Bent Seesaw

Jim Schwiegerling

The University of Arizona, USA

John taught geometrical optics to multiple generations of graduate students. For several of those years, I taught the undergraduate version of his course. We had many discussions regarding teaching the material. While we could never agree on the correct way of writing the imaging equation, we did agree that the students who struggled with the material seem to struggle with the most fundamental concepts. Lacking the proper foundation meant that these students were lost with more advanced concepts. Many of our discussions were coming up with easy visualizations for this fundamental material. One of John’s favorites was the bent seesaw.

The bent seesaw is a visualization to illustrate the object and image locations. While these relationships are easily determined from the imaging equation, students may have difficulty visually analyzing a system, especially when the object or image is virtual. For positive-powered lenses, the seesaw is bent such that, when the left side is horizontal, the right side is bent toward, the ground as shown below.

The planks of the seesaw schematically represent the upper marginal ray before and after the lens. The lens serves as the fulcrum of the seesaw. The optical axis lies along the ground plane, with the front and rear focal points placed as shown. In this particular case, the left plank is horizontal, so the object is located at infinity. The right plank is bent downward, passing through the rear focal point.

As the seesaw tips through different angles, the object and image locations follow from the orientations of the left and right planks. When the left plank is tilted downward, the object is real. If the right plank is tilted downward, the image is real. If the object is to the left of the front focal point, the real image is to the right of the rear focal point. If the object is to the right of the front focal point, the virtual image is to the left of the front focal point. From these illustrations, the various object/image relationships are easily visualized for a positive-powered lens.

For a lens with negative power, the seesaw simply needs to be bent in the opposite direction. When the left plank is horizontal, the right plank is now bent upward to appear as if it is angled toward the rear focal point (which is now on the left side).

Tilting the seesaw now gives the various combinations of object and image relationships. As the left plank moves up, denoting a virtual object, the right plank moves down, giving a virtual image. When the left plank appears to pass through F, the right plank is horizontal. Further raising the left plank now tilts the right plank downward to give a real image. Tilting the left plank downward means that a real object is formed with a virtual image to its right.

As I take over John’s class in the Fall of 2022, I only hope that I can match his dedication and enthusiasm. His efforts will be continued.