Bruce Torrence
A mathematician with longtime passion for photography, I've been exploring recent developments in digital imagery which allow me to utilize mathematics and computer programming to solve visual problems. In particular, I've been looking for ways to depart from the standard rectilinear projection of ordinary lenses in order to widen the field of view. A wider field of view enables the reader to understand architectural structure from a vantage point inside that structure. It challenges the viewer to make sense of the scene. And it is my hope that images employing mathematical projections will lead the curious viewer to a better understanding of the underlying mathematical principles.
The images shown here were created by shooting a sequence of photographs from the same vantage point, with the camera pointing in a different direction for each shot. The overlapping images were then stitched together pairwise, and a "spherical" panorama of the complete scene was created. Stereographic projection was then applied to the spherical panorama to create the artwork shown here. Stereographic projection has three beautiful properties. First, it allows a far wider field of view than an ordinary rectilinear photograph. Second, lines in the original scene become either lines or circles in the projected image. And finally, the mapping is conformal, so that small features in the scene may be scaled or rotated, but otherwise they are not distorted. Distortion occurs only at a global scale---the more you zoom in on any feature, the more ordinary it looks.
This is a projection made from a panorama of 13 photographs. The individual photos were shot from precisely the same point in space, and when stitched together they comprise the entire "viewable sphere" centered at that vantage point. That is, the panorama has complete coverage of the scene---360 degrees around, and 180 degrees from top to bottom. Stereographic projection was then applied to the spherical panorama, with the projection taken from the North Pole so that the point directly overhead becomes the point at infinity. This produces a lovely "little planet" effect, with the geometry of the roof structure framing the scene. The panorama was shot at Oriente Station in Lisbon, Portugal.
This is a projection made from a panorama of 13 photographs. The scene is a courtyard at Pena Palace in Sintra, Portugal. The individual photos were shot from precisely the same point in space, and when stitched together they comprise the entire "viewable sphere" centered at that vantage point. That is, the panorama has complete coverage of the scene---360 degrees around, and 180 degrees from top to bottom. Stereographic projection was then applied to the spherical panorama, with the projection taken from a point just above the equator. The sky and the stone floor of the courtyard form two disks at the center of the image, with the decorated wall of the palace filling out the scene. As your eye moves upward from the center of the image, past the doorway at the top, the scene wraps around to the bottom of the image (which shows a decorative facade above the doorway), upward to the sky.