THE FROBENIUS INTEGRABILITY THEOREM AND …moskow/papers/curlWdotW.pdfThe Frobenius integrability...

THE FROBENIUS INTEGRABILITY THEOREM AND THEBLIND-SPOT PROBLEM FOR MOTOR VEHICLES

MEREDITH L. COLETTA,∗, R. ANDREW HICKS† , AND SHARI MOSKOW‡

Abstract. We consider the problem of designing a passenger-side automotive mirror that hasno blind-spot or distortion. While reasonably good solutions have been found for the analogousproblem for the driver-side mirror, a reasonable mirror for the passenger-side problem has not yetbeen found. Our model requires us to find surfaces perpendicular to a given vector field determinedby the data. This is in general impossible, which leads us to investigate estimates and error formu-las for approximate solutions to the problem. If the vector field does not satisfy the integrabilitycondition, we give a bound on how non-perpendicular any surface must be to the given vector field.Furthermore, we show that if the integrability condition holds approximately, then there will bea good approximating integral surface and we provide a construction method with an exact errorformula. We apply this method to the construction of a wide-angle passenger-side mirror. Our workindicates that a satisfactory passenger-side mirror may not exist.

Key words. geometric optics, optical design, integral surface

AMS subject classifications. 78A05

1. Introduction. Our problem is the design of passenger-side mirrors for motorvehicles that provide a sufficiently wide field of view so as to remove the blind-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 andminimal distortion appears in Fig. (1.1). Here we see a conventional flat driver-sidemirror compared with a aluminum prototype designed using the method described in[7]. For the passenger-side mirror no solution has yet been found that gives such asmall amount of distortion, despite the numerous attempts using different methods[6, 8, 7]. In this paper we describe a new method which gives comparable results tothese previous ones for the passenger-side problem, but for which we can provide anerror formula. We also present an estimate which is a first step in showing why it ispossible that no “good” solution to the passenger-side mirror problem exists.

As most people are intuitively aware, the issue of the driver’s field of view is acrucial one for automotive safety. According to a U.S. Department of Transportationreport, merging and lane changing accidents led to 827 fatalities and 58,000 injuriesin 2007 [1]. The crux of the problem is that flat mirrors do not provide a wide enoughfield of view. When an observer gazes at a flat mirror, the field of view, (measured interms of an angle in the horizontal plane) is exactly the same angle as produced bythe rays connecting the observer’s eye to the mirror surface. As a result, the smallerthe mirror the smaller the field of view for the observer. Additionally, if a mirroris moved away from an observer the field of view decreases. For example based onmeasurements by the authors, a driver-side mirror that is flat, and of average size ona US production sedan, yields a paltry 17◦ field of view, and an even more distantflat passenger-side mirror would yield a 5◦ field of view. A familiar solution used bytrucks and buses, where the problem is more severe due to vehicle size, is to employ

∗Gateway Ticketing Systems, 315 East Second Street, Boyertown, PA 19512([email protected] ).†Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, 19104

([email protected]).‡Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, 19104

([email protected]).

1

2 M.L. Coletta, R.A. Hicks and S. Moskow

A. B.

Fig. 1.1. A. A view of a parking lot through a conventional driver-side mirror, that hasapproximately a 17◦ field of view. B. A view of the same scene as in (A), using a mirror designedby the second author, which has a 45◦ field of view, yet has little distortion.

spherical mirrors, but these introduce considerable distortion. Thus the problem is tofind mirror shapes that yield a wide field of view without distorting the 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’seye. The basic geometry is depicted in Fig. (1.2). The essential difference betweenthese two problems is that the angle of deflection of the optical axis, θ is 90◦ in thepassenger-side case, while for the driver-side the corresponding angle ψ is close to65◦. For reasons that will become clearer below, the more extreme problem of thepassenger-side mirror results in inferior approximate solutions1

DriverObjectplane

θψ

Fig. 1.2. The geometry of the blind-spot problem.

While some countries require that driver-side mirrors be flat on production modelvehicles, many allow curved driver-side and passenger-side mirrors. The conventionalU.S. passenger-side mirror is slightly curved, giving a view of approximately 27◦.Without this curvature, the field of view would be that which is subtended by thedriver’s eye, which is typically about 5◦ for a passenger-side mirror, as mentionedabove. One drawback of convex curved mirrors is the problem of depth perception; ifthe view of the driver is increased by introducing a curved mirror then the apparentsizes of some objects in the reflection must decrease, since more of the scene is beingimaged. This results in the need for the familiar “Objects in Mirror are Closer thenthey Appear” warning that is often printed on curved mirrors. The danger is of coursethat if an object appears smaller then normal to an observer that there is a chance

1We will use the term “solution” throughout this paper to refer to a solution to our problem, i.e.a surface, which would be used as a mirror.

