Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses

The following gives the derivation for Standard and Implicit Coordinate Equations for any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(X\)-Axis.

Let's consider a Real Ellipse having it's Semi-Major Axis Length as \(a\), Semi-Minor Axis Length as \(b\), Coordinates of Center at \((x_c,y_c)\), and Coordinates of 2 Foci at \((x_{f1},y_f)\) and \((x_{f2},y_f)\). Now, as per definition of Real Ellipse, the Sum of the Distance from 2 Foci to Any Point \((x,y)\) on the Ellipse is equal to the Length of it's Major Axis \(2a\). Hence

\(\sqrt{{(x-x_{f1})}^2 + {(y-y_f)}^2} + \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2} =L=2a\)

\(\Rightarrow \sqrt{{(x-x_{f1})}^2 + {(y-y_f)}^2} = 2a - \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)   ...(1)

Squaring Both Sides of equation (1) above we get

\({(x-x_{f1})}^2 + {(y-y_f)}^2 = 4a^2 + {(x-x_{f2})}^2 + {(y-y_f)}^2 - 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow 4a^2 + {(x-x_{f2})}^2 - {(x-x_{f1})}^2 = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow 4a^2 + x^2 + {x_{f2}}^2 - 2x_{f2}x - x^2 - {x_{f1}}^2 + 2x_{f1}x = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow 4a^2 + {x_{f2}}^2 - {x_{f1}}^2 - 2x_{f2}x + 2x_{f1}x = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow 4a^2 + ({x_{f2} - x_{f1}})({x_{f2} + x_{f1}}) - 2x(x_{f2}-x_{f1}) = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)   ...(2)

Now, \({x_{f2} - x_{f1}}=F=2c\) (i.e Distance Between the 2 Foci) and \({x_{f2} + x_{f1}}=2x_c\) (i.e 2 times the \(X\)-Coordinate of the Center of Ellipse). Substituting these in equation (2) above we get

\(\Rightarrow 4a^2 + (2c)(2x_c) - 2x(2c) = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow 4a^2 + 4cx_c - 4cx = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)

\(\Rightarrow a^2 + cx_c - cx = a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2}\)   ...(3)

Squaring Both Sides of equation (3) above we get

\({(a^2 + cx_c - cx)}^2 = {(a \sqrt{{(x-x_{f2})}^2 + {(y-y_f)}^2})}^2\)

\(\Rightarrow a^4 + c^2{x_c}^2 + c^2x^2 + 2a^2cx_c - 2a^2cx - 2c^2x_cx = a^2{(x-x_{f2})}^2 + a^2{(y-y_f)}^2\)

\(\Rightarrow c^2x^2 - 2a^2cx - 2c^2x_cx + a^4 + c^2{x_c}^2 + 2a^2cx_c = a^2x^2 + a^2{x_{f2}}^2 - 2a^2x_{f2}x + a^2{(y-y_f)}^2\)

\(\Rightarrow c^2x^2 - a^2x^2 + 2a^2x_{f2}x - 2a^2cx - 2c^2x_cx + a^4 + c^2{x_c}^2 + 2a^2cx_c - a^2{x_{f2}}^2 = a^2{(y-y_f)}^2\)

\(\Rightarrow (c^2 - a^2)x^2 + (2a^2x_{f2} - 2a^2c - 2c^2x_c)x + a^4 + c^2{x_c}^2 + 2a^2cx_c - a^2{x_{f2}}^2 = a^2{(y-y_f)}^2\)   ...(4)

Now, for any Ellipse we know that \(a^2 - b^2 = c^2\hspace{3mm}\Rightarrow -b^2=c^2-a^2\). Also, since the Major Axis of Ellipse is Parallel to the \(X\)-Axis therefore \(y_f=y_c\). Substituting these in equation (4) above we get

