The Frobenius Integrability Theorem and the Blind-Spot Problem for Motor Vehicles

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.