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

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. IMAGING SCIENCES c© 2013 Society for Industrial and Applied MathematicsVol. 6, No. 3, pp. 1367–1384

The Frobenius Integrability Theorem and the Blind-Spot Problem for MotorVehicles∗

Meredith L. Coletta†, R. Andrew Hicks‡, and Shari Moskow‡

Abstract. We consider the problem of designing a passenger-side automotive mirror that has no blind-spotor distortion. While reasonably good solutions have been found for the analogous problem for thedriver-side mirror, the passenger-side problem has not yet yielded a reasonable solution. Our modelrequires us to find surfaces perpendicular to a given vector field determined by the data. This isin general impossible, which leads us to investigate estimates and error formulas for approximatesolutions to the problem. We show that if the integrability condition holds approximately, thenthere will be a good approximating integral surface, and we provide a construction method withan exact error formula. We apply this method to the construction of a wide-angle passenger-sidemirror. Furthermore, if the vector field does not satisfy the integrability condition, we also give abound on how nonperpendicular any surface must be to the given vector field.

Key words. geometric optics, optical design, integral surface

AMS subject classification. 78A05

DOI. 10.1137/100793347

1. Introduction.

1.1. The blind-spot problem for motor-vehicles. Our problem is the design of passenger-side mirrors for motor vehicles that provide a sufficiently wide field of view so as to remove theblind-spot, while simultaneously presenting the driver with an undistorted (i.e., perspective)view. An example of a previous result of a driver-side mirror with no blind-spot and minimaldistortion appears in Figure 1. Here we see a conventional flat driver-side mirror comparedwith an aluminum prototype designed using the method described in [7]. For the passenger-side mirror no solution has yet been found that gives such a small amount of distortion, despitethe numerous attempts made using different methods [6, 8, 7]. In this paper we describe a newmethod which gives comparable results to these previous ones for the passenger-side problem,but for which we can provide an error formula. We also present an estimate which is a firststep in showing why it is possible that no “good” solution to the passenger-side mirror problemexists.

As most people are intuitively aware, the issue of the driver’s field of view is a crucial onefor automotive safety. According to a U.S. Department of Transportation report, merging and

∗Received by the editors April 26, 2010; accepted for publication (in revised form) April 29, 2013; publishedelectronically July 16, 2013.

http://www.siam.org/journals/siims/6-3/79334.html†Gateway Ticketing Systems, 315 East Second Street, Boyertown, PA 19512 ([email protected]). The re-

search of the first author was supported by NSF IIS-0413012.‡Department of Mathematics, Drexel University, Philadelphia, PA 19104 ([email protected], moskow@math.

drexel.edu). The research of the second author was supported by NSF IIS-0413012 and NSF DMS-0908299. Theresearch of the third author was supported by NSF DMS-0605021 and NSF DMS-1108858.

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http://www.siam.org/journals/siims/6-3/79334.html

mailto:[email protected]





1368 M. L. COLETTA, R. A. HICKS, AND S. MOSKOW

A. B.

Figure 1. A. A view of a parking lot through a conventional driver-side mirror, which has approximately a17◦ field of view. B. A view of the same scene as in (A), using a mirror designed by the second author, whichhas a 45◦ field of view, yet has little distortion.

lane changing accidents led to 827 fatalities and 58,000 injuries in 2007 [1]. The crux of theproblem is that flat mirrors do not provide a wide enough field of view. When an observergazes at a flat mirror, the field of view (measured, say, in terms of an angle in the horizontalplane) is exactly the same angle as produced by the rays connecting the observer’s eye tothe mirror surface. As a result, the smaller the mirror, the smaller the field of view for theobserver. Additionally, if a mirror is moved away from an observer, the field of view decreases.For example, based on measurements by the authors, a driver-side mirror that is flat, and ofaverage size on a U.S. production sedan, yields a paltry 17◦ field of view, and an even moredistant flat passenger-side mirror would yield a 5◦ field of view!

A familiar solution used by trucks and buses, where the problem is more severe due tothe vehicle size, is to employ spherical mirrors, but these introduce considerable distortion.Thus the problem is to find mirror shapes that yield a wide field of view without distortingthe image. The design problems for the driver-side mirror and the passenger-side mirror areclearly related, but the difference lies in the path of the optical axis of the driver’s eye. Thebasic geometry is depicted in Figure 2. The essential difference between these two problemsis that the angle of deflection of the optical axis θ is 90◦ in the passenger-side case, while forthe driver side the corresponding angle ψ is close to 65◦. For reasons that will become clearerbelow, the more extreme problem of the passenger-side mirror results in inferior approximatesolutions.