\(\Rightarrow -b^2x^2 + (2a^2x_{f2} - a^2(x_{f2}-x_{f1}) - 2(a^2-b^2)x_c)x + a^4 + (a^2-b^2){x_c}^2 + 2a^2cx_c - a^2{x_{f2}}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + (2a^2x_{f2} - a^2x_{f2} + a^2x_{f1} - 2a^2x_c + 2b^2x_c)x + a^4 + a^2{x_c}^2 + 2a^2cx_c - a^2{x_{f2}}^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + (a^2(x_{f2}+ x_{f1}) - 2a^2x_c + 2b^2x_c)x + a^4 + a^2{(\frac{x_{f2} + x_{f1}}{2})}^2 + \frac{a^2}{2}(x_{f2}- x_{f1})(x_{f2}+ x_{f1}) - a^2{x_{f2}}^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + (2a^2x_c - 2a^2x_c + 2b^2x_c)x + a^4 + a^2(\frac{{x_{f1}}^2 + {x_{f2}}^2 + 2x_{f1}x_{f2}}{4}) + \frac{a^2}{2}({x_{f2}}^2- {x_{f1}}^2) - a^2{x_{f2}}^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 + a^2(\frac{{x_{f1}}^2 + {x_{f2}}^2 + 2x_{f1}x_{f2}}{4}) + a^2 (\frac{{x_{f2}}^2- {x_{f1}}^2 - 2{x_{f2}}^2}{2}) -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 + a^2(\frac{{x_{f1}}^2 + {x_{f2}}^2 + 2x_{f1}x_{f2}}{4} + \frac{-{x_{f2}}^2- {x_{f1}}^2}{2}) -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 + a^2(\frac{{x_{f1}}^2 + {x_{f2}}^2 + 2x_{f1}x_{f2} -2{x_{f2}}^2 -2{x_{f1}}^2}{4}) -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 - a^2(\frac{{x_{f2}}^2 + {x_{f1}}^2 - 2x_{f1}x_{f2}}{4}) -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 - a^2{(\frac{x_{f2}- x_{f1}}{2})}^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^4 - a^2c^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^2(a^2-c^2) -b^2{x_c}^2 = a^2{(y-y_c)}^2\)

\(\Rightarrow -b^2x^2 + 2b^2x_cx + a^2b^2 -b^2{x_c}^2 = a^2{(y-y_c)}^2\)   ...(5)

Now, Dividing By \(-a^2b^2\) on Both Sides of equation (5) above we get

\(\frac{x^2}{a^2} - \frac{2x_cx}{a^2} + \frac{{x_c}^2}{a^2} - 1 = -\frac{{(y-y_c)}^2}{b^2}\)

\(\Rightarrow \frac{{(x-x_c)}^2}{a^2} + \frac{{(y-y_c)}^2}{b^2} =1 \)   ...(6)

The equation (6) above gives the Standard Coordinate Equation for any Real Ellipse having it's Major Axis Parallel to the \(X\)-Axis.

Unlike a Real Elllipse, an Imaginary Ellipse cannot be plotted or traced in a Real Space (2D Plane or 3D Space). And hence, the Standard Coordinate Equation for any Imaginary Ellipse Cannot be Derived in similar manner. However, the Standard Coordinate Equation of any Imaginary Ellipse having it's Major Axis Parallel to the \(X\)-Axis is similar to that of a Real Ellipse as given in equation (6) above and is given as

\(\frac{{(x-x_c)}^2}{a^2} + \frac{{(y-y_c)}^2}{b^2} =-1 \)   ...(7)

Therefore in general, the Standard Coordinate Equation for any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(X\)-Axis can be given as

\(\frac{{(x-x_c)}^2}{a^2} + \frac{{(y-y_c)}^2}{b^2} = \pm\hspace{1mm}1 \)   ...(8)

where the Right Hand Side of the equation (8) above is \(1\) for Real Ellipse and \(-1\) for Imaginary Ellipse

For any Ellipse having its Center at the Origin equation (8) becomes

\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = \pm\hspace{1mm}1 \)   ...(9)

