Planimeters and Isoperimetric Inequalities

Robert L. FooteWabash College

Isoperimetric ProblemGiven a fixed length for the boundary of a region in the plane, what shape has the largest area?

Answer: A circle, attributed to Dido, Queen of Carthage, Greek and Roman mythology, appears in Virgil’s Aeneid, ca. 25 BC

Dido’s problem. To enclose the largest possible area by the sea using an ox hide. Solution: Cut hide into thin strips and form a semi-circle.

Not proven until early 1900s!

How are area and perimeter related for a circle?

Isoperimetric Inequality

For every region,

Furthermore, if and only if the boundary is a circle.

First proved by Hurwitz (1902) using Fourier series.

The inequalityIs “sharp.”

• The inequality implies that circles are solutions• The sharp part implies that circles are the only solutions

This solves the isoperimetric problem

Polar and Linear Planimeters Jacob Amsler, 1854

Polar Planimeter

• The wheel rolls and slides – it measures the component of its motion perpendicular to the tracer arm.

• The area is proportional to the net roll of the wheel (not obvious!)

Linear Planimeter

Line segment sweeping out a signed area

Positive Negative

Component of dm in direction of N

A Formula for Signed Area

p

qm

N

dmMoving segment has four degrees of freedom

• Motion of midpoint (2 degrees)• Rotation about midpoint• Change length

Roll (signed distance) of a wheel at m

Planimeter: Moving segment sweeping out area with a wheel attached

p

qm

N

dmdσm

The Area Difference Theorem

If the endpoints of a moving segment each go around a region CCW, the signed area swept out by the segment is

Intuitivereason

Recall (if you’ve seen Green’s Theorem)

Proof of the Area Difference Theorem …

To show: the signed area here is

Proof of the Area Difference Theorem …

p

qm

N

dm

dp

dq

Integrates to

Integrates to 0

Planimeter: A moving segment of fixed length with a wheel attached

Location of wheel

Roll of wheel determined by λ

Now three degrees of freedom• Translation forward/backward, measured by dσλ

• Rotation about wheel, measured by dθ• Translation sideways, doesn’t contribute to dA

Consider

Consider

No rotation:

Rotation about wheel:

How a Planimeter Works!

Integrate to get area:

Ω

• Area Difference Theorem:• Right endpoint goes around ∂Ω: AR = AΩ • Left endpoint goes around no area: AL = 0• No net rotation of segment:

• Area is proportional to roll of wheel• Net roll of wheel doesn’t depend on location!

The Prytz “hatchet” Planimeter

1875, Holger Prytz, Danish mathematician and cavalry officer

Behaves like a bicycleFront wheel: tracer pointRear wheel: chisel edge

Economical alternative toAmsler’s planimeters

It can measure area!

σ is the net disp. of the chisel (red arc)

Error

To Prove: Isoperimetric Inequality

if and only if the boundary is a circle

We’ll do better:

Isoperimetric Defect

Put wheel at right endpoint (λ = 1)so it rolls along ∂Ω.

New: Have planimeter make one CCW rotation.This and size of region put geometric constraints on ℓ.

As before, AR = AΩ and AL = 0.

Trace differently …

ℓ = half-width

ℓ = circumradiusℓ = inradius

Goal:

Currently have

Really, just complete the square in ℓ and rearrange!

q

N

dq

Key Observation

is a component of ds

This impliesNeed a more geometrically

meaningful expression

Solve for and substitute …

When do we have ?

Suppose

Then

So ∂Ω is the circle.

But Ω is contained in the circle.The radius of the circumscribing circle determines ℓ.

Ω

“Sharp” part of theIsoperimetric Inequality

Isoperimetric Inequality

if and only if the boundary is a circle

• B has geometric significance• B = 0 iff the boundary is a circle

Bonnesen found several of these, 1920'sOsserman, Amer. Math. Monthly, Jan 1979

Bonnesen type of Isoperimetric Inequality

Isoperimetric Inequality in Spherical and Hyperbolic Geometries

K is the Gaussian curvature of the geometry; for a sphere.

Get equality if and only if the region is circular.

p q

pq

Segments sweep out area differently than in Euclidean geometry …

… but similar proofs work for planimeters and the isoperimetric inequality.

Amsler's Spherical Polar Planimeter, 1884

Never manufactured. One prototype built.

Many other types of isoperimetric inequalities: higher dimensions, other geometries, in physics …

A Plethora ofPlanimeters!

What’s forsale on eBay?

Thanks! Check out the planimeters on display

Construction of tracks