DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 245–248

AREA AND LENGTH FOR ROULETTES VIA CURVATURE

Momčilo Bjelica

Abstract. Formula for the area bounded by a roulette and centroid of curvatures are given.

Introduction

A roulette R(c, l, P) is a curve traced out by a point P in fixed position with respect to a rolling curve c which rolls without slipping along a fixed base curve l. Area bounded by the roulette and base curve, area R(c, l, P) in some cases is minimal if tracing point is barycenter B of curve c (which is) pondered by a difference of curvature measures κ_{c} − κ_{l}. Moreover,

area R(c, l, P) = area R(c, l, B) + area sector(BP, κ_{c}^{T} − κ_{l}^{T}) + area □P C_{1}BC_{2},

where the sector is circular with radius BP and the angle of rotation of curve c, i. e., difference of total curvatures κ_{c}^{T} − κ_{l}^{T}; also c = C_{1}C_{2}^{⌢}. If the rolling curve is closed, then the last addend vanishes and points from the circle with center B trace equal areas. By this formula the areas for cycloid, cardioid, epi and hypocycloid, trochoid etc. simply achieve. Given formula follows from

area R(c, l, P) = area coneP c + \frac{1}{2} ∫_{c∋X} P X^{2} d(κ_{c} − κ_{l}),

1991 Mathematics Subject Classification. 53A04.

Key words and phrases. Roulette, area, barycenter of curvatures.

the coneP c is bounded by c and straight segments P C_{1} and C_{2}P. For an ellipse e and straight line L obtains area R(e, L, P) = π(a^{2} + b^{2} + OP^{2}), a and b are half-axes, and O is center of the ellipse, cf. E. Ehrhart [1]. For symmetric closed curves κ_{c} = −κ_{l}, the cardioid—cycloid proportion holds: area R(c, l, P) + area c = 2area R(c, L, P). The area between two roulettes traced by points P and Q is

area R(c, l, P Q) = \frac{1}{2}(κ_{c}^{T} − κ_{l}^{T})(BP^{2} − BQ^{2}).

Finally,

length R(c, l, P) = ∫_{c∋X} P X d(κ_{c} − κ_{l}).

Definitions

Let c = C_{1}C_{2}^{⌢} and l = L_{1}L_{2}^{⌢} be oriented piecewise regular curves with parametric equations r(t) = (x(t), y(t)), a ≤ t ≤ b, and ρ(t) = (u(t), v(t)), c ≤ t ≤ d, respectively. Tangent vector ṙ(t) = (ẋt), ẏ(t)) at regular point X(t) is non-zero. A parametric representation is regular if vector function r is of class C^{1} and ṙ(t) ≠ 0 for all a ≤ t ≤ b. Tangent vectors can be translated so that they have the same origin. The total curvatre κ_{c}^{T} is difference θ_{1}−θ_{0} in the values of inclination θ of the tangent to the curve at the end points of the curve. For closed curve c (C_{1} = C_{2}) it holds κ_{c}^{T} = 2zπ, z is an integer. Natural parameter is length of arc s = ∫ |ṙ(t)| dt; then tangent vectors have length 1. If c is smooth in some neighborhood of a point X, then curvature of the curve c at the point X is

κ(X) = lim \frac{Δα}{Δs} = \frac{dα}{ds},  Δs → 0,  Y → X,

Δα is angle between tangent vectors at X and Y, Δs is length of arc XY^{⌢} (limes of quotient of tangent inclination and increase of curve length). Classical formula for curvature is

κ(X) = \frac{ẋ(t)ÿ(t) − ẍ(t)ẏ(t)}{(ẋ(t)^{2} + ẏ(t)^{2})^{3/2}}.

At singular point S the point curvature is κ(S) = ∡(ṙ_{−}(S), ṙ_{+}(S)), so that −π ≤

κ(S) ≤ π. We take κ(S) = ±π if lim ∠(ṙ_{−}(S), SX^{⌢}) = ±π, X → S, x ∈ S_{2}^{⌢}. If the

whole arc SC_{2}^{⌢} is straight segment opposite to ṙ_{−}(s), we can choose either π or −π. Mention that curvature κ as function of length of arc is natural equation of curve. If two curves have equal curvatures as functions of length, then they can coincide by motion.

Main results

Definition 1. Curvature measure κ on curve c is defined on regular segments by the curvature functional κ(t), and at singular points the point-measure is equal to point-curvature.