The Frobenius integrability theorem and the blind-spot problem for motor vehicles 3

that the driver will judge the distance to the object to be further than it really is,resulting in a collision. Nevertheless, in the US it is apparently felt by regulators thatthis is an important trade-off: a 27◦ field of view curved passenger-side mirror thatproduces distorted object shapes and sizes is considered safer than a 5◦ flat mirrorthat honestly represents object sizes. (We do not claim to solve the problem of depthperception here.) But as already mentioned, on the other hand, in the US, flat mirrorsare required on the driver-side. References for the history and scientific basis for thesedecisions are unknown to the authors.

To remove the blind-spot on the passenger-side we would want the field of viewto be about 40◦ − 45◦, and as undistorted as possible.

This paper is organized as follows: Section 2 contains two subsections - a de-scription of our model of the problem and our assumptions, followed by a descriptionof the corresponding pure mathematical problem. Next, in Section 3 we present anoverview of the relevant optical technology and its history. Section 4 contains a dis-cussion of the Frobenius theorem in the special case of R3. This is the key motivationfor the error estimates in this paper. In Section 5, we state and prove Theorem (5.1),a result (lower bound) demonstrating how unsolvable the problem is when the Frobe-nius theorem does not apply. Following this, in Section 6 we discuss how to applyTheorem (5.1), and in particular study the cases of two far-field approximations. Sec-tion 7 contains Theorem (7.1), a result (error formula) showing how good one mayexpect a mirror to be if Frobenius is not satisfied. We use this theorem along withnumerical evidence to make the case that there do not exist approximate solutions tothe passenger-side problem that are as good as for the driver-side problem. Finally,in Section 8 we see numerical simulations of mirrors designed using the constructiveproof of Theorem (7.1). These are negative results in the sense that they do notappear to be good enough mirrors for actual use on a car, but the evidence points tothe possibility that no truly good solution exists.

2. Statement of the Problem.

2.1. The Model. At this point we must discuss a small amount of optics inorder to explain our model. We work in the realm of geometric optics, where a lightray is represented by a straight line. Geometric optics is reversible, and so for ourpurposes we may think of light rays as entering an observer’s eye or as emanatingfrom that eye. We assume that the observer has a single eye, and that the eye is apoint. Viewed as a source, we would like the rays that exit the eye to reflect off ofthe passenger-side mirror and strike a designated target plane that lies behind andto the side of the vehicle.

Suppose we fix a plane in front of the driver’s eye, which we will refer to as theimage plane, through which the emanating rays must pass. We may label each rayby the point in this plane through which the ray passes. Given a mirror S, the raysthat leave the driver’s eye and reflect off of the mirror S induce a transformation TMfrom a subset of the image plane to the target plane, as in Fig. (2.1). Computing TMgiven S is a well understood forward problem. What interests us here is the inverseproblem, that is, to find a mirror S given a prescribed transformation T , such thatTM = T . In our specific problem the prescribed transformation T is essentially ascaling between the two planes. For example, a checkerboard pattern in the imageplane should be transformed onto a checkerboard pattern in the target plane by themirror.

We next restate the above inverse problem so that it is a statement about vectorfields. One wishes the rays to exit the observer’s eye, reflect off of a mirrored surface


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

M(x, y, z)

(1, y/x, z/x)

Observer’s Eye

Target Plane

Image Plane

n

Fig. 2.1. Given a mirror, S, which is viewed by an observer from behind an image plane anda target plane, the rays emanating from the eye may reflect off of S and strike the target plane. Inthat case we have a transformation TM , between a subset of the image plane and a subset of thetarget plane. Here we will always take the observers eye to be at the origin and the image plane tobe at x = 1, thinking of the mirror as having positive x-coordinates.

and go to prescribed destinations in the target plane. How a ray is reflected depends onthe normal to the surface at the point of intersection. Therefore we will be attemptingto find a surface whose normal vector field reflects the rays in the “correct” directions.

(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)

Fig. 2.2. Given a correspondence, T , between points in an image plane and points on a targetplane, one can define a vector field W that is hopefully normal to a mirror surface that realizes thecorrespondence.

To do this, we calculate what the normal to the ideal mirror containing (x, y, z)should be: one has incoming and outgoing rays, so the normal should be in the


direction of the sum of the two unit vectors In(x, y, z) and Out(x, y, z) as depicted inFig. (2.2). Here, and throughout this paper we have chosen for simplicity to have theeye at the origin (0, 0, 0) and to take the plane x = 1 to be the image plane. If onehas a line from the eye to (x, y, z) then the point in the image plane that lies on thatline is (1, y/x, z/x). This is the form of the points that T acts upon. So a ray that“exits” the eye and strikes the mirror at (x, y, z) should end up at T (1, y/x, z/x), ifit is reflecting off of a mirrored surface S that is a solution to our problem. In otherwords we have that TM = T . This tells us that

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)| . (2.1)

Therefore, if one is given T , then a vector field (which depends on the prescribedT )

W(x, y, z) = In(x, y, z) + Out(x, y, z) (2.2)

is defined at any point on a ray emanating from the eye with the exception of thosethat would cause a (rather blatant) singularity in the definition of In and Out. (Thenotation WT rather than W is entirely appropriate, but we suppress the T here forthe sake of compactness.) Consequently we have a vector field defined on a subset ofR3. Any surface contained in this subset whose normals were pointwise multiples ofW would reflect the rays striking it in such a way that the transformation T wouldbe physically realized. That is, it would solve our problem, at least for those rays thatstrike it.