DriverObjectplane

θψ

Figure 2. The geometry of the blind-spot problem.

While some countries require that driver-side mirrors be flat on production model vehicles,

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INTEGRABILITY AND THE BLIND-SPOT PROBLEM 1369

e.g., the U.S. [13], many allow curved passenger-side mirrors, following U.N. regulations [2].The conventional U.S. passenger-side mirror is slightly curved, giving a view of approximately27◦. Without this curvature, the field of view would be that which is subtended by the driver’seye as mentioned above, which is typically about 5◦ for a passenger-side mirror. The drawbackis a problem of depth perception; if the field of view of a system is increased by introducing acurved mirror, then the apparent sizes of some objects in the reflection must decrease, sincemore of the scene is being imaged. This results in the need for the familiar “Objects in Mirrorare Closer than They Appear” warning that is often printed on curved mirrors. The dangeris of course that if an object appears smaller than normal to an observer, there is a chancethat the driver will judge the distance to the object to be farther than it really is, resultingin a collision. Nevertheless, in the U.S. it is apparently felt by regulators that this is animportant trade-off: a 27◦ field of view curved passenger-side mirror that produces distortedobject shapes and sizes is considered safer than a 5◦ flat mirror that honestly represents objectsizes.

To remove the blind-spot on the passenger-side we would want the field of view to beabout 40◦–45◦, and as undistorted as possible.

1.2. The model. At this point we must discuss a small amount of optics in order toexplain our model. We work in the realm of geometric optics, where a light ray is representedby a straight line. Geometric optics is reversible, and so for our purposes we may think oflight rays as entering an observer’s eye or as emanating from that eye. We assume that theobserver has a single eye, and that the eye is a point. Viewed as a source, we would like therays that exit the eye to reflect off of the passenger-side mirror and strike a certain targetplane that lies behind and to the side of the vehicle.

Suppose we fix a plane in front of the driver’s eye, through which the emanating raysmust pass. We may label each ray by the point in this plane through which the ray passes.Given a mirror M , the rays that leave the driver’s eye and reflect off of the mirror M inducea transformation from the plane that labels the rays to the target plane, as in Figure 3.The general inverse problem is to find a mirror M so that TM is equal to a given prescribedtransformation, T . In our specific problem, the prescribed transformation T is essentiallya scaling between the two planes. For example, a checkerboard pattern in the image planeshould be transformed into a checkerboard pattern in the target plane by the mirror.

We next restate the above inverse problem so that it is a statement about vector fields.One wishes the rays to exit the observer’s eye, reflect off of a mirrored surface and go toprescribed destinations in the target plane. How a ray is reflected depends on the normalto the surface at the point of intersection. Therefore we will be attempting to find a surfacewhose normal vector field reflects the rays in the “correct” directions.

To do this, we assume that a ray is to reflect off of a surface at the point (x, y, z) andtravel to its ideal destination. To do this, calculate what the normal to the ideal mirrorcontaining (x, y, z) should be: one has incoming and outgoing rays, so the normal should bein the direction of the sum of the two unit vectors In(x, y, x) and Out(x, y, z) as depicted inFigure 4. Here and throughout this paper, we have chosen for simplicity to have the eye atthe origin (0, 0, 0) and to take the plane x = 1 as our image plane. If one has a line from theeye to (x, y, z), then the point in the image plane that lies on that line is (1, y/x, z/x). This is

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TM (1, y/x, z/x)

M(x, y, z)

(1, y/x, z/x)

Observer’s Eye

Target Plane

Image Plane

n

Figure 3. Given a mirror, M , which is viewed by an observer from behind an image plane and a targetplane, the rays emanating from the eye may reflect off of M and strike the target plane. In that case we have atransformation TM , between a subset of the image plane and a subset of the target plane. Here we will alwaystake the observers eye to be at the origin and the image plane to be at x = 1, thinking of the mirror as havingpositive x coordinates.

(x, y, z)

(1, y/x, z/x)

Observer’s Eye

Target Plane

Image Plane

W(x, y, z)

In(x, y, z)Out(x, y, z)

T (1, y/x, z/x)

Figure 4. Given a correspondence, T , between points in an image plane and points on a target plane, onecan define a vector field W that is hopefully normal to a mirror surface that realizes the correspondence. Weshould emphasize that one does not use a mirror in any way to define W , but only the correspondence T .

the form of the points that T acts upon. So a ray that “exits” the eye and strikes the mirrorat (x, y, z) should end up at T (1, y/x, z/x), if it is reflecting off of a surface that is a solution

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to our problem. This tells us that

(1.1) In = − (x, y, z)√x2 + y2 + z2

, Out =T (1, y/x, z/x) − (x, y, z)

|T (1, y/x, z/x) − (x, y, z)| .

