Date: 03/11/99 at 10:24:25 From: Patrick Roberts Subject: Equation of an Ellipse How do I get the equation of an ellipse from four points and an orientation?
Date: 03/11/99 at 15:19:48 From: Doctor Rob Subject: Re: Equation of an Ellipse I am a little confused about the word 'orientation' in this context. I will assume you mean the inclination of the major axis of the ellipse. Denote the angle the major axis of the ellipse with the positive x-axis by alpha. Then you can rotate your system of coordinates about the origin by an angle alpha using the transformation x = X*cos(alpha) - Y*sin(alpha), y = X*sin(alpha) + Y*cos(alpha). Then, because the new coordinate axes are parallel to the major and minor axes of the ellipse, the equation of the ellipse has the form A*X^2 + C*Y^2 + D*X + E*Y + F = 0 (A*C > 0) Substitute in the new coordinates of your four points, and you will have a system of four equations in the five unknowns A, C, D, E, and F. Solve them for C, D, E, and F in terms of A. Set A equal to any convenient nonzero value. That gives you the equation of the ellipse with numerical values for the coefficients. Now undo the coordinate rotation by substituting X = x*cos(alpha) + y*sin(alpha) Y = -x*sin(alpha) + y*cos(alpha) Expand and simplify, and you will have the equation of the ellipse you desired. Example: (11,0), (11,-12), (19/5,42/5), (-53/5, -54/5), inclination 36.8699 degrees. Then, cos(alpha) = 4/5 and sin(alpha) = 3/5. In the rotated coordinate system, the four points become (44/5,33/5), (16,-3), (-2,9), and (-2,-15), respectively. The four linear equations are (1936/25)*A + (1089/25)*C + (44/5)*D + (33/5)*E + F = 0 256*A + 9*C + 16*D - 3*E + F = 0 4*A + 81*C - 2*D + 9*E + F = 0 4*A + 225*C - 2*D - 15*E + F = 0 Solving, I got C = (9/4)*A D = 4*A E = (27/2)*A F = (-1199/4)*A Setting A = 4 to make all values integers, A = 4 C = 9 D = 16 E = 54 F = -1199 4*X^2 + 9*Y^2 + 16*X + 54*Y - 1199 = 0 4*(X+2)^2 + 9*(Y+3)^2 = 1296 (X+2)^2/18^2 + (Y+3)^2/12^2 = 1 Unrotating the coordinate system, the equation of this ellipse in the old coordinate system becomes 29*x^2 + 24*x*y + 36*y^2 + 226*x + 168*y - 5995 = 0 You can check that the four points given do satisfy this equation. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
Date: 03/11/99 at 16:01:18 From: Patrick Roberts Subject: Equation of an ellipse Earlier, I asked how to get the equation of an ellipse from 4 points and an orientation, and you replied, but I needed it for an ellipse *not* necessarily centred at the origin. Please show me how to do that.
Date: 03/16/99 at 16:29:35 From: Doctor Rob Subject: Re: Equation of an ellipse This answer does apply to an ellipse *not* centered at the origin. If the ellipse were centered at the origin, you would have D = E = 0, that is, no first-degree terms in x or y appearing. Since I allow nonzero values of D and E in the above answer, and those first-degree terms can appear, the ellipse is not necessarily centered at the origin. The example above was centered at (-2,-3), not the origin. - Doctor Rob, The Math Forum http://mathforum.org/dr.math
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