Orbital Intersections
Orbital Intersections (Finding the ellipse)
Let the basis vectors b1 and b2 as defined above form the coordinate system of the plane so that:
File:B1b1 basis as plane coordinates.gif (1.17)
The equation of the ellipse is a two dimensional curve where the sum of the distance to each foci is constant for any two points on the plane. If we consider the two points located with the position vectors b, m the ellipse of the transfer orbit is given by:
File:Ellips foci f and s point b sum d.gif (1.18)
File:Ellips foci f and s point m sum d.gif (1.19)
note the norm of (b-f) is the distance between the position vectors b and f is given by:
File:Norm b minus f.gif (1.20)
Equations 1.18 says that f lies in a circle centered at b with radius d-||b-s||. Similarly equation (1.19) says f lies in a circle centered at m with radius d-||m-s||. Thus the second foci can be found by finding the intersection between two circles. The result is the distance from the center of the big circle along the line between the center of the two the point on the line that is perpendicular to where the circles intersect is given by:
File:Intersect two circles 1.gif (1.21)
where R is the radius of the big circle r is the radius of the small circle and d is the distance between the center of each circle. The distance the point of intersection is from the line between the two circles is given by:
File:Intersect two circles 2.gif (1.22)
