The most efficient maneuver for transferring from one circular orbit to another circular orbit of roughly the same size is the Hohmann transfer orbit. It requires two burns: one to leave the initial circular orbit into an elliptical orbit, and another to leave the elliptic orbit for the new circular orbit.
If the new orbit is much larger than the original, it can be more efficient to do three burns, inserting two elliptical orbits between the two circular orbits.
In the graph below, the Hohmann transfer orbit moves a satellite from the blue circle to the green circle via the orange ellipse. The orange ellipse fits inside the green circle. The bi-elliptic transfer maneuver replaces the orange ellipse with two half ellipses, both of which extend far beyond the green circle [1].
The dashed curve completes the ellipse but isn’t part of the orbit.
Below is a graph of a bi-elliptic transfer orbit where the radius of the outer circle is 13 times larger than the inner circle and the distance from the furthest point on the ellipse to the center of the circles is 50 times the radius of the inner circle. This is the orbit described in footnote [1].
The meaning of the dotted orange curve is different in this plot. Now it is part of the orbit.
The abrupt change in curvature is at the furthest point of the elliptical orbits is striking, but I believe the graph is correct.
I tried to compare the plot above with other sources. None of my reference books include a plot. When I searched online, I found examples where the two circular orbits are roughly the same size, which defeats the point of the bi-elliptic transfer orbit. Perhaps this was done for aesthetic reasons, but it is not realistic.
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[1] How far is “far” is a matter of efficiency. A bi-elliptic transfer orbit could extend slightly outside the target circular orbit, but in this case the maneuver would be less efficient than the Hohmann transfer. To move from one circular orbit to another with a radius 13 times greater, the apogee of the transfer ellipses would have to be about 50 times larger than the radius of the inner circular orbit.
The bi-elliptical transfer orbit need not extend past the outer circle at all. It could just kiss the target circular orbit, in which case it reduces to the Hohmann transfer orbit, i.e. the Hohmann transfer orbit is a special case of the bi-elliptic transfer orbit.
