S: The Viewable Sphere
BC: Ashland Station, stereographic projection, 2011
FC: The Viewable Sphere | Panoramic Photographs by Bruce Torrence
2: Geometry of the Viewable Sphere
3: In an ordinary photograph, straight lines appear straight.This basic fact is easy to understand if we regard an ordinary camera as if it were a pinhole camera. In an idealized pinhole camera, light rays pass through a single point (that we will call the node). An "upside down and backwards" version of the image is captured on the image sensor inside the camera. Imagine an invisible plane, parallel to the plane of the image sensor, and located between the lens and the subject. Each light ray passes through this projection plane on its way to the pinhole, and we may think of this plane as a perfect copy of the image captured by the sensor. A moment's thought reveals why a straight line in the scene becomes a straight line on the projection plane. The line in the scene, together with the node (a point not on it), determine a plane. This plane intersects the projection plane in a line, and this is the line that we see in the photograph.
4: Train passing by Randolph-Macon College, equirectangular panorama, 2011
5: In the image on the facing page, straight lines do not appear straight! So this is not an ordinary photograph. What is it, then? Technically, it's a stereographic projection of an equirectangular panorama. More simply, it's a little planet. Let's investigate exactly what this means. The central idea is this: Imagine a pinhole camera where the projection screen is not a flat plane, but rather is spherical. That is, imagine an invisible sphere centered at the node of the lens. Each light ray from the scene passes through this sphere on its way to the node, and so we have a projection of our scene onto this sphere. The spherical version of the photo from the previous page is shown in the top image one the right. Of course, we can't print a spherical photograph. Rather, we store it in a two-dimensional format using a longitude-latitude coordinate system. That is, the x-axis indicates the longitude position and the y-axis measures the latitude position of each point on the sphere. When stored this way, a spherical image is called an equirectangular projection. The equirectangular projection of our sample image is shown in the bottom image on the right. Note that straight lines in the scene are not straight.
6: The beautiful thing about a spherical projection screen is that if you shoot two overlapping photographs (while keeping the node at the exact same position), the two images will match perfectly, allowing you to assemble a panorama. The reason the images will match is simple: they share the same projection screen! They are simply being projected onto different portions of the exact same sphere. The overlapping regions are identical. In reality, we accomplish this with an ordinary camera, using software to transform each ordinary image into an equirectangular projection.
7: With our ability to build a panorama, we next shoot several overlapping images, keeping the node of the lens in the exact same position, until we have completed a full 360 x 180 degree equirectangular panorama. We now have a photograph of the entire viewable sphere, much like the more familiar equirectangular map of the Earth.
8: The final step in producing a little planet is to apply stereographic projection to the spherical image. Imagine our equirectangular panorama as a true spherical image, oriented so that the north pole on the sphere corresponds to the point directly above the camera. Place the sphere on top of your photo paper, so that the bottom of the sphere is sitting at the center of your paper. For each point on the sphere (other than the north pole), draw the line from the north pole through that point, and extend it until it hits the paper, like this: | When this is done to each point on the spherical image, we get a little planet:
10: It appears that there are circular arcs in the final image. Are they true circles? Let's determine how straight lines in the original scene will appear in the final image. Recall that we first project the scene onto the viewable sphere. We then apply stereographic projection to the spherical image. We need to follow a line in the original scene through each stage of this process. For the first stage, observe that a line in the original scene is mapped to a great circle on the viewable sphere. To see why, note that the line in the scene, together with the node (a point not on it), determine a plane. This plane passes through the center of the viewable sphere (the node is the center), and hence intersects the sphere in a great circle. | For the second stage, we note (a) that stereographic projection maps a circle through the north pole to a line in the final image:
11: Finally, we note (b) without proof, that stereographic projection maps any circle not passing through the north pole to a circle in the final image: | This is easy to see: A line in the final image, together with the north pole (a point not on it), determine a plane. And a plane cuts the sphere in a circle. | This leads us to conclude that a vertical line in the original scene, whose extension will of necessity pass through the north pole in the viewable sphere, corresponds to a "meridian" circle that also passes through the south pole. In the stereographic projection it becomes a radial line through the center of the image, like the radial lines in the image to the right. | Thus a nonvertical line in the original scene will correspond to a great circle not passing through the north pole on the viewable sphere. And this will be mapped to a circle in the stereographic projection. So nonvertical lines in the original scene become circles in the stereographic projection, like the edge of the crown molding, highlighted in red below.
13: Train passing by Randolph-Macon College, equirectangular panorama, 2011
15: Poor Farm Park, equirectangular panorama, 2011
16: Poor Farm Park, stereographic projection from a point near the horizon, 2011
17: Poor Farm Park, stereographic projection, 2011
19: Poor Farm Park II, equirectangular panorama, 2011
21: Living Room, equirectangular panorama, 2011
22: Living Room, stereographic projection, 2011
23: Living Room, stereographic projection from south pole, 2011
24: Living Room, stereographic projection from a point on the horizon, 2011
25: MAA Carriage House, stereographic projection, 2011
27: MAA Carriage House, equirectangular panorama, 2011
28: Belle Isle Pedestrian Bridge, stereographic projection from a point on the horizon, 2011
29: Belle Isle Pedestrian Bridge, stereographic projection, 2011
31: Belle Isle Pedestrian Bridge, equirectangular panorama, 2011
33: Sound Stage at the Kirk Williams Rollout, equirectangular panorama, 2011
34: Sound Stage at the Kirk Williams Rollout, stereographic projection, 2011
35: Stadium Planet, stereographic projection, 2011
37: Patriot's Stadium, equirectangular panorama, 2011
38: Train Blossom Planet, stereographic projection, 2011
39: Ashland Station under the Blossoms, stereographic projection from a point on the horizon, 2011
40: Zoe's World, Panini projection, 2011
41: Zoe's World, stereographic projection, 2011