Lambert's Problem - Initial Geometrical Analysis

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.