Therefore, if one is given T , then a vector field

(1.2) W(x, y, z) = In(x, y, z) +Out(x, y, z)

is defined at any point on a ray emanating from the eye with the exception of those that wouldcause a (rather blatant) singularity in the definition of In and Out. Consequently we have avector field defined on a (generally open) subset of R3. Any surface contained in this subsetwhose normals were pointwise multiples of W would reflect the rays striking it in such a waythat the transformation T would be physically realized. That is, it would solve our problem,at least for those rays that strike it.

Let us now consider the problem of constructing the vector field W that would be per-pendicular to an ideal passenger-side mirror. Assume that the mirror will pass through thepoint (x0, 0, 0), where we choose our units to be centimeters, and so x0 = 180 cm is a commondistance in cars between the driver’s eye and the passenger-side mirror. The target plane willbe of the form y = −k, where k > 0. Taking k = 1000 cm would be reasonable. A schematiclayout appears in Figure 5. Then

(1.3) T (1, y, z) = (−αy + x0,−k, αz)

is then the desired form of T . Taking α = x0 + k gives the scaling produced by a flat mirror.Thus it is convenient to generally take α = λ(x0 + k), so that λ = 1 provides the field of aflat mirror, and λ > 1 gives a bigger field of view. One must take λ to be approximately 10in order to create a field of view of 40◦ (as opposed to the “natural” one of 5◦).

Our above definition of W gives

W(x, y, z) =− (x, y, z)√x2 + y2 + z2

+T (1, y/x, z/x) − (x, y, z)

|T (1, y/x, z/x) − (x, y, z)|

(1.4)

=− (x, y, z)√x2 + y2 + z2

+(−λ(x0 + k)y/x+ x0,−k, λ(x0 + k)z/x) − (x, y, z)

|(−λ(x0 + k)y/x+ x0,−k, λ(x0 + k)z/x) − (x, y, z)| .(1.5)

1.3. The mathematical problem. The mathematical problem addressed in this paper,which we are led to by the above discussion, is the following ill-posed inverse problem:

Given a nonvanishing vector field W on an open set U in R3, do there exist surfaces in

U that are perpendicular to W?

Requiring that the surfaces be C1 is adequate. The problem rarely has a solution; formost choices of W it is ill-posed. Thus the question becomes the following:

How close can one come to finding a surface perpendicular to a given W?

The crucial example the reader should keep in mind throughout is when W is a gradient,W = ∇φ for some differentiable potential φ defined on U . In this case we know that W is

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x

y

x =1 cm (image plane)

y =-1000 cm

x =180 cm

mirror

(x,y,z)

(1,y/x,z/x)

T(1,y/x,z/x)

W

Figure 5. A schematic view of the layout for the coordinates of the passenger-side mirror problem.

perpendicular to the level surfaces of φ, i.e., solutions of φ(x, y, z) = C. Assuming that thesesolution sets are C1 surfaces in U , then we have an example of a foliation of U , which is acollection of surfaces that disjointly decompose U . In general we say that W is integrable ifit is perpendicular to a foliation of U .1

We will assume throughout that the open sets on which our vector fields are defined havecompact closure. This paper contains the following results:

1. an estimate showing that if curl(W) ·W is bounded away from 0 on U , that W mustdeviate from at least one normal to any given surface in U by a quantity bounded away from0;

2. an application of (1) showing that no isolated exact solutions exist for the passenger-sidemirror problem;

3. a method for constructing an approximate integral surface of W with an error formulasuch that the error goes to 0 linearly with curl(W) ·W;

4. an application of the constructive method of (3), with simulations showing the distortionthat would be viewed by the driver employing the resulting mirrors.

2. Some history of optical engineering: Free-form surfaces. Here we discuss the tech-nological heritage of our work. It is not a prerequisite for understanding the application. Themain point is that technological advances have been made in the last ten years that make itpossible to create optical quality surfaces of essentially any shape.

Optical design, which for the most part takes place in the realm of geometric optics,

1This definition of integrable may sound strange, since for the most part reasonable vector fields alwayspossess integral curves. The terminology used here comes from considering, instead of W, the field of planeson U that is perpendicular to W (the plane field “dual” to W). This viewpoint is useful in geometry andtopology, and although we will not use it here, preferring the more familiar notion of the vector field, we do,however, borrow the term “integral surface” for convenience.

