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The image below shows a very thin translucent plannar section being swept through a translucent connic. The result shows two traces on the plane, both of which are hyperbolas. The use of "motion" allows the author to use 3D; a 2D image changing in time. The result is an example showing how a hyperbola is the result of plane intersecting a cone when the plane is parallel to the axis of the cone. I hope that this 3D example is much more intuitive than a static 2D image or the previous sentence.
