Ellipses and Conic Sections

• An ellipse is one of the conic sections (intersections of a right circular cone with a crossing plane). The others are hyperbolas and parabolas (circles are special cases of ellipses). All conic sections are actually possible orbits, not just ellipses.

• One possible equation for an ellipse is:
  

  $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (1)

  
  
  The larger parameter ($a$ above) is called the semimajor axis; the smaller ($b$) the semiminor; they lie on the similarly defined major and minor axes, where the foci of the ellipse lie on the major axis.

• Not all ellipses have major/minor axes that can be easily chosen to be $x$ and $y$ coordinates. Another general parameterization of an ellipse that is useful to us is a parametric cartesian representation:
  

  $$x(t) = x_0 + a \cos(\omega t + \phi_x)$$ (2)

  $$y(t) = y_0 + b \cos(\omega t + \phi_y)$$ (3)

  
  
  This equation will describe any ellipse centered on $(x_0,y_0)$ by varying $\omega t$ from 0 to $2\pi$. Adjusting the phase angles $\phi_x$ and $\phi_y$ and amplitudes $a$ and $b$ vary the orientation and eccentricity of the ellipse from a straight line at arbitrary angle to a circle.

• The foci of an ellipse are defined by the property that the sum of the distances from the foci to every point on an ellipse is a constant (so an ellipse can be drawn with a loop of string and two thumbtacks at the foci). If $f$ is the distance of the foci from the origin, then the sum of the distances must be $2d = (f + a) + (a - f) = 2a$ (from the point $x = a$, $y = 0$. Also, $a^2 = f^2 + b^2$ (from the point $x = 0$, $y = b$). So $f = \sqrt{a^2 - b^2}$ where by convention $a \ge b$.