Blind-spot problem for motor vehicles

Abstract

We consider the problem of designing a passenger side mirror for an automobile that does not have a blind spot and that does not distort the image. Our model consists of a coupled pair of partial differential equations that do not have a common solution. Using a best mean-square-error functional, we find approximate solutions using nonlinear optimization. In one case a local minimum provides a mirror that solves the problem, but it does not reverse the image.

Fast tool servo diamond turning of optical freeform surfaces for rear-view mirrors

“…Thus, the challenge is to find shapes that obtain a wide field of view without distorting the image. R. Andrew Hicks [8][9][10][11][12] presents a new technique for finding such approximations and applies it to the design of a rear-view mirror for an automobile that has no blind spot and minimal distortion. With this method, we design a rear-view mirror with a scale factor of 2.…”

Surfaces refracting and reflecting collimated beams

“…For mirrors, we also give a necessary and sufficient condition on the image transformation for the problem to be solvable, condition (3.9). In particular, when the image transformation is the scaling magnification this condition is not satisfied, and we obtain as in [HP05] that there is no solution in this case. In contrast with this, the scaling magnification does verify (2.5) and therefore there is a refracting surface in this case.…”

“…When h 1 (x, y) = λ x and h 2 (x, y) = λ y, it is easy to verify that (3.9) does not hold, which shows as in [HP05], that the reflector problem does not have a solution for constant scaling.…”

“…Likewise, we also study the corresponding problem for mirrors or reflectors that was considered by R. A. Hicks and collaborators in [HB01] and [HP05] for certain important applications such us the design of a driver mirror without a blind spot. For mirrors, we also give a necessary and sufficient condition on the image transformation for the problem to be solvable, condition (3.9).…”

Designing coupled free-form surfaces

“…One can think of this as designing a periscope with curved mirrors to achieve a prescribed purpose such as having wide angle of view. This problem was considered in [1,2] by the first author and R. Perline. An example of this problem in which only one reflector is used is the design of a driver-side mirror for a motor vehicle, which has no blind-spot and yet provides the observer with a perspective view of the scene [3].…”