Multiplying By \(a^2b^2\) on Both Sides of equation (8) above we get

\(b^2{(x-x_c)}^2+ a^2{(y-y_c)}^2=\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow b^2(x^2-2x_cx+{x_c}^2)+ a^2(y^2-2y_cy+{y_c}^2)=\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow b^2x^2-2b^2x_cx+b^2{x_c}^2+ a^2y^2-2a^2y_cy+a^2{y_c}^2=\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow b^2x^2+ a^2y^2-2b^2x_cx-2a^2y_cy+b^2{x_c}^2+a^2{y_c}^2\pm\hspace{1mm}a^2b^2=0 \)   ...(10)

Setting \(b^2=\mathbf{A}\), \(a^2=\mathbf{C}\) in equation (10) above we get

\(Ax^2+ Cy^2- 2Ax_cx - 2Cy_cy +A{x_c}^2+C{y_c}^2\pm\hspace{1mm}AC=0 \)   ...(11)

Setting \(-2Ax_c=\mathbf{D}\), \(-2Cy_c=\mathbf{E}\) and \(A{x_c}^2+C{y_c}^2\pm\hspace{1mm}AC=\mathbf{F}\) in equation (11) above we get

\(Ax^2+ Cy^2 + Dx + Ey + F=0 \)   ...(12)

The equations (10), (11) and (12) above give the Implicit Coordinate Equation of any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(X\)-Axis.

For any Ellipse having its Center at the Origin equation (12) becomes

\(Ax^2+ Cy^2+F=0 \)   ...(13)
The following gives the derivation for Standard and Implicit Coordinate Equations for any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(Y\)-Axis.

Let's consider a Real Ellipse having it's Semi-Major Axis Length as \(a\), Semi-Minor Axis Length as \(b\), Coordinates of Center at \((x_c,y_c)\), and Coordinates of 2 Foci at \((x_{f1},y_f)\) and \((x_{f2},y_f)\). Now, as per definition of Real Ellipse, the Sum of the Distance from 2 Foci to Any Point \((x,y)\) on the Ellipse is equal to the Length of it's Major Axis \(2a\). Hence

\(\sqrt{{(x-x_f)}^2 + {(y-y_{f1})}^2} + \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2} =L=2a\)

\(\Rightarrow \sqrt{{(x-x_f)}^2 + {(y-y_{f1})}^2} = 2a - \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)   ...(14)

Squaring Both Sides of equation (14) above we get

\({(x-x_f)}^2 + {(y-y_{f1})}^2 = 4a^2 + {(x-x_f)}^2 + {(y-y_{f2})}^2 - 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow 4a^2 + {(y-y_{f2})}^2 - {(y-y_{f1})}^2 = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow 4a^2 + y^2 + {y_{f2}}^2 - 2y_{f2}y - y^2 - {y_{f1}}^2 + 2y_{f1}y = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow 4a^2 + {y_{f2}}^2 - {y_{f1}}^2 - 2y_{f2}y + 2y_{f1}y = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow 4a^2 + ({y_{f2} - y_{f1}})({y_{f2} + y_{f1}}) - 2y(y_{f2}-y_{f1}) = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)   ...(15)

Now, \({y_{f2} - y_{f1}}=F=2c\) (i.e Distance Between the 2 Foci) and \({y_{f2} + y_{f1}}=2y_c\) (i.e 2 times the \(Y\)-Coordinate of the Center of Ellipse). Substituting these in equation (15) above we get

\(\Rightarrow 4a^2 + (2c)(2y_c) - 2y(2c) = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow 4a^2 + 4cy_c - 4cy = 4a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)

\(\Rightarrow a^2 + cy_c - cy = a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2}\)   ...(16)

Squaring Both Sides of equation (16) above we get

\({(a^2 + cy_c - cy)}^2 = {(a \sqrt{{(x-x_f)}^2 + {(y-y_{f2})}^2})}^2\)