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traditionally made use of spherical surfaces or other conics. With the appearance of computercontrolled machining, it became possible to make a surface on a lathe with any profile, but forthe most part designers continued to design systems that consisted of rotationally symmetriccomponents. The extra degrees of freedom afforded by “aspheres” allow for compact designs,and we see the benefits now, for example, in small consumer digital cameras, which havevery flexible lenses in terms of zoom and depth of field. For example, the Canon SD1000contains six lenses, two of which are aspheres. The next step was to consider the use ofarbitrarily shaped optical elements, both mirrors and lenses. A surface that is not a surfaceof revolution, or portion of one, is referred to in the optical design community as a free-formsurface. Free-form surfaces have historically been almost impossible to machine to opticalquality. Early free-form designs include a progressive spectacle lens [9] designed in the late1950s by Kanolt, which to the knowledge of the authors was not implemented at the time.On the other hand, the Polaroid SX-70 folding camera [15] is another example, and it washugely successful. In this case the molds for the lenses were essentially made by hand. Only inthe last ten years has technology existed that can machine optical quality free-form surfaces.This technology was developed as part of the DARPA conformal optics program [10]. Sinceit was never possible to fabricate these surfaces until recently, no design theory was everdeveloped, and little has been developed to date. In particular, unlike many other engineeringdesign problems, until recent years, optical design has rarely been formulated as a problem inpartial differential equations (PDEs), probably due to the technological restriction of havingto work with spherical surfaces.2 The traditional approach has been to use optimization, butthe number of parameters needed to model free-form surfaces is significantly greater than inthe rotationally symmetric case, and optimization is best augmented with direct methods.Certainly, many commercial optimization packages have been unable to meet the needs ofoptical designers [16].

It would appear that free-form surfaces could play a role in numerous applications that bytheir nature lack rotational symmetry, but methods for the design of free-form surfaces are intheir infancy. The design of illumination systems is one area where such problems often arise,and it has recently attracted the attention of the PDE community. Systems for illuminationare examples of nonimaging optics, i.e., the goal is not to form an image but to redistributethe light from a source or sources with prescribed intensity distribution onto a target [21]. Yetthe theory of controlling even a single point source or collimated beam is fairly complicated.Rubinstein and Wolansky in [17] describe a means of designing free-form lenses to controlthe intensity of a collimated beam. For a point source, construction and existence have beenconsidered by Oliker and Kochengin [14] and Oliker and Glimm [3, 4]. In general the problemof controlling multiple bundles simultaneously is unsolved, as is discussed in [21]. Probablythe most recent and popular application of illumination optics has been due to the widespreaduse of light emitting diodes, which is a technology that could help cut energy consumptiondrastically worldwide. Note that illumination design has applications to areas such as laserbeam shaping [18] and solar collector design [20]. A more recent application has been thedesign of light pipes, which are used, for example, to light buildings with natural light “pipedin” from the roof [21]. Many of these applications are very timely in that they are linked to

2Of course PDEs play an important role in optics in general. Some design instances exist, such as in [5, 12].

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energy efficiency in some way.

In previous work [6, 8], the second author showed that some design problems for free-formsurfaces can be reduced to the problem of finding a surface which is perpendicular, or nearlyperpendicular, to a given vector field. The design method described in this paper will almostalways give a free-form surface.

3. The Frobenius integrability theorem. We do not need the general theorem of Frobe-nius.3 For our needs it suffices to state a version for open sets in R

3.

Theorem 3.1 (Frobenius in R3). Suppose W is a C∞ vector field defined on an open subset

of R3. Then

curl(W) ·W = 0

if and only if W is integrable. That is, curl(W) · W = 0 if and only if U has a foliation ofintegral surfaces perpendicular to W.

Again, the reader should consider gradient fields to gain intuition for the Frobenius the-orem. Assuming that W is defined on a simply connected open set, we then have thatcurl(W) = 0 implies that W = gradφ for some φ. Then any level surface φ(x, y, z) = Cthat is in fact a true smooth surface4 is a solution to the problem, in the sense that the raysstriking the surface are reflected to the target plane so as to induce the mapping T . These arethe integral surfaces appearing in the conclusion of the Frobenius theorem, since curl(W) = 0implies that curl(W) · W = 0. Suppose though that W is not a gradient, but a multipleof a gradient by a function, W = β(x, y, z) grad φ. (For our application this would be sat-isfactory, since we don’t care about the length of W, only its direction, since that is whatdetermines the direction of reflected light.) The level surfaces of φ are solutions, but it isunlikely that curl(W) = 0. It is a straightforward calculation to show that if W = β gradφ,then curl(W) ·W = 0. Proving the other direction amounts to proving the Frobenius theoremin R

3.