Let us now consider the problem of explicitly determining T and hence the vectorfield W that would be perpendicular to an ideal passenger-side mirror. Assume thatthe mirror will pass through the point (x0, 0, 0), where we choose our units to becentimeters. A common distance in cars between the driver’s eye and the passenger-side mirror is x0 = 180 cm. The target plane will be of the form y = −k, where k > 0.Taking k = 1000 cm would be reasonable. A schematic layout appears in Fig. (2.3).Then

T (1, y, z) = (−αy + x0,−k, αz) (2.3)

is then the desired form of T . Taking α = x0 + k gives the scaling produced by aflat mirror. Thus it is convenient to generally take α = λ(x0 + k), so that λ = 1provides the field of a flat mirror, and λ > 1 gives a bigger field of view. One musttake λ to be approximately 10 in order to create a field of view of 40◦, assuming thatthe mirror surface we seek will have approximately the usual size of a passenger-sidemirror, which we take to be 18× 10 cm.

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)| (2.4)

=− (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)|(2.5)


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

Fig. 2.3. A schematic view of the layout for the coordinates of the passenger-side mirror problem.

2.2. The Mathematical Problem. The above leads us to the following ill-posed inverse problem:

Given a non-vanishing vector field W on an open set U in R3, do there exist surfacesin U that are perpendicular to W?

In this paper we require that all vector fields be at least C1 and that all surfaces beat least differentiable. (In practice they will be C∞ or better.) The problem rarelyhas a solution; for most choices of W it is ill-posed. Thus the question becomes

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 agradient, i.e W = ∇φ for some scalar function φ defined on U . In this case we knowthat W is perpendicular to the level surfaces of φ, i.e. solutions of φ(x, y, z) = C.Assuming that these solution sets are differentiable surfaces in U , then we have anexample of a foliation of U , which is a collection of surfaces that disjointly decomposeU (like the pages of a book). In general, an integral surface of W is a differentiablesurface S whose normal field has the same direction as W at each point of S, i.e.an integral surface of W is a surface that is perpendicular to W2. A foliation is adisjoint collection of integral surfaces of W whose union decomposes U . We say thatW is integrable if it is perpendicular to a foliation of U .

This paper contains the following results:

1. An estimate showing that if curl(W) ·W is bounded away from zero on U , thatW must deviate from the normal at at least one point on any given surface in U bya quantity bounded away from zero.

2This definition of integrable may sound strange, since one more commonly speaks of integralcurves of a vector field. The terminology used here comes from considering, instead of W, the thefield of planes on U that is perpendicular to W. (The plane field ‘dual’ to W.) This viewpointis useful in geometry and topology, and although we will not use it here, preferring with the morefamiliar notion of the vector field, we do however borrow the term “integral surface” for convenience.


2. An application of the above showing that no isolated exact solutions exist for thepassenger-side mirror problem.

3. A method for constructing an approximate integral surface of W with an errorformula such that the error goes to zero linearly with curl(W) ·W.

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

3. Some History of Optical Engineering: Free-form surfaces. Here wediscuss the technological heritage of our work. It is not a pre-requisite for understand-ing the application. The main point is that technological advances have been made inthe last ten years that make it possible to create optical quality surfaces of essentiallyany shape.

Optical design, which for the most part takes place in the realm of geometricoptics, traditionally made use of spherical surfaces or other conics. With the appear-ance of computer controlled machining, it became possible to make a surface on alathe with any profile, but for the most part designers continued to design systemsthat consisted of rotationally symmetric components. The extra degrees of freedomafforded by “aspheres” allow for compact designs, and we see the benefits now, forexample, in small consumer digital cameras, which have very flexible lenses, in termsof zoom and depth of field. For example, the Canon SD1000 contains six lenses, twoof which are aspheres. The next step was to consider the use of arbitrary shapedoptical elements, both mirrors and lenses. A surface that is not a surface of revolu-tion, 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 tooptical quality. Early free-form designs include a progressive spectacle lens[9] de-signed in the late 1950’s by Kanolt, which to the knowledge of the authors was notimplemented at the time. On the other hand the Polaroid SX-70 folding camera [14]is another example, and was hugely successful. In this case the molds for the lenseswere essentially made by hand. Only in the last ten years has technology existedthat can machine optical quality free-form surfaces. This technology was developedas part of the DARPA conformal optics program [10]. Since it was never possibleto fabricate these surfaces until recently, no design theory was ever developed, andlittle has been developed to date. In particular, unlike many other engineering designproblems, until recent years, optical design has rarely been formulated as a problemin partial differential equations(PDEs), probably due to the technological restrictionof having to work with spherical surfaces3. The traditional approach has been to useoptimization, but the number of parameters needed to model free-form surfaces issignificantly greater than in the rotationally symmetric case, and optimization is bestaugmented with direct methods. Certainly, many commercial optimization packageshave been unable to meet the needs of optical designers [15].

