Initial Geometrical Analysis
The three points
- The centre of attraction
- The point corresponding to vector
- The point corresponding to vector
form a triangle in the plane defined by the vectors and as illustrated in figure 1. The distance between the points and is, the distance between the points and is and the distance between the points and is . The value is positive or negative depending on which of the points and that is furthest away from the point . The geometrical problem to solve is to find all ellipses that go through the points and and have a focus at the point
The points, and define a hyperbola going through the point with foci at the points and . The point is either on the left or on the right branch of the hyperbola depending on the sign of . The semi-major axis of this hyperbola is and the eccentricity is . This hyperbola is illustrated in figure 2.
Relative the usual canonical coordinate system defined by the major and minor axis of the hyperbola its equation is
with
For any point on the same branch of the hyperbola as the difference between the distances to point and to point is
For any point on the other branch of the hyperbola corresponding relation is
i.e.
But this means that the points and both are on the ellipse having the focal points and and the semi-major axis
The ellipse corresponding to an arbitrary selected point is displayed in figure 3.
Read more about this topic: Lambert's Problem
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