MATEMATIQKI VESNIK 49 (1997), 23–26

UDK 514.752.2

originalni nauqni rad research paper

PEDALS, AUTOROULETTES AND STEINER’S THEOREM

Momčilo Bjelica

Abstract. The area of the autoroulette of a closed curve is double the area of the roulette of that curve on a straight base line taken with respect to the same generating point. This is an equivalent to Steiner’s theorem for pedals. Also, an extension of Steiner’s theorem is asserted.

1. Pedals and Steiner’s theorem

Definition 1. The pedal of a given curve with respect to a point P is the locus of the foot of the perpendicular from P to a variable tangent line to the curve.

Steiner’s theorem states as follows.

Theorem 1. [2], [3] When a closed curve rolls on a straight line, the area between the line and the roulette generated in a complete revolution by any point on the rolling curve is double the area of the pedal of the rolling curve, this pedal being taken with respect to the generating point.

The arc of the roulette on a straight base line and the arc of the corresponding pedal have equal lengths.

For the sake of simplicity, Steiner’s theorem have be stated only when P lies on the curve. This is not an essential restriction. In Theorem 4 an extension is given to the non-closed curves.

2. Roulettes

Definition 2. The roulette R(c, l, P) is the 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.

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)) and ρ(t) = (u(t), v(t)), a ≤ t ≤ b, respectively. A parametric representation is regular if vector function r is of class C^{1} and ṙ(t) = (ẋ(t), ẏ(t)) ≠ 0 for all a ≤ t ≤ b. Tangent vectors ṙ can be translated so that they have same origin. The total curvature κ_{c}^{T} is the 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 the closed curve we have κ_{c}^{T} = 2zπ, where z is an integer. The natural parameter is the length of the arc s = ∫ |ṙ(t)| dt, in which case tangent vectors have length 1. If c is smooth in some neighborhood of a point X, then the curvature κ(X) of c at X is

AMS Subject Classification: 53 A 04

Keywords and phrases: Pedal, autoroulettes, barycenter of curvatures

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

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

where Δα is the angle between tangent vectors at X and Y, Δs is the length of the arc XY^{⌢}. The classical formula for the curvature is

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

At a singular point S ∈ c 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 ∈ SC_{2}^{⌢}.

If the whole arc SC_{2}^{⌢} is a straight segment opposite to ṙ_{−}(s), we can choose either π or −π. Mention that the curvature κ as a function of the length of the arc is the natural equation of the curve.

Definition 3. The curvature measure κ on a curve c is defined at the regular points by the curvature functional κ(t), and at the singular points the point-measure is equal to the point-curvature. The centroid B of curvature of a curve c is the barycenter of c while c is by the curvature measure κ pondered.

Note that κ_{c}^{T} = ∫_{c} dκ. Also, the difference κ_{c}^{T} − κ_{l}^{T} is equal to the angle ρ of rotation of c in tracing the roulette.

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

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

Theorem 2. [1] Let B be the barycenter of a curve c, let c be pondered by the difference of curvature measures κ_{c} − κ_{l}, and let κ^{T} denote the total curvature. Then, the area bounded by the roulette and the 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 the closed rolling curve area cone P c = area c and area □P C_{1}BC_{2} = 0. Further,

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

If c is a closed curve then points from the circle with center B trace roulettes with equal areas. Note that (κ_{c}^{T} − κ_{l}^{T})BP^{2}/2 = area sector (BP, κ_{c}^{T} − κ_{l}^{T}), where the sector is circular, with radius BP, and which angle is equal to the angle of rotation of c.

3. Autoroulettes

Recall that the area under a cycloid cusp is 3πr^{2}, and the area bounded by a cardioid is 6πr^{2}. This cardioid-cycloid proportion also holds in the more general case.

Definition 3. The autoroulette AR(c, P) is the roulette R(c, l, P), where c and l are symmetric curves κ_{l}(t) = −κ_{c}(t), a ≤ t ≤ b.

If c is a closed curve, then AR(c, P) is closed also, and, obviously,

area AR(c, P) = area l + area R(c, l, P).

Theorem 3. Let c and l be closed symmetric curves κ_{l}(t) = −κ_{c}(t), a ≤ t ≤ b, and let L be a straight line; then

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

Proof. By Theorem 2 we have

area AR(c, P) = area l + area R(c, l, P)

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

= 2 (area c + \frac{1}{2} ∫_{c} P X^{2} dκ_{c})

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

Theorem 3 is an equivalent to Steiner’s Theorem 1.

Proposition. The autoroulette of a closed curve is double the corresponding pedal and

area pedal : area straight-roulette : area autoroulette = 1 : 2 : 4
length pedal : length straight-roulette : length autoroulette = 1 : 1 : 2.

4. Extensions

Theorem 4. (a) Extended straight roulette — autoroulette proportion. Let c be piecewise regular curve and L be a straight line, then

area AR(c, P) − area cone P c = 2 [area R(c, L, P) − area cone P c] .

(b) Extended Steiner’s theorem. When a piecewise regular curve rolls on a straight line, the area between the line and the roulette, whereby from the end points of the roulette are taken perpendiculars to the line, is double the area of the cone which vertex is P and which base is the pedal.

Proof. (a) It follows immediately from Theorem 2.

(b) Let c = C_{1}C_{2}^{⌢}, the pedal denote by p = P_{1}P_{2}^{⌢}, and let AR(c, P) = A_{1}A_{2}^{⌢}. Notice that the all areas are oriented.

area R(c, L, P) = \frac{1}{2} [area AR(c, P) + area cone P c]

= \frac{1}{2} area P C_{1}A_{1}A_{2}^{⌢}C_{2}P

= \frac{1}{2} [ area coneP (A_{1}A_{2}^{⌢}) + 2ΔP C_{1}P_{1} − 2ΔP C_{2}P_{2}]

= \frac{1}{2} [4 area cone P p + 2ΔP C_{1}P_{1} − 2ΔP C_{2}P_{2}]

= 2 area cone P p + ΔP C_{1}P_{1} − ΔP C_{2}P_{2}.

If Area R(c, L, P) denote the area bounded by the roulette, the base straight line and two perpendiculars from the end points of the roulette to the line, then

area R(c, L, P) = area R(c, L, P) + ΔP C_{1}P_{1} − ΔP C_{2}P_{2},

so that

Area R(c, L, P) = 2 area cone P p,

what ends the proof.

These two definitions of the area of roulettes, by connecting the endoints of the roulette by the endpoints of the base straight line, and taking perpendiculars from the end points to the base straight line, for closed curves give equal values.

REFERENCES

[1] M. Bjelica, Area and length for roulettes via curvature, Differential geometry and applications, Proc. Conf. 1995, Brno, 1996, 245–248.

[2] E. Goursat, Cours D’Analyse Mathématique, Tome I Gauthier-Villars et C^{ie}, Paris, pp. 240.

[3] M. S. Klamkin, Solution of problem 10254, Amer. Math. Monthly, 102, 4 