It would appear that free-form surfaces could play a role in numerous applicationsthat by their nature lack rotational symmetry, but methods for the design of free-formsurfaces are in their infancy. The design of illumination systems is one area where suchproblems often arise and has recently attracted the attention of the PDE community.Systems for illumination are examples of non-imaging optics, i.e. the goal is not toform an image but to redistribute the light from a source or sources with prescribedintensity distribution onto a target [20]. Yet the theory of controlling even a single

3Of course PDEs play an important role in optics in general. Some design instances exist, suchas in [4, 12].


point source or collimated beam is fairly complicated. Rubinstein and Wolanskyin [16] describe a means of designing free-form lenses to control the intensity of acollimated beam. For a point source, construction and existence have been consideredby Oliker and Koshegin, [13] and Oliker and Glimm [2], [3]. In general the problemof controlling multiple bundles simultaneously is unsolved, as is discussed in [20].Probably the most recent and popular application of illumination optics has been dueto the wide-spread use of light emitting diodes, which is a technology that couldpossibly help cut energy consumption drastically worldwide. Note that illuminationdesign has applications to areas such as laser beam shaping [17] and solar collectordesign [19]. A more recent application has been the design of light pipes, which areused for example to light buildings with natural light “piped in” from the roof [20].Many of these applications are very timely in that they are linked to energy efficiencyin some way.

In previous work, [6, 8], the second author showed that some design problemsfor free-form surfaces can be reduced to the problem of finding a surface which isperpendicular, or nearly perpendicular, to a given vector field. The design methoddescribed in this paper will almost always give a free-form surface.

4. The Frobenius Integrability Theorem. We do not need the general the-orem of Frobenius4. For our needs it suffices to state a version for open sets in R3:

Theorem 4.1. (Frobenius in R3) Suppose W is a C∞ vector field defined on anopen subset U of R3. Then

curl(W) ·W = 0

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

Therefore, if we are trying to design a mirror to realize a given transformation Tthen we should compute W and then check to see if curl(W) ·W = 0 in our openset of interest U . If it is, then we have found infinitely many mirrors that solve ourproblem, as each surface in the foliation will exactly realize the transformation T .

To gain intuition for the Frobenius theorem the reader should again considergradient fields. Assuming that W is defined on a simply connected open set, wethen have that curl(W) = 0 implies that W = gradφ for some φ. Then any levelsurface φ(x, y, z) = C that is in fact a true differentiable surface is a solution to theproblem. These are the integral surfaces appearing in the conclusion of Frobenius’theorem, since curl(W) = 0 of course implies that curl(W) ·W = 0. Suppose thoughthat W is not a gradient, but a multiple of a gradient by a scalar function, i.e.W = β(x, y, z) gradφ. For our application this would be satisfactory since we don’tcare about the length of W, only its direction, since that is what determines thedirection of reflected light. Hence the level surfaces of φ are still solutions since theyare perpendicular to W. If W has the form β(x, y, z) gradφ then it is unlikely thatcurl(W) = 0 and so the “curl” test does not detect this situation. Nevertheless, it isa straightforward calculation to show that if W = β gradφ then curl(W) ·W = 0.Proving the other direction amounts to proving the Frobenius’ theorem in R3.

The hypothesis that curl(W) ·W = 0 in all of U is indeed not satisfied when Warises from the passenger-side mirror problem. There are then two natural cases toconsider, which correspond to our two main results.

4The general version addresses the problem of finding integral surfaces to distributions, in thesetting of an arbitrary d-dimensional differentiable manifold Md (See Lee page 500 [11]. Stoker [18]page 392 presents another viewpoint.)


First, rather then having an entire foliation of surfaces(which is much more thanis needed), it could be that there are isolated surfaces in U that are perpendicular toW. This situation does sometimes arise. To find such surface it is natural look at thepoints in U where curl(W) ·W does vanish, i.e. to consider the equation

curl(W) ·W = 0 (4.1)

and hope that any surface in the solution set would be perpendicular to W. Unfor-tunately this approach does sometimes produce surfaces that are not perpendicularto W. That is, just because curl(W) ·W vanishes on a surface doesn’t mean thatthat surface is perpendicular to W. In fact, an example occurs when W is the vec-tor field that we derived above for the passenger-side mirror problem, as we will seebelow. It is true though that if a a surface is perpendicular to W then curl(W) ·Wmust vanish on it? For unit length vector fields, this is Corollary (5.2). Therefore, ifone does want to find isolated solution surfaces, then one should normalize W andconsider the solution set the equation (4.1). (Since W and the normalization of Ware multiples of each other, no solutions surfaces will be created or destroyed by thenormalization of W.) Additionally, if W is unit and curl(W) ·W never vanishes inU then no solutions surfaces exist of any type.

