Geometry

Ellipses

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Directrix Property

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• Directrix: $d$

• Each of the two lines parallel to the minor axis, and the following distance from it:

  • $$d=\frac{a^2}{c}=\frac{a}{e}$$

• Relationships:

  • For an arbitrary point $P$ on the ellipse, the quotient of the distance between a focus and its directrix is equal to the eccentricity:
    • $$e=\frac{|P{F}_1|}{|P{l}_1|}=\frac{|P{F}_2|}{|P{l}_2|}$$
  • $$\frac{p}{e}=x+\frac{r}{e}$$

Eccentric Anomaly

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• Eccentric Anomaly: $E$
• Angle at the center of the ellipse between the periapsis and the s/c’s current position
• $$\tan \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+e}{1-e}}\tan \left(\frac{E}{2}\right)$$

Triangles

Small Angle Approximation

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• Angle must be measured in radians
• $$\sin \left(\theta \right)\approx \tan \left(\theta \right)\approx \theta$$
• $$\cos \left(\theta \right)\approx 1-\frac{1}{2}{\theta }^2\approx 1$$