Pole and Polar 1
- Ryan Hirst
A and A' are conjugate points. (OA)(OA') = r² http://www.geogebratube.org/material/show/id/30937 _____ Pole and Polar (1 of 3): →1. Step-by-step Construction for a pole inside and outside the circle. 2. Reduced Construction and vectors: http://www.geogebratube.org/material/show/id/34585 3. Ruslan and Ludmila: http://www.geogebratube.org/material/show/id/34586
- Case 1: A falls outside the circle. OA is the diameter of the new circle. A' is the midpoint of the common chord of the two circles (DE).
- Case 2: A falls inside the circle. A is the midpoint of DE, and OA' the diameter.
- Case 3: A is on the circumference of the given circle. The two circles are tangent: A = A' = D = E. The transformation is continuous across the boundary of the circle.
- Case 4: A = O The two tangents are at opposite ends of a diameter: they never meet. However, we can let the line DE cross over the diameter undisturbed. From A's point of view, in the limit the curvature of the second circle passes indistinguishably into a straight line.