We consider the problem of designing a passenger-side automotive mirror that has no blind-spot or distortion. While reasonably good solutions have been found for the analogous problem for the driver-side mirror, the passenger-side problem has not yet yielded a reasonable solution. Our model requires us to find surfaces perpendicular to a given vector field determined by the data. This is in general impossible, which leads us to investigate estimates and error formulas for approximate solutions to the problem. We show that if the integrability condition holds approximately, then there will be a good approximating integral surface, and we provide a construction method with an exact error formula. We apply this method to the construction of a wide-angle passenger-side mirror. Furthermore, if the vector field does not satisfy the integrability condition, we also give a bound on how nonperpendicular any surface must be to the given vector field.

Catadioptric sensors are visual sensors that employ lenses (dioptrics) and mirrors (catoptrics). We present a general method of catadioptric sensor design for realizing prescribed projections. Our method makes use of geometric distributions in. 3-dimensional space, which are generalizations of vector fields. The main idea is this: if one desires a reflective surface that will image the world in a certain way, then this condition determines the orientation of the tangent planes to the surface. Analytically, this means that the surface will then be determined by a pair of partial differential equations, which may or may not have a common solution. We show how to check if a common solution exists. If no common solution exists, we describe a method for obtaining optimal approximate solutions in a least-squares sense. As an example application, we construct a mirror that will give a panoramic view of a scene without any digital unwarping.

We present two families of reflective surfaces that are capable of providing a wide field of view, and yet still approximate a perspective projection to a high degree. These surfaces are derived by considering a plane perpendicular to the axis of a surface of revolution and finding the equations governing the distortion of the image of the plane in this surface. We then view this relation as a differential equation and prescribe the distortion term to be linear. By choosing appropriate initial conditions for the differential equation and solving it numerically, we derive the surface shape and obtain a precise estimate as to what degree the resulting sensor can approximate a perspective projection. Thus these surfaces act as computational sensors, allowing for a wide-angle perspective view of a scene without processing the image in software. The applications of such a sensor should be numerous, including surveillance, robotics and tradition photography.