$$ r(\theta) = \pm a \, \frac{1-e^2}{1 \pm e \cos \theta} $$ What's the difference between the one with denominator $1+e \, \cos\theta$ and $1-e \, \cos\theta$ , and between +a and -a?
I see that most people use the one with $1+e \, \cos\theta$, which makes me confused.