\(\Rightarrow a^4 + c^2{y_c}^2 + c^2y^2 + 2a^2cy_c - 2a^2cy - 2c^2y_cy = a^2{(y-y_{f2})}^2 + a^2{(x-x_f)}^2\)

\(\Rightarrow c^2y^2 - 2a^2cy - 2c^2y_cy + a^4 + c^2{y_c}^2 + 2a^2cy_c = a^2y^2 + a^2{y_{f2}}^2 - 2a^2y_{f2}y + a^2{(x-x_f)}^2\)

\(\Rightarrow c^2y^2 - a^2y^2 + 2a^2y_{f2}x - 2a^2cy - 2c^2y_cy + a^4 + c^2{y_c}^2 + 2a^2cy_c - a^2{y_{f2}}^2 = a^2{(x-x_f)}^2\)

\(\Rightarrow (c^2 - a^2)y^2 + (2a^2x_{f2} - 2a^2c - 2c^2x_c)y + a^4 + c^2{y_c}^2 + 2a^2cy_c - a^2{y_{f2}}^2 = a^2{(x-x_f)}^2\)   ...(17)

Now, for any Ellipse we know that \(a^2 - b^2 = c^2\hspace{3mm}\Rightarrow -b^2=c^2-a^2\). Also, since the Major Axis of Ellipse is Parallel to the \(Y\)-Axis therefore \(x_f=x_c\). Substituting these in equation (17) above we get

\(\Rightarrow -b^2y^2 + (2a^2y_{f2} - a^2(y_{f2}-y_{f1}) - 2(a^2-b^2)y_c)y + a^4 + (a^2-b^2){y_c}^2 + 2a^2cy_c - a^2{y_{f2}}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + (2a^2y_{f2} - a^2y_{f2} + a^2y_{f1} - 2a^2y_c + 2b^2y_c)x + a^4 + a^2{y_c}^2 + 2a^2cy_c - a^2{y_{f2}}^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + (a^2(y_{f2}+ y_{f1}) - 2a^2y_c + 2b^2y_c)x + a^4 + a^2{(\frac{y_{f2} + y_{f1}}{2})}^2 + \frac{a^2}{2}(y_{f2}- y_{f1})(y_{f2}+ y_{f1}) - a^2{y_{f2}}^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + (2a^2y_c - 2a^2y_c + 2b^2y_c)y + a^4 + a^2(\frac{{y_{f1}}^2 + {y_{f2}}^2 + 2y_{f1}y_{f2}}{4}) + \frac{a^2}{2}({y_{f2}}^2- {y_{f1}}^2) - a^2{y_{f2}}^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 + a^2(\frac{{y_{f1}}^2 + {y_{f2}}^2 + 2y_{f1}y_{f2}}{4}) + a^2 (\frac{{y_{f2}}^2- {y_{f1}}^2 - 2{y_{f2}}^2}{2}) -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 + a^2(\frac{{y_{f1}}^2 + {y_{f2}}^2 + 2y_{f1}y_{f2}}{4} + \frac{-{y_{f2}}^2- {y_{f1}}^2}{2}) -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 + a^2(\frac{{y_{f1}}^2 + {y_{f2}}^2 + 2y_{f1}y_{f2} -2{y_{f2}}^2 -2{y_{f1}}^2}{4}) -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 - a^2(\frac{{y_{f2}}^2 + {y_{f1}}^2 - 2y_{f1}y_{f2}}{4}) -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 - a^2{(\frac{y_{f2}- y_{f1}}{2})}^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^4 - a^2c^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^2(a^2-c^2) -b^2{y_c}^2 = a^2{(x-x_c)}^2\)

\(\Rightarrow -b^2y^2 + 2b^2y_cy + a^2b^2 -b^2{y_c}^2 = a^2{(x-x_c)}^2\)   ...(18)

Now, Dividing By \(-a^2b^2\) on Both Sides of equation (18) above we get

\(\frac{y^2}{a^2} - \frac{2y_cy}{a^2} + \frac{{y_c}^2}{a^2} - 1 = -\frac{{(x-x_c)}^2}{b^2}\)

\(\Rightarrow \frac{{(y-y_c)}^2}{a^2} + \frac{{(x-x_c)}^2}{b^2} =1 \)   ...(19)

The equation (19) above gives the Standard Coordinate Equation for any Real Ellipse having it's Major Axis Parallel to the \(Y\)-Axis.