Secondly, what if curl(W) ·W 6= 0 in U , but curl(W) ·W is, in some sense, small?Does this imply the existence of good approximate solutions? Theorem (7.1) answersthis question in the affirmative.

5. Obstructions to the existence of good designs.

Theorem 5.1. Suppose that U ⊂ R3 is an open set with compact closure U . LetW : U → R3 be a differentiable vector field on U with curl(W) ·W ≥ ε > 0. LetS ⊂ U be a C1 compact, orientable surface with boundary and unit normal field n.Then

maxp∈S|W(p)− n(p)| ≥ εA

MA+ L> 0. (5.1)

where A is the area of S, L is the length of the boundary of S and M = maxp∈U |curl(W)|.

Remark If instead of the above one has that curl(W) ·W ≤ −ε < 0 on U , whereε > 0, then same conclusions follow. Generally of course, curl(W) ·W may changesign on U , which is the case in the passenger-side mirror problem.

Proof of Theorem (5.1). We begin by noting that∫S

curl(W) ·Wdσ ≥ εA (5.2)

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

maxp∈S|W(p)− n(p)| (5.3)

Applying Stoke’s theorem yields the following inequality


∫S

curl(W) ·Wdσ =

∫S

curl(W) · (W − n)dσ +

∫S

curl(W) · n dσ

≤∫S

|curl(W) · (W − n)|dσ +

∫∂S

W · ds

≤∫S

|curl(W)||W − n|dσ +

∫∂S

W · ds−∫∂S

n · ds

≤∫S

M |W − n|dσ +

∫∂S

|W − n|ds

where

M = maxp∈U|curl(W)(p)|. (5.4)

Taking

k = maxp∈S|W(p)− n(p)| (5.5)

we have that ∫S

Mkdσ +

∫∂S

kds = MkA+ kL ≥ εA > 0 (5.6)

where L is the length of ∂S.

Therefore

k = maxp∈S|W(p)− n(p)| ≥ εA

MA+ L> 0. (5.7)

�Corollary 5.2. Suppose that W is a C1 unit vector field on an open set U ⊆ R3,

and let S ⊆ U be a differentiable surface, possibly with boundary. If W is perpendic-ular to S at every point of S then curl(W) ·W must vanish on S.

Remark Corollary (5.2) also follows from basic facts about pullbacks of differentialforms (without the unit length assumption, in fact), but we choose to give a self-contained proof.

Proof. Suppose that there is a point p ∈ S at which curl(W) ·W 6= 0. Then on Sabout p, there is a small closed topological disk D, on which curl(W) ·W 6= 0, sinceW is C1. Thus on D curl(W) ·W is bounded away from zero, and so Theorem (5.1)may be applied to D, for either choice of normal field on D. Since W is perpendicularto D and is unit length, on D it should coincide exactly with one of the two choicesof normal field of D. But then with that choice of n we have that W − n is zero atevery point of D, which contradicts the inequality of Theorem (5.1). Therefore nosuch p exists. �

6. Applications of Theorem (5.1).


6.1. Passenger-side mirror. Formula (2.5) gives the vector field correspondingthe the passenger-side mirror problem. To apply Corollary 5.2, we first normalize Wto have unit length (but continue to denote it by W). The next step is to check ifcurl(W) ·W vanishes on an open set, in which case the Frobenius theorem wouldapply. The resulting formula is somewhat large:

curl(W) ·W = −5900z(P1 + r

√QS1)

(P2 + r√QS2)r

√Q

(6.1)

where r =√x2 + y2 + z2,

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

+ 1032400x2 + x2y2 + z2x2 − 23600 z2x+ 139240000 z2, (6.2)

P1 = −8676000x4 + 116640000x2y2 + 118764000 z2x2 + 2 y2z2x2 − 20600 y2z2x

+820 z2x2y−140151600 z2yx+2x3yz2 +12620x4y−140151600 yx3 +2x4y2 +2 z2x4

−33580 z2x3+12620 y3x2−8980 y2x3−140151600 y3x+y4x2+252756000 y2z2+z4x2

− 22600 z4x+x5y+ 2x3y3 + 2000xy4 +x6− 10980x5 + 125316000 y4 + 127440000 z4

− 11800 z2y3 − 11800 z4y + 2 z2y3x+ z4yx+ xy5, (6.3)

S1 = x2y + z2y + xy2 + z2x − 10800x2 − 10800 y2 − 10800 z2 + y3 + x3, (6.4)