Definition 2. Center B of curvature of curve c is barycenter of curve c which is by the curvature measure κ pondered.

Hence, κ_{c}^{T} = ∫_{c} dκ. Alternative terms for B are barycenter (centroid) of curvature or κ-center of curve. Obviously, if c is axially symmetric curve, then B belongs to the axis of symmetry; also B ∈ convc. κ_{c}^{−}κ_{l}^{T} = ρ is the angle of rotation of curve c.

Let R(c, l, P) = P_{1}P_{2}^{⌢}, and let l = L_{1}L_{2}^{⌢} where L_{1} and L_{2} are first and last contact point respectively. The area R(c, l, P) is oriented area L_{1}L_{2}P_{2}P_{1} bounded

by the base curve l = L_{1}L_{2}^{⌢}, the roulette P_{2}P_{1}^{⌢}, and straight segments L_{2}P_{2} and P_{1}L_{1}.

Theorem 1. Let B be a barycenter of curve c by difference of curvature measures κ_{c} − κ_{l} pondered, and let κ^{T} be total curvature. Then, area bounded by a roulette and base curve l is

area R(c, l, P) = area cone P c + \frac{1}{2} ∫_{c} P X^{2} d(κ_{c} − κ_{l}),  X ∈ c

= area R(c, l, B) + \frac{κ_{c}^{T} − κ_{l}^{T}}{2} BP^{2} + area □P C_{1}BC_{2}.

For closed rolling curve area cone P c = area c and area □P C_{1}BC_{2} = 0.

For closed rolling curve c points from the circle with center B trace roulettes which bound equal areas.

The equivalence between two given formulas from theorem 1. follows from

∫_{c} P X^{2} dκ_{c} = ∫_{c}(P B^{2} + BX^{2}) dκ_{c} = κ_{c}^{T} P B^{2} + ∫_{c} BX^{2} dκ_{c}.

This theorem is used in solving the problem [1] proposed by E. Ehrhart, Université de Strasbourg: ”The curve traced out by a fixed point of a closed convex curve as that curve rolls without slipping along a second curve will be called a ”roulette”. Let S be the area of one arc of a roulette traced out by an ellipse of area s rolling on a straight line. Prove or disprove that S ≥ 3s, with equality only if the ellipse is a circle.”

Let x = a cos t, y = b sin t, 0 ≤ t ≤ 2π be parametric equations of an ellipse, L be a straight line, and let BP = h. Then

area R(e, L, P) = area R(e, L, B) + \frac{1}{2}2πBP^{2}

= area e + \frac{1}{2} ∫_{0}^{2π} \frac{a^{2} cos^{2} t + b^{2} sin^{2} t}{a^{2} sin^{2} t + b^{2} cos^{2} t} | \begin{matrix}−a sin t & b cos t \\ −a cos t & −b sin t \end{matrix} | dt + πh^{2}

= πab + 2ab [\frac{a^{2} + b^{2}}{ab} arctan \frac{a}{b} tan t − t]|_{0}^{π/2} + πh^{2}

= π(a^{2} + b^{2} + h^{2}).

For cycloid, trochoid, epicycloid, hypocicloid, cardioid, and evolvent of circle, the areas are simply obtained.

The area under the cycloid cusp is 3πr^{2}, and the area bounded by a cardioid is 6πr^{2}. This relation holds in more general case.

Corollary. Cardioid—cycloid proportion. Let c and l be closed symmetric curves κ_{c}(t) = −κ_{l}(t), a ≤ t ≤ b, and L be a straight line, then

area R(c, l, P) + area l = 2area R(c, L, P).

From corollary and solution of the problem one obtains the area bounded by an epiellipse.

Theorem 2. Let area R(c, l, P Q) be the area traced by a segment P Q in fixed position with respect to the rolling curve c, i. e., the area between two roulettes traced by points P and Q which ends are connected by straight segments. Then

area R(c, l, P Q) = \frac{1}{2}(κ_{c}^{T} − κ_{l}^{T})(BP^{2} − BQ^{2}).

Especially,

area R(c, l, BP) = \frac{1}{2}(κ_{c}^{T} − κ_{l}^{T})BP^{2}.

Note that

area R(c, l, P) = area R(c, l, B) + area R(c, l, BP) + area □P C_{1}BC_{2}.

References

1. E. Ehrhart, Problem 10254, Amer. Math. Monthly, no. 8, 99 (1992), 782.

University of Novi Sad ”M. Pupin”, Zrenjanin 23000 Yugoslavia