Unlike a Real Elllipse, an Imaginary Ellipse cannot be plotted or traced in a Real Space (2D Plane or 3D Space). And hence, the Standard Coordinate Equation for any Imaginary Ellipse Cannot be Derived in similar manner. However, the Standard Coordinate Equation of any Imaginary Ellipse having it's Major Axis Parallel to the \(Y\)-Axis is similar to that of a Real Ellipse as given in equation (19) above and is given as

\(\frac{{(y-y_c)}^2}{a^2} + \frac{{(x-x_c)}^2}{b^2} = -1 \)   ...(20)

Therefore in general, the Standard Coordinate Equation for any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(Y\)-Axis can be given as

\(\frac{{(y-y_c)}^2}{a^2} + \frac{{(x-x_c)}^2}{b^2} = \pm\hspace{1mm}1 \)   ...(21)

where the Right Hand Side of the equation (21) above is \(1\) for Real Ellipse and \(-1\) for Imaginary Ellipse

For any Ellipse having its Center at the Origin equation (21) becomes

\(\frac{y^2}{a^2} + \frac{x^2}{b^2} = \pm\hspace{1mm}1 \)   ...(22)

Multiplying By \(a^2b^2\) on Both Sides of equation (21) above we get

\(b^2{(y-y_c)}^2 + a^2{(x-x_c)}^2 =\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow b^2(y^2-2y_cy+{y_c}^2) + a^2(x^2-2x_cx+{x_c}^2) =\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow a^2x^2-2a^2x_cx+a^2{x_c}^2+ b^2y^2-2b^2y_cy+b^2{y_c}^2=\pm\hspace{1mm}a^2b^2 \)

\(\Rightarrow a^2x^2+ b^2y^2-2a^2x_cx-2b^2y_cy+a^2{x_c}^2+b^2{y_c}^2\pm\hspace{1mm}a^2b^2=0 \)   ...(23)

Setting \(a^2=\mathbf{A}\), \(b^2=\mathbf{C}\) in equation (23) we get

\(Ax^2+ Cy^2- 2Ax_cx - 2Cy_cy +A{x_c}^2+C{y_c}^2\pm\hspace{1mm}AC=0 \)   ...(24)

Setting \(-2Ax_c=\mathbf{D}\), \(-2Cy_c=\mathbf{E}\) and \(A{x_c}^2+C{y_c}^2\pm\hspace{1mm}AC=\mathbf{F}\) in equation (14) we get

\(Ax^2+ Cy^2 + Dx + Ey + F=0 \)   ...(25)

The equations (23), (24) and (25) above give the Implicit Coordinate Equation of any Ellipse (Real or Imaginary) having it's Major Axis Parallel to the \(Y\)-Axis.

For any Ellipse having its Center at the Origin equation (25) becomes

\(Ax^2+ Cy^2+F=0 \)   ...(26)
Following are the Properties of Implicit Equation of Axis Aligned Ellipses
1. The Sign of Co-efficients \(A\) and \(C\) are always same (i.e. either Both are Positive or Both are Negative).
2. If \(|A| > |C|\) the Implicit Equations represent a \(Y\)-Major Ellipse. If \(|A| < |C|\) the Implicit Equations represent a \(X\)-Major Ellipse.
3. In equations (13) and (26) if Co-efficients \(A\) and \(C\) and the Constant \(F\) have the Same Sign (i.e. either all are Positive or all are Negative) then the Equation represents an Imaginary Ellipse. However, if the Sign of the Constant \(F\) is different than the Sign of Co-efficients \(A\) and \(C\) the Equation represents a Real Ellipse.

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