P2 = 1032400x4 + 140272400x2y2 + 140272400 z2x2 + 2 y2z2x2 − 23600 y2z2x

+25600 z2x2y−4248000 z2yx+25600x4y−4248000 yx3+2x4y2+2 z2x4−23960 z2x3

+25600 y3x2−360 y2x3−4248000 y3x+y4x2+278480000 y2z2+z4x2−23600 z4x+x6

− 360x5 + 139240000 y4 + 139240000 z4, (6.5)

and

S2 = xy2 + z2x+ 12800 yx− 180x2 − 11800 z2 + x3. (6.6)

Since curl(W) ·W is not identically zero in an open set, the Frobenius theoremtells us that there does not exist a foliation of any open set by integral surfaces ofW, which would have provided an infinite number of mirrors each of which wouldsolve our problem. Note though, that due to the z term occurring in equation 6.1,curl(W) ·W vanishes if z = 0. This set is not an integral surface, since clearly it isnot perpendicular to W. Thus we are have with an example of a surface on whichcurl(W) ·W vanishes but which is not and integral surface. Whether W has isolatedintegral surfaces elsewhere is not clear.

We consider the box U = [175, 185]× [−9, 9]× [−5, 5] to contain the constructionof our mirror. As mentioned before, the units are in centimeters, and these numbersare chosen to be typical of the distance between a driver with eye at the origin viewinga typical size passenger-side mirror.

Numerical evidence suggests that curl(W) ·W vanishes only on the intersectionof U with the plane z = 0, and so from Corollary (5.2) we are led to believe that


no isolated surfaces exist in U that are perpendicular to W. This was determinedby computing curl(W) ·W on a lattice of points in U with a spacing of .2 cm. Thisamounted to a total of 120666 lattice points in the z ≥ 0 portion of the box, whichis sufficient to consider since the problem is symmetric about z = 0. A plot ofcurl(W) ·W over the y − z plane with z > 0 and where x has been fixed to be 180cm appears in Fig. (6.1). Empirically we have found that this plot is typical of whatresults when a value of x is chosen in the interval [175, 185].

z

y

curl(W) ·W

Fig. 6.1. A plot of curl(W) · W where x is fixed at 180 cm and −9 ≤ y ≤ 9, 0 ≤ z ≤ 5.

The max of |curl(W) ·W| is approximately 0.020 in U , which seems to be small.However, it is not clear how small we really need it to be to allow for a good mirrordesign. This is the motivation for Theorem (7.1), which gives an error formula for aparticular construction algorithm that we will describe.

6.2. Far-field approximations. In this section we observe that some of thecomplexity of W may be reduced by changing the problem from a near-field problemto a far-field problem. There are two different ways to do this.

The first approach is to assume that the driver is at a great distance from themirror, i.e. that the mirror subtends a small angular view of the driver. This is areasonable assumption for the passenger-side mirror, which as mentioned subtends a5◦ degree field of view for the driver. In such a case the approximation is going toamount to taking the In vector to be constant. The terminology used for this modelof imagine is that the projection is orthographic, as opposed to perspective. In thiscase that means choosing In = (−1, 0, 0). As we will see in Section 7, this will nothelp in solving the problem.

A second approach is to assume that the distance between the mirror and thetarget plane, k is large in comparison to the other distances in the problem, e.g. thesize 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 is helpful to take the target plane to be “at infinity”. What we mean by this is thatwe compute 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

(6.7)

=− (x, y, z)√x2 + y2 + z2

+(−λy,−x, λz)√λ2y2 + x2 + λ2z2

, (6.8)

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

W∞ =(−x− y,−y − x, 0)√

x2 + y2 + z2(6.9)

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 onemight hope. The authors have found that for our purposes W∞ gives essentially thesame results as W. (Although we will continue to use W throughout the paper sincefor numerical computations the simplification doesn’t make a noticeable difference.)

Notice, that as one might expect, that W∞ is invariant under scaling. This isbecause the In vector certainly is, and whenever one moves a target to infinity, theresulting Out will also be invariant under scaling. Thus if a single integral surface toW∞ existed, one could create an entire foliation of solutions by simply scaling thatone surface. In that case curl(W∞) ·W∞ would vanish everywhere. This is not thecase. We have a (somewhat) compact non-zero expression for curl(W∞) ·W∞:

zλ(λ− 1)

d

(√x2 + y2 + z2

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

)(6.10)

where

d =(x2 + z2λ2 + λ2y2

)3/2√x2 + z2 + y2. (6.11)

The fact that the expression (6.10) is not identically zero proves that in this far-fieldversion of our problem there are no isolated solution surfaces.

7. An error formula in the case of non-integrability. If a vector field doesnot have any integral surfaces, one may ask for the next best thing: approximateintegral surfaces. Here we will describe a means of constructing an approximateintegral surface, and show how the error is related to the quantity curl(W)·W. We seethat an upper bound for curl(W) ·W gives an error bound for the difference betweenW and the normal field of the particular surface that we construct. This formulashows that a characteristic of the given construction is that as curl(W) ·W→ 0 thenthe error of the resulting surfaces will also tend to zero.

