General 3Given a tangent line with corresponding point of tangency (tangent and T) and three other points (P1, P2, P3) on curve, construct the conic section.  Suppose this conic had already been constructed using Newton’s five-point construction with poles P2 and T. And suppose point P3 was the fifth point, through which two lines were constructed: j1 parallel to P1P2, and j2 parallel to P1T. The next step would be to construct some tangent lines. In order to construct the tangent through T, line P2T would be constructed, intersecting j1 at some point A. Then through A a line would be constructed in a predetermined direction to intersect j2 on the required tangent line. But that tangent line is already given. Let it intersect j2 at B. So AB must be in the required direction.  Now, back to the conic section itself, construct a circle through P2, and let it be the path of point C. Let line P2C meet j1 at D. Construct DE parallel to AB, meeting j2 at E. Line TE meets line P2CD at F, a point on the curve. Construct the locus of F as C travels its path.  There are now five known points on the curve: P1, P2, P3, T, and F. The curve itself has already been constructed, but those five points can be used to construct two more tangent lines and complete the construction of the associated parts. See Newton for the tangent line constructions, and Conic Parameters for completion of the other associated parts. Drag the given objects in the websketch below, and confirm that the construction holds up for every positioning. |