Driver-side Mirrors without Blind-spots
R. Andrew Hicks
Department of Mathematics
Most people are familiar with the problem of having a blind-spot when they are driving a car. This means when relying on your mirrors there is some important place that you don't see. In particular, when you look into your driver-side mirror, there could be cars approaching from behind that you don't see. In the US, it is required that cars coming off the production line have flat mirrors. You are free as an owner to change it to a different shaped mirror (although I've never found where the details of these laws written down).
Here is a photo taken of a (flat) driver-side mirror from the driver's seat of my former graduate student's car. (Thanks Meredith Coletta!) These photos were taken in a parking lot, so the cars that appear in the mirror are parked, and facing away from the driver.The typical driver-side mirror will give you something like a 17 degree field of view.
You can avoid this problem by using a curved mirror, such as the ones on buses and trucks. But the resulting view is distorted, even though it is a very wide-angle view. Here is a comparison of the above flat mirror with such a curved mirror. (My brother-in-law found this mirror on the South street bridge back when it had a lot of potholes.)
This is not so satisfactory. But, to be fair, that mirror wasn't designed to be a driver-side mirror for this situation. Nevertheless, the point is that you can see more stuff easily with a spherical mirror like the above, but you pay in terms of distortion.
For many years now I've been looking at the problem of designing a mirror that would give a prescribed distortion to the viewer. In other words you tell me what you want to see when you look in the mirror and how it should be distorted, and I have an algorithm that spits out a mirror shape that achieves this. (Well not quite. The problem is not always solvable with one mirror, so the method does the best it can.)
If you ask then for a driver-side mirror that has, say a 45 degree field of view (thus annihilating the blind-spot) and no distortion (the view is perspective: straight lines look straight), you've got exactly the kind of problem I've got a way to solve. Here is the result I got when I put the above constraints into my code:
Then you can cut one out of aluminum, and you have a prototype:
Now you can use this as a driver-side mirror, in the parking lot setting, and compare the results to the very first image shown above:
The picture on the right is of the mirror I designed. The black car now looks smaller, but you can see lots more stuff to the left of it. That's the blindspot. Things have to be smaller btw, since if you ask for a wider field of view you need to pack more into the image. And the main point is that the image is not too distorted.
Here is a side by side comparison:
A question I am often asked when I show these images is how sensitive the view is to the position of the driver's head. Here is a little simulation that shows it is not sensitive:
The paper that describes this work is
Controlling a ray bundle with a free-form reflector
R. Andrew Hicks
Optics Letters, Volume 33, No. 15, pages 1672-1674, July 2008.
Last modified Sat Nov 12 12:31:22 EST 2011