The hypothesis that curl(W) ·W = 0 on an open set is rarely satisfied. This is the casefor the W of the passenger-side mirror problem. There are two natural points to take then.

First, could it be that solutions to the equation curl(W)·W = 0 might be isolated surfacesthat are solutions to our problem? It turns out that isolated solutions may exist for some Wand it is a necessary condition for a surface to be perpendicular to W that on that surface,curl(W) ·W must vanish. This is Corollary 4.2 of Theorem 4.1. It is not, however, a sufficientcondition. That is, a set of points that is a solution to curl(W) ·W = 0 and is a surface maynot be perpendicular to W. We consider these questions below.

Second, what if curl(W) · W �= 0, but curl(W) · W is in some sense small? Does thisimply the existence of approximate solutions? Theorems 4.1 and 6.1 answer this question inthe affirmative.

3The general version addresses the problem of finding integral surfaces to distributions, in the setting ofan arbitrary d-dimensional differentiable manifold Md. (See Lee [11, page 500]. Stoker [19, page 392] presentsanother viewpoint.)

4Of course degenerate situations can occur, i.e., the solution set may not be a manifold.

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4. Obstructions to the existence of good designs.Theorem 4.1. Suppose that U ⊂ R

3 is an open set with compact closure U . Let W : U → R3

be a differentiable vector field on U with curl(W)·W ≥ ε > 0. Let S ⊂ C be a smooth, compact,orientable surface with boundary and unit normal field n. Then

(4.1) maxp∈S

|W(p) − n(p)| ≥ εA

MA+ L> 0,

where A is the area of S, L is the length of the boundary of S, and M = maxp∈U |curl(W)|.An immediate corollary of Theorem 4.1 is as follows.Corollary 4.2. Given the conditions and notations of Theorem 4.1, any S will be nonper-

pendicular to W at at least one of its points. Thus no isolated solutions can exist under theseconditions.

Remark. Generally W is defined on a larger set in R3 than the set U in which one wishes

to construct a surface. Almost always W will be smooth, or even analytic, in applications. Inany case, the subset of U on which curl(W) ·W > 0 will be open, as will the subset on whichcurl(W) ·W < 0. If the latter is true on all of U , then the below proof can be modified in anobvious way to make a similar conclusion. In applications, though, it may be the case thaton U , curl(W) ·W may take on all three signs. This is the case in the example given below.

Proof of Theorem 4.1. We begin by noting that

(4.2)

∫Scurl(W) ·Wdσ ≥ εA,

where the integral is the usual surface integral taken over S and A is the area of S.We would like to be able to find a lower bound on the quantity

(4.3) maxp∈S

|W(p)− n(p)|.

Applying Stokes’ theorem yields the following inequality:∫Scurl(W) ·Wdσ =

∫Scurl(W) · (W − n)dσ +

∫Scurl(W) · n dσ

≤∫S|curl(W) · (W − n)|dσ +

∫∂S

W · dc

≤∫S|curl(W)||W − n|dσ +

∫∂S

W · dc−∫∂S

n · dc

≤∫SM |W − n|dσ +

∫∂S

|W − n|ds,

where M = maxp∈U |curl(W)(p)|.If k = maxp∈S|W(p) − n(p)|, then

(4.4)

∫SMkdσ +

∫∂Skds =MkA+ kL ≥ εA > 0,

where L is the length of ∂S.

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Therefore

(4.5) k = maxp∈S

|W(p)− n(p)| ≥ εA

MA+ L> 0.

In terms of the angle between W and n, we have

(4.6) ∠(W,n) ≥ 2 arctan

(εA

2MA+ 2L

).

5. Application of Theorem 4.1. In the introduction we computed the vector field cor-responding to the passenger-side mirror problem. The next step is to check if curl(W) orcurl(W) · W vanish, although this is probably best done with a computer algebra system.The resulting formula is somewhat large:

(5.1) curl(W) ·W = −11800z

rQ3/2

× [x4 + x3y − 10980x3 + 12620x2y + x2y2 + z2x2 − 8676000x2 − 140151600 yx

+ r√Qx− 22600 z2x+ 2000xy2 + y3x+ yz2x+ r

√Qy − 10800 r

√Q+ 127440000 z2

− 11800 yz2 + 125316000 y2 ],

where r =√x2 + y2 + z2 and

(5.2) Q = x4 + 25600x2y − 360x3 + 139240000 y2 − 4248000 yx

+ 1032400x2 + x2y2 + z2x2 − 23600 z2x+ 139240000 z2 .