Theorem 7.1. Let U ⊂ R3 be an open set with compact closure and let W :U → R3 be a differentiable vector field . Then if we assume that W is scaled so that

W = (1, F (x, y, z), H(x, y, z)),


Problem Max of curl(W) ·W Average of curl(W) ·WPassenger-side 0.047 0.019Passenger-side, target at infinity 0.048 0.014Passenger-side, driver at infinity 0.051 0.020Driver-side 0.012 0.005

Table 7.1A comparison of values of the maximum and averages of curl(W) ·W for four different vector

fields corresponding to four different problems, normalized according to Theorem (7.1). Note thatthe driver-side problem has considerably lower values than the others.

then given any point p of U , we have that locally about p there is a surface x = g(y, z)whose graph contains p and lies in U with normal

n = (1,−gy,−gz)

such that the first two components are exactly equal, i.e.,

F = −gyand the error in the third component is given by

E(y, z) ≡ H − (−gz) =

∫ y

0

e−∫ ytFx(g(τ,z),τ,z)dτ (curl(W) ·W)(g(t, z), t, z)dt, (7.1)

that is,

E =

∫ y

0

e−∫ ytFxdτcurl(W) ·Wdt

Remarks1. An important assumption in the theorem is that W resembles the gradient

of the graph of a function. This “normalization” prevents curl(W) and curl(W) ·Wfrom having small magnitude merely because |W| is small. (Recall that it is sufficientto solve the problem for any scaling of W. )

2. Note that |E| = |W − n|. Additionally, if curl(W) ·W = 0 in U , then theerror |E| = 0, which proves a local version of the Frobenius’ theorem.

3. The vector field W for the passenger-side problem considered in section 5.1 isnot of the required form for this theorem. Therefore one must scale it by the inverse ofthe first component (which in this case never vanishes). Given that, if we then performnumerical sampling of |curl(W) ·W| in U we obtain a maximum value of 0.047 whilefor the similarly scaled driver-side mirror problem the maximum is 0.012. For the twofar-field cases of the passenger-side problem we find that the corresponding numbersare close to the near-field case, and so making these approximations is not clearlyhelpful from a numerical standpoint. A comparison of four different problems/vectorfields appears in Table (7.1).

Proof of Theorem (7.1). Our proof is constructive, and this will allow usto compute the difference between W and the normal n to the constructed surface.WLOG, assume that U is a neighborhood of the origin. Define our initial curve φ(z)to be the solution of the differential equation

(φ′(z), 0, 1) ·W(φ(z), 0, z) = 0, (7.2)


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

φ′(z) = −H(φ(z), 0, z) (7.3)

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

gy(y, z) = −F (g(y, z), y, z), (7.4)

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

gz(0, z) = φ′(z) = −H(φ(z), 0, z), (7.5)

which will play an important role later.

The normal of the surface that is the graph of g has direction (1,−gy,−gz), whichwe wish to compare with W(g(y, z), y, z) = (1, F (g(y, z), y, z), H(g(y, z), y, z)) . Fromthe above definition it follows that on the surface

|W(g(y, z), y, z)− (1,−gy(y, z),−gz(y, z))| = |(0, 0, H(g(y, z), y, z) + gz(y, z))|, (7.6)

i.e. we want to estimate H + gz.Note that

gyz(y, z) = −Fx(g(y, z), y, z)gz(y, z)− Fz(g(y, z), y, z), (7.7)

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

gz(y, z) =

∫ y

0

−Fx(g(t, z), t, z)gz(t, z)− Fz(g(t, z), t, z)dt+ gz(0, z). (7.8)

Suppressing the (t, z) variables in the expression, we may re-write this as

gz(y, z) =

∫ y

0

−Fxgz − Fzdt+ gz(0, z) +H(g(y, z), y, z)−H(g(y, z), y, z). (7.9)

Next we write H as an integral in order to combine the above, and cause the appear-ance of a curl(W) ·W term. Thus

H(g(y, z), y, z) =

∫ y

0

Hx(g(t, z), t, z)gy(t, z) +Hy(g(t, z), t, z)dt+H(g(0, z), 0, z)

(7.10)So that (again suppressing the variables in some selected places)

gz(y, z) =

∫ y

0

−Fxgz − Fzdt+ gz(0, z) +

∫ y

0

Hxgy +Hydt+H(g(0, z), 0, z)−H(g(y, z), y, z)

(7.11)

=

∫ y

0

−Fxgz − Fz +Hxgy +Hydt−H (7.12)


where the terms gz(0, z) and H(g(0, z), 0, z) cancelled each other due to (7.5). Thekey observation is then that this is equal to

gz(y, z) = −H(g(y, z), y, z) +

∫ y

0

curl(W)(g(y, z), y, z) · (1,−gy,−gz)dt (7.13)

