**Robert L. Foote**

**Selected Papers**

Written for the conference "Exploring Undergraduate Algebra & Geometry with Technology" held at DePauw University, June 1996.

with L. D. Drager, C. F. Martin, and J. Wolper,

Tractrices,
Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old
Conjecture

with M. Levi and S. Tabachnikov

*Amer. Math. Monthly*, 120 (2013)
199-216.

with M. Levi and S. Tabachnikov

Geometry of the tracks left by a bicycle is closely related to the so-called Prytz planimeter and with linear fractional transformations of the complex plane. We describe these relations, along with the history of the problem, and give a proof of a conjecture made by Menzin in 1906. Reprint

Infinitesimal
Isometries
Along Curves and Generalized Jacobi Equations

with C.K. Han and J.W. Oh

J Geometric Analysis, 23
(2013) 377–394

**The Volume Swept Out by a Moving Planar Region
**

A result is presented, evidently due to Courant, about the connection between the volume swept out by a moving planar region and the motion of its centroid. This generalizes both the Theorem of Pappus and Cavalieri's Principle. Reprint

**The Dynamics of Pendulums on Surfaces of
Constant Curvature**

with P. Coulton and G. Galperin

**Integral-Geometric Formulas for Perimeter in S^{2},
H^{2}, and Hilbert Planes**

with Ralph Alexander and I.D. Berg

Rocky Mountain Journal of Mathematics, 35 (2005) 1825-1860

**Geometry of the Prytz Planimeter**

*Reports on Mathematical Physics*, 42 (1998) 249-271

The configuration space of the planimeter is a non-principal circle bundle acted on by SU(1,1) ( » SL(2,R)). The motion of the planimeter is realized as parallel translation for a connection on this bundle and for a connection on a principal SU(1,1)-bundle. The holonomy group is SU(1,1). As a consequence, the planimeter is an example of a system with a phase shift on the circle that is not a simple rotation.

There is a qualitative difference in the holonomy when tracing large regions as opposed to small ones. Generic elements of SU(1,1) act on S

Presented at the Pacific Insitute for the Mathematical Sciences Workshop on Non-holonomic Constraints in Dynamics, Calgary, August 1997, and at the Lehigh University Geometry & Topology Conference, June 1998.

**Planimeters and Isoperimetric Inequalities on Constant
Curvature Surfaces**

In preparation. Preliminary
version
available.

Abstract.A planimeter is a mechanical instrument used to determine the area of a region in the plane. As the boundary of the region is traced, a wheel attached to the instrument partially rolls and partially slides, recording a component of its motion on the plane. The area of the region is a simple function of the net roll of the wheel. We show how the analogue of this instrument works on the sphere and the hyperbolic plane, and then use the results to give a simple proof of one of the Bonnesen isoperimetric inequalities for these surfaces. The rolling of the wheel can be interpreted as parallel translation for a connection in a certain bundle over the surface. The isoperimetric inequality can then be viewed as a statement about the holonomy of this connection.

Presented at Lehigh Geometry/Topology Conference, June 1997

**Dynamic, Interactive Hyperbolic Geometry with PoincaréDraw**

An account of the motivation for and development of the computer programPoincaréDraw.Unpublished. Written for the conference "Exploring Undergraduate Algebra & Geometry with Technology" held at DePauw University, June 1996.

**Close Encounters with Mathematical Cranks of the Third Kind**

review of

*Mathematical Cranks* and *A Budget of Trisections *(renamed
*The Trisectors*) by Underwood Dudley

*Skeptical Inquirer*,
Vol.18, Winter 1994, 182-186.

For better or for worse, pseudo-mathematics does not share the popularity among cranks that other subjects seem to have. It just doesn't have the same appeal as parapsychology, astrology, or UFOs. Nevertheless, mathematical cranks do exist, as mathematics departments at many major universities are well aware. When I was a graduate student at the University of Michigan, it was not uncommon for the departmental mailboxes to be stuffed periodically with the local crank's most recent proof of the rationality of p or the square root of 2.

While most crank mathematics winds up in the wastebasket (frequently after serving as departmental entertainment), occasionally a well-meaning mathematician makes the mistake of responding to a crank, pointing out his errors, hoping to set him straight. Thus may begin a series of communications between crank and mathematician (mostlyfromcranktomathematician), usually ending when the mathematician is forced to be rude, or the crank becomes angry and goes in search of a more "tolerant" mathematician. The moral of the story is, don't get involved with a crank.

While frequently giving this advice to others, Underwood Dudley, professor of mathematics at DePauw University in Greencastle, Indiana, has made it his professional hobby to collect the works of modern mathematical cranks, to correspond with them, and even to meet them. In these two books, he shares his fascinations and frustrations with people bent on pseudo-mathematics.

What types of things do mathematical cranks work on? Most of Dudley's examples can be placed into two broad categories: 1) real mathematics that has been twisted, misunderstood, or has had its significance horribly exaggerated, and 2) impenetrable nonsense.Excerpt from the review.

**Vector Bundles Over Homogeneous Spaces and Complete Locally
Symmetric Spaces**

with Lance
D.
Drager

*Proceedings of Symposia in Pure Mathematics*, 54
(1993), Part 2, Green and Yau, eds., 183--189.

**A Geometric Solution to the Cauchy Problem for the
Homogeneous Monge-Amperè Equation**

Proc. Workshops
in Pure Math. Vol. 11,
1991, 31--39, Korean Acad. Council.