Clearly curl(W) ·W vanishes if z = 0. This is not a physical solution, though, since theobserver’s eye is placed at the origin. And it is not an integral surface, since clearly W doesnot have constant direction (0, 0, 1) on it. Whether curl(W) ·W vanishes away from this planeis less clear. In this paper we consider the box U = [175, 185]× [−9, 9]× [−5, 5] to contain ourconstruction of our mirror. The units are in centimeters, and these numbers are chosen to betypical of the distance between a driver with eye at the origin viewing a passenger-side mirrorof typical size. Numerical evidence suggests that curl(W)·W vanishes only on the intersectionof U with the plane z = 0. This was determined by computing curl(W) · W on a lattice ofpoints U with a spacing of .2 units. This amounted to a total of 120666 lattice points in thez ≥ 0 portion of the box, which is sufficient to consider since the problem is symmetric aboutz = 0. It is interesting to compare the magnitude of curl(W) · W for our current problemwith that of the driver-side mirror problem, for which a fairly successful solution exists, asdepicted in Figure 1.

Within U , the above described numerical sampling for the passenger-side mirror gave amaximum value of 0.038 for |curl(W)·W| and an average value of 0.016. A plot of curl(W)·Wover the y-z plane with z > 0 where x has been fixed to be 180 appears in Figure 6. It isinteresting to note that for the driver-side mirror problem, the maximum of |curl(W) ·W| wasfound to be much smaller, 0.024, as was the average value of 0.010. This is numerical evidencefor why the driver-side mirror problem appears to have a solution while the passenger-sideproblem does not.

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z

y

curl(W) ·W

Figure 6. A plot of curl(W) ·W where x is fixed at 180 and −9 ≤ y ≤ 9, 0 ≤ z ≤ 5.

Our error estimate derived above, though, depends on a lower bound for curl(W) · W.This is slightly problematic here since curl(W) · W vanishes in U . But if we consider aportion of U , [175, 185] × [−9, 9] × [2.5, 5], then on this region the minimum of curl(W) ·Wis approximately .009.

For both problems, one is therefore tempted to perhaps say that curl(W) · W is smallin the region of interest. But what is a small enough value to allow for a good approximatesolution is not clear. This is the motivation for our next theorem, in which an upper boundfor curl(W) ·W gives an error formula for the difference between W and the normal field ofa surface that may be constructed according to a particular algorithm that we will describe.This formula shows that, using this construction, as curl(W)·W → 0, the error of the resultingsurfaces will also tend to zero.

One drawback to the above result, though, is that it is difficult to apply directly. For onething, knowing the difference between W and n does not completely determine the error inthe image, i.e., the distortion. For example, that depends on the orientation of W and n withrespect to each other, in addition to the angle of the incoming ray.

Finally, note that some of the complexity of W may be reduced by the following trick:changing the problem from a near-field problem to a far-field problem. The distance betweenthe mirror and the target plane k is large in comparison to the other distances in the problem,e.g., the size of the mirror. The authors have found that the values of k beyond a few meterswill give vector fields that are essentially indistinguishable for numerical calculations. It ishelpful to take the target plane to be “at infinity.” What we mean by this is that we compute

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the vector field that is the limiting vector field as k → ∞, which gives

W∞ =− (x, y, z)√x2 + y2 + z2

+(−λy/x,−1, λz/x)√λ2y2/x2 + 1 + λ2z2/x2

(5.3)

=− (x, y, z)√x2 + y2 + z2

+(−λy,−x, λz)√λ2y2 + x2 + λ2z2

,(5.4)

assuming that x > 0, so that |x| = x. W∞ is more manageable than W. For one thing, if wetake λ = 1, we have that

(5.5) W∞ =(−x− y,−y − x, 0)√

x2 + y2 + z2.

Thus, as a check of our model, in the case of λ = 1, W∞ has constant direction, namely(−1,−1, 0), i.e., the planes x = −y + x0 (flat mirrors) are all solutions, as one might hope.The authors have found that for our purposes W∞ gives essentially the same results as W.(Although we will continue to use W throughout the paper, since for numerical computationsthe simplification doesn’t make a noticeable difference.)

