the equation for any ellipse by applying rotations and translations to the standard equation of an ellipse. Then every ellipse can be obtained by rotating and translating an ellipse in the standard position. For the most general formulation, we can include rotations through an angle of 0 (that is, no rotation at all) and translations by the zero vector (no translation at all). But such an ellipse can always be obtained by starting with one in the standard position, and applying a rotation and/or a translation. In general, an ellipse may be centered at any point, or have axes not parallel to the coordinate axes. The standard equation for an ellipse, x 2 / a 2 + y 2 / b 2 = 1, represents an ellipse centered at the origin and with axes lying along the coordinate axes. The Most Marvelous Theorem in Mathematics, Dan Kalman General Equation of an Ellipse The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
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