Ellipse 1Given two vertices on the same axis (V1 and V2) and one other point on curve (P), construct the ellipse.  Construct line V1V2, one axis. Let center O be the midpoint of V1 and V2. Construct the second axis perpendicular to this one through the center. Construct the circle c1 centered on O and passing through the given vertices.  Now construct the line through P perpendicular to V1V2, intersecting it at A. The challenge has a solution only if A falls between V1 and V2. Let PA intersect the circle at B. It actually intersects the circle twice, but it does not matter which is used. Construct line OB. Construct the line through P parallel to V1V2, and let it meet OB at C. Construct circle c2 with center O, passing through C. The second axis intersects this circle at vertices V3 and V4.  Construct ray OK, where K is a point on c1. Let the ray intersect circle c2 at L. From K construct a line perpendicular to V1V2, and from L construct a line perpendicular to V3V4. Let these lines intersect at M. Construct the locus of M as K travels on its path. This locus is the required ellipse.  The construction above will work whether the given vertices define the major axis or the minor. In this case they are the major vertices. Translate the circle c1 by vector OV3. The image circle intersects the major axis at the two foci, labeled F1 and F2. The locus is hidden below to show construction of the foci and directrices.  Let the minor axis meet circle c1 at E. Construct lines EV1 and EF1. Reflect EF1 across EV1, and let the reflection image intersect the major axis at point G. As a result, GF1 divides V1V2 harmonically, so one directrix passes through G. Construct it perpendicular to the major axis, and reflect it across the minor axis for the other directrix.  In the interactive sketch below drag the given objects to confirm that the solution holds up. Notice that it continues to work even if the given vertices are minor. Another thing to note is that there is no solution at all if point P cannot be projected orthogonally onto line segment V1V2. None of the constructed objects, not even the center point, should exist in that case. Those conditions are not covered in the construction as given above. |