Notice (as one might expect) that W∞ is invariant under scaling. This is because the Invector certainly is, and whenever one moves a target to infinity, the resulting Out will also beinvariant under scaling. Thus if a single integral surface to W∞ existed, one could create anentire foliation of solutions by simply scaling that one surface. In that case curl(W∞) ·W∞would vanish everywhere. This is not the case. We have a (somewhat) compact nonzeroexpression for curl(W∞) ·W∞:

(5.6)zλ(λ− 1)

d

(√x2 + y2 + z2

√x2 + λ2y2 + λ2z2 − λz2 − λy2 + (λ+ 1)yx+ x2

),

where

(5.7) d =(x2 + z2λ2 + λ2y2

)3/2 √x2 + z2 + y2.

6. An error formula in the case of nonintegrability. Since most smooth vector fields in R3

do not have perpendicular surfaces, one may ask for the next best thing: approximate integralsurfaces. Here we will describe a means of constructing an approximate integral surface andshow how the error is related to the quantity curl(W) ·W.

Theorem 6.1. Let W : C → R3 be a differentiable vector field and let U ⊂ R

3 be an openset with compact closure. Then if we assume that W is scaled so that W = (1, U, V ), thengiven any point p of U , locally about p there is a surface x = g(y, z) whose graph contains pand lies in U with normal n = (1,−gy ,−gz) such that

U = −gyand

(6.1) E(y, z) ≡ V − (−gz) = e−∫ y0 Uxdτ

∫ y

0e∫ t0 Uxdτ (curl(W) ·W)dt.

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Remarks.1. Note that |E| = |W− n|. Additionally, if curl(W)·W = 0 in V , then the error |E| = 0,

which proves the Frobenius theorem. It should also be noted that our construction is local—itis not clear when the constructed surface may stray out of V .

2. It is possible to perform the construction using a horizontal rather than vertical initialcurve. The resulting formula is of course very similar. The two approaches are compared inthe application below to the passenger-side mirror. One could also choose other coordinatesystems to perform the construction, but we do not consider other coordinate systems here.

Proof of Theorem 6.1. Our proof is constructive, and this will allow us to compute thedifference between W and the normal n to the constructed surface. Without loss of generality,assume that U is a neighborhood of the origin. Define φ(z) to be the solution of the differentialequation

(6.2) (φ′(z), 0, 1) ·W(φ(z), 0, z) = 0,

with the initial condition φ(0) = x0. Written out this says that

(6.3) φ′(z) = −V (φ(z), 0, z).

Next, define our surface g(y, z), dependent on φ, to be the solution of the differentialequation

(6.4) gy(y, z) = −U(g(y, z), y, z),

with the initial condition that g(0, z) = φ(z). It follows from this that

(6.5) gz(0, z) = φ′(z) = −V (φ(z), 0, z),

which will play an important role later.The normal of the surface that is the graph of g has direction (1,−gy,−gz), which we

wish to compare with W(g(y, z), y, z) = (1, U(g(y, z), y, z), V (g(y, z), y, z)) . From the abovedefinition it follows that on the surface

(6.6) |W(g(y, z), y, z) − (1,−gy(y, z),−gz(y, z))| = |(0, 0, V (g(y, z), y, z) + gz(y, z))|;i.e., we want to estimate V + gz.

Note that

(6.7) gyz(y, z) = −Ux(g(y, z), y, z)gz(y, z) − Uz(g(y, z), y, z),

and so by integrating both sides with respect to y we have

(6.8) gz(y, z) =

∫ y

0−Ux(g(t, z), t, z)gz(t, z) − Uz(g(t, z), t, z)dt + gz(0, z).

Suppressing the (t, z) variables in the expression, we may rewrite this as

(6.9) gz(y, z) =

∫ y

0−Uxgz − Uzdt+ gz(0, z) + V (g(y, z), y, z) − V (g(y, z), y, z).

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Next we write V as an integral in order to combine the above and cause the appearance of acurl(W) ·W term. Thus,

(6.10) V (g(y, z), y, z) =

∫ y

0Vx(g(t, z), t, z)gy(t, z) + Vy(g(t, z), t, z)dt + V (g(0, z), 0, z)

so that (again suppressing the variables in some selected places)

gz(y, z) =

∫ y

0−Uxgz − Uzdt+ gz(0, z) +

∫ y

0Vxgy + Vydt+ V (g(0, z), 0, z) − V (g(y, z), y, z)

(6.11)

=

∫ y

0−Uxgz − Uz + Vxgy + Vydt− V,(6.12)

where the terms gz(0, z) and V (g(0, z), 0, z) cancelled each other due to (6.5). The key obser-vation is then that this is equal to

gz(y, z) = −V (g(y, z), y, z) +

∫ y

0(curl(W))(g(y, z), y, z) · (1,−gy,−gz)dt(6.13)

= −V +

∫ y

0curl(W) ·Wdt+

∫ y

0curl(W)(g(y, z), y, z) · (0, 0,−gz − V )dt.(6.14)

If we define E(y, z) = gz(y, z) + V (g(y, z), y, z), then we have that

(6.15) E =

∫ y

0curl(W) ·Wdt−

∫ y

0UxEdt.

