Parabola 11Construct a parabola through four given points.  The construction steps are given below. For justification of the solution see Parabola 11 Support, which presents the solution of Heinrich Dörrie. ConstructionBe aware, the solution to this challenge is not possible for all arrangements of four given points. The points must form vertices of a convex quadrilateral, and the quadrilateral must not be a parallelogram. Here the given points, P1, P2, P3, and P4 satisfy those conditions.  Construct lines P1P3 and P2P4, and let them intersect at point O. In this sketch the lines are the diagonals of quadrilateral P1P2P3P4. That is not actually necessary. They could otherwise be opposite sides, but in that case their intersection might fall far off the page.  Construct the circle centered on O, having radius equal to the geometric mean of OP2 and OP4. Let it intersect line P2P4 at E1 and E2.  Now construct the circle centered on O, having radius equal to the geometric mean of OP1 and OP3. Let it intersect line P1P3 at F1 and F2.  The sides of E1F1E2F2 are diameters of the solutions. But notice that this is a parallelogram. Opposite sides are diameters of the same parabola. To avoid redundancy, eliminate point E2 and use the remaining lines, E1F1 and E1F2. These are diameters of the only two solutions.  For completion of the solutions, see the challenge Parabola 9, a parabola given any diameter and three points on the curve. Word to the wise: that construction uses only three of the four points, which means it will always render a parabola fitting those three points, even if the four given points do not satisfy the conditions. If they do not, then the fourth point will not fit, because there would be no solution fitting all four. With considerable effort it is possible to cause intersection point O to exist only when the convex quadrilateral condition is satisfied. Everything remaining in the construction depends on that point, so no bogus solutions appear. In the interactive sketch below is the completed construction from the four points given in the Sketchpad document. |