![]() ![]() Theorem 6: Lines which are parallel to the same line are parallel to each other. Technically, to complete the proof you have to show that under orthogonal projection onto the $x,y$ plane, the preimage of an ellipse is also an ellipse (or possibly a circle, although you should also be able to show that the preimage is a circle only if $m = 0$). We know that if a transversal intersects two lines such that the pair of alternate interior angles are equal, then the lines are parallel. It does not immediately prove that the intersection itself is an ellipse. Which is the condition you need in order for Equation $(1)$ to be the equation of an ellipse.Ī word of caution: all of this proves only that the orthogonal projection of the intersection of the cone and plane onto the $x,y$ plane is an ellipse. So let's suppose you have set up your cone and plane so that If $\lvert m\rvert \geq \left\lvert\frac hr\right\rvert$ then in fact you will not get an ellipse, but rather a parabola or hyperbola. The sides of the cone have slope $\frac hr$ relative to the $x,y$ plane. \iff \alpha y^2 \beta y \gamma
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