**Observing the Heat Equation on a Torus Along a
Dense Geodesic**

with L. D. Drager and C. F. Martin

Sys. Sci. and Math. Sci., 4 (1991) 186--192

**Abstract.**
Several
authors have considered observability problems for the heat
equation and related partial differential equations. A basic
problem is to determine what kinds of sampling provide sufficient
information to uniquely determine the initial heat distribution.
We address the case where the temperature is measured while
traveling along a curve.

We consider the special
case where the space is a flat torus (of arbitrary dimension) and
the curve is a geodesic. It is shown that, in this case, the
observed temperature is sufficient information to uniquely
determine the initial heat distribution if and only if the
geodesic is dense in the torus.

In the case of a torus,
Fourier analysis techniques can be used to write down the solution
of the heat equation. This allows us to derive an explicit
representation of the observed temperature in terms of the initial
distribution. We use this representation and some ideas from the
theory of almost periodic functions to show that the Fourier
coefficients of the initial distribution can be recovered from the
observation.

**Homogeneous Complex Monge-Amperè Equations and
Algebraic Embeddings of Parabolic Manifolds**

*Indiana Univ Math. J.*, 39 (1990) 1245-1273

**Controllability of Linear Systems, Differential Geometry of
Curves in**

**Grassmannians and Generalized Grassmannians, and Riccati
Equations**

with Lance D. Drager, Clyde F. Martin, and James Wolper

*Acta Appl. Math.*, 16 (1989) 281-317

**Observing Ergodic Translations on Compact Abelian Groups
with Discontinuous Functions**

with L.D. Drager and D. McMahon

IMA J. Math.
Control & Info., 6 (1989)
441--463

**Abstract.**
Several authors have considered the problem of global
observability for minimal dynamical systems, and particularly the
example of ergodic (equivalently, minimal) translations on compact
abelian groups. The global observability problem for ergodic
translations is considered here in the case where the output
function may be discontinuous. The result for continuous output
functions, essentially due to McMahon, is described. This result
shows that a continuous output function observes all ergodic
translations if and only if it has no nontrivial symmetries. This
result is extended here in two directions.

First, functions
continuous except on a meagre set and symmetries modulo
meagre sets are considered. A result analogous to the continuous
case is obtained that includes and generalizes the continuous case
and results of Balogh, Bennett, and Martin on observing ergodic
translations with the characteristic functions of certain subsets
of the group. Functions continuous except on a set of measure zero
are also considered.

Secondly, ergodic
theory and harmonic analysis techniques are employed to consider
the case of integrable output functions and symmetries modulo sets
of measure zero. It is shown that an integrable function observes
ergodic translations modulo sets of measure zero if and only if it
has no nontrivial symmetries modulo sets of measure zero.

The group of symmetries
of the output function modulo sets of measure zero can be
determined from its Fourier series. The results here show that
classical observability can be determined from the Fourier series
when the function is continuous almost everywhere, but only
observability modulo sets of measure zero can be determined when
the output function is merely integrable.

**Differential Geometry of Real Monge-Amperè Foliations**

*Mathematische Zeitschrift*, 194 (1987) 331-350

Abstract.The foliations studied in this paper are those that arise from the Monge-Amperè condition requiring the Hessian of a function defined on an affine manifold to have rank that is one less than maximal. The real, homogeneous Monge-Amperè equation is detH(u) = 0, whereuis a real-valued function on an open subset ofR. Solutions that satisfy rank^{n}H(u) =n-1 induce a foliation of the domain ofuby lines. The standard solution isu(x) = ||x||, which satisfies rankH(u) =n-1. The standard solution is characterized in three ways using the geometric "twist" of the foliation, the covariant derivatives ofH(u), and strictly convex exhaustions of the domain ofu. The Cauchy problem for detH(u) = 0 is treated.

**The Contraction Mapping Lemma and the Inverse Function
Theorem in Advanced Calculus**

with Lance D. Drager

*Amer. Math. Monthly*, 93 (1986) 52-54

Abstract.The most elegant and general proof of the inverse function theorem uses the contraction mapping lemma. In an advanced calculus course, time does not generally permit a proof of the contraction mapping lemma as it is usually formulated. We give a simple proof of the contraction mapping lemma inRbased on the max/min principle, familiar to calculus students -- a continuous, real-valued function defined on a non-empty, closed, bounded subset of^{n}Rattains its maximum and minimum on that set -- thus allowing the general proof of the inverse function theorem usually left for more advanced courses.^{n}

**Regularity of the Distance Function**

*Proc. Amer. Math. Soc.*, 92 (1984) 153-155

**Curvature Estimates for Monge-Amperè Foliations**

Ph.D. Dissertation, University of Michigan, 1983

**Abstract.**
The foliations studied in this paper are those that arise from the
Monge-Amperè condition that requires the Hessian of a
real-valued function to have rank that is one less than maximal.

In the first chapter,
the geometry of the foliations of R^{n} is studied. The
main theorems are the two characterizations of the standard
foliation.

In the second chapter,
the foliations of complex manifolds are studied. The *locally
Reinhardt* foliations are those to which the geometry and
methods of the first chapter can be applied directly. Hörmander's L^{2}-∂ technique
is used to relate the geometry and the analysis of a Stein
manifold that admits such a foliation outside a compact set. The
main result is the embedding theorem of the last section.

Last updated 14 May 2014