Conic Section - Cartesian Coordinates - Modified Form

Modified Form

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For some practical applications, it is important to re-arrange the standard form so that the focal-point can be placed at the origin. The mathematical formulation for a general conic section is then given in the polar form by

and in the Cartesian form by

\begin{alignat}{7} \sqrt{x^{2}+y^{2}} = \left(l+e x\right) \\ \Rightarrow\left(\frac{x-\frac{le}{1-e^{2}}}{\frac{l}{1-e^{2}}}\right)^{2}+\frac{\left(1-e^{2}\right)y^{2}}{l^{2}} = 1 \end{alignat}

From the above equation, the linear eccentricity (c) is given by .

From the general equations given above, different conic sections can be represented as shown below:

  • Circle:
  • Ellipse:
  • Parabola:
  • Hyperbola:

Read more about this topic:  Conic Section, Cartesian Coordinates

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