= −H +

∫ y

0

curl(W) ·Wdt+

∫ y

0

curl(W)(g(y, z), y, z) · (0, 0,−gz −H)dt

(7.14)

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

E =

∫ y

0

curl(W) ·Wdt−∫ y

0

FxEdt (7.15)

We then differentiate with respect to y to give

Ey = curl(W) ·W − FxE, (7.16)

which is a first order differential equation for E. Note that our initial condition isE(0, z) = 0 due to (7.5). Using the integrating factor

exp

(∫ y

0

Fx(g(τ, z), τ, z)dτ

)(7.17)

we have that

d

dy

(exp

(∫ y

0

Fxdτ

)E

)= exp

(∫ y

0

Fxdτ

)(curl(W) ·W), (7.18)

which gives an exact expression for E

E(y, z) = e−∫ y0Fxdτ

∫ y

0

e∫ t0Fxdτ (curl(W) ·W)dt (7.19)

=

∫ y

0

e−∫ ytFxdτ (curl(W) ·W)dt. (7.20)

�Remark. It is possible to perform the construction such that H = −gz, and theerror is in the second component. The resulting formula is of course very similar.This corresponds to taking an initial curve φ(y) instead of φ(z). The two approachesare compared in the application below to the passenger-side mirror. One could alsochoose other coordinate systems to perform the construction, but we do not considerother coordinate systems here.


Fig. 8.1. A simulated view, via ray-tracing, of a checkerboard covered wall through thepassenger-side mirror designed with the method of Theorem (7.1).

8. Applying the construction. Taking for W the vector field defined for thepassenger-side mirror problem described in section 2.1, then we may apply the con-struction described above and test the result in ray-tracing simulation. So we considera design that is a graph over the y values from −9 cm to 9 cm, and the z from −5cm to 5 cm with an initial condition at x = 180.

When applying the exact construction described above, the initial (vertical) curveφ(z) is computed from equation (7.3) using Runge-Kutta with a step size of .05.When computing the horizontal curves that form the rest of the surface, again we useRunge-Kutta with a step size of .05. Smaller step sizes did not change the resultssignificantly. Using this method we generated the height values of the surface on a30x30 grid and a standard triangular mesh is formed. This data is used to representthe surface in a ray-tracing simulation written in the POV-Ray scene descriptionlanguage. The simulation code reads the file consisting of the triangular mesh andsmooths the triangulation. Built-in POV-Ray primitives are used to create a scenein which the mirror is placed in the center of a cube shaped room with checkeredwalls. When executed, POV-Ray performs ray-tracing to create a simulated view ofthe mirror in this scene. The resulting image appears in Fig. (8.1). Ideally the viewershould see in the mirror a perfect white and green checkered pattern. The amountof distortion seems somewhat high. This is consistent with the numerical evidencediscussed above and we view this as a negative result. The error formula in Theorem(7.1) and the plot in Fig. (6.1) suggest that it might be better to take the initialcurve to be horizontal, i.e. a function over the y-axis, rather than over the z-axis.This gives a surface whose plot appears in Fig. (8.2). A simulation appears in Fig.(8.3). While this approach appears to give a better result than what is achieved withthe vertical initial curve, again, we view this as a negative result.

9. Conclusions. We have demonstrated the role that the quantity curl(W) ·Wplays in the existence of exact and approximate integrals surfaces. Our first theorem


x

y

z

Fig. 8.2. A plot of a passenger-side mirror designed with the method of Theorem (7.1), butwith a horizontal initial curve φ(y), rather than a vertical one φ(z). The units are centimeters.

shows that if curl(W) ·W is bounded away from zero in a region, then we have alower bound on the max norm of the error between the normal field on any surfaceand W. This is the first result, as far as the authors know, of a bound on how non-perpendicular a surface must be to a given vector field. A corollary provides a toolfor finding isolated surfaces or ruling out their possible existence.

In our second theorem we showed that if curl(W) ·W is small in a region, thenthere should be a approximating integral surface whose error depends on curl(W) ·W. We give an exact error formula for a particular construction method. We thenapplied this method in two forms to the passenger-side mirror problem. The errorformula explains why one mirror is significantly better than the other, but overallthe simulations are a negative result, that is consistent with earlier indications thatthe problem of designing a “good” passenger-side mirror is impossible. This, andprevious evidence points to an underlying obstruction to the existence of a satisfactorypassenger-side mirror.

Acknowledgments. M. Coletta acknowledges support from NSF IIS-0413012.R.A. Hicks thanks Christopher Croke and acknowledges support from NSF IIS-0413012and NSF DMS-0908299. Shari Moskow acknowledges support from NSF DMS-0605021.

REFERENCES


Fig. 8.3. A simulated view of a checkerboard covered wall through the passenger-side mirrordesigned with the method of Theorem (7.1), but with a horizontal initial curve φ(y) , rather than avertical one φ(z).

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