We then differentiate with respect to y to give

(6.16) Ey = curl(W) ·W− UxE,

which is a first order differential equation for E. Note that our initial condition is E(0, z) = 0due to (6.5). Using the integrating factor

(6.17) exp

(∫ y

0Ux(g(τ, z), τ, z)dτ

),

we have that

(6.18)d

dy

(exp

(∫ y

0Uxdτ

)E

)= exp

(∫ y

0Uxdτ

)(curl(W) ·W),

which gives an exact expression for E:

E(y, z) = e−∫ y0Uxdτ

∫ y

0e∫ t0Uxdτ (curl(W) ·W)dt(6.19)

=

∫ y

0e∫ ty Uxdτ (curl(W) ·W)dt.(6.20)

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7. Applying the construction. Taking for W the vector field defined for the passenger-side mirror problem described in section 1.2, we may then apply the construction describedabove and test the result in ray-tracing simulation. So we consider a design that is a graphover the y values from −9 to 9, and the z from −5 to 5, with an initial condition at x = 180.

The problem is symmetric about the z-axis, and to get a sense of the size of curl(W) ·W,we consider the value of curl(W) · W on the y-z plane, i.e., when x = 0 and with z > 0. Aplot appears in Figure 6. For other values of x in U the results are very similar.

Figure 7. A simulated view, via ray-tracing, of a checkerboard covered wall through the passenger-sidemirror designed with the method of Theorem 6.1.

When applying the exact construction described above, the initial vertical curve φ is com-puted from (6.3) using Runge–Kutta with a step size of .05. When computing the horizontalcurves that form the rest of the surface, again we use Runge–Kutta with a step size of .05.Smaller step sizes did not change the results significantly. Using this method we generated theheight values of the surface on a 30× 30 grid, and a standard triangular mesh is formed. Thisdata is used to represent the surface in a ray-tracing simulation written in the POV-Ray scenedescription language. The simulation code reads the file consisting of the triangular mesh andsmooths the triangulation. Built-in POV-Ray primitives are used to create a scene in whichthe mirror is placed in the center of a cube shaped room with checkered walls. When executed,POV-Ray performs ray-tracing to create a simulated view of the mirror in this scene. Theresulting image appears in Figure 7. Ideally the viewer should see in the mirror a perfectwhite and green checkered pattern. The amount of distortion seems somewhat high. This isconsistent with the numerical evidence discussed above and we view this as a negative result.The error formula in Theorem 6.1 and the plot in Figure 6 suggest that it might be betterto take the initial curve to be horizontal, i.e., a function over the y-axis, rather than over thez-axis. This gives a surface whose plot appears in Figure 8. A simulation appears in Figure9. While this approach appears to give a better result than what is achieved with the vertical

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initial curve, again, we view this as a negative result.

8. Conclusions. We have demonstrated the role that the quantity curl(W) · W playsin the existence of approximately perpendicular surfaces. Our first theorem shows that ifcurl(W) ·W is bounded away from zero in a region, then we have a lower bound on the maxnorm of the error between the normal field on any surface and W. This is the first result, asfar as the authors know, of a bound on how nonperpendicular a surface must be to a givenvector field.

In our second theorem we showed that if curl(W)·W is small in a region, then there shouldbe a good approximating integral surface. We gave an exact error formula for a particularconstruction method. We then applied this method in two forms to the passenger-side mirrorproblem. The error formula explains why one result is significantly better than the other,but overall the result in a negative one. This, and previous evidence, points to an underlyingobstruction to the existence of a solution. Our first result is in this theme, although it is notclear how to directly apply it to this example to conclude what the lower bound on the opticaldistortion of any given mirror is forced to be.

x

y

z

Figure 8. A plot of a passenger-side mirror designed with the method of Theorem 6.1, but with a horizontalinitial curve, rather than a vertical one. The units are centimeters.

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Figure 9. A simulated view of a checkerboard covered wall through the passenger-side mirror designed withthe method of Theorem 6.1, but with a horizontal initial curve rather than a vertical one.

Acknowledgment. The authors would like to thank Christopher Croke.

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Documents

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