Three in One Page 9
It is instructive to imagine what happens to the dark red polar plane as the pole P moves in a circle around the flat circular surface shown. As the pole transcribes the circle, so the polar will ‘follow’ the point by tilting about the point S. Hence the surface of the red disc will sweep out a surface shaped like two cones (this still represents a surface and not a volume). The red disc is just a section of the actual plane, which extends continuously throughout space, so the cones surface will extend continuously. So the polar equivalent of a circle in a plane is a cone in a point.
Decreasing the radius of the circle in the pink plane would result in a cone with a broader and broader apex, eventually becoming an endless horizontal plane. As the radius of the circle in the pink plane tends to the infinitely large, so the cone tends to become so sharp that it coincides with the vertical axis. Once again there is a relationship of expansion and contraction.
An interesting example is created when the shape traced out by the point in the pink plane is a logarithmic spiral. This starts at an infinite outer distance and moves towards an infinitude within. The cone will, correspondingly, spiral out from infinitude within (an infinitude tending towards a vertical line) towards a flat plane, all about a fixed apex. This closely resembles the form of uncurling leaves that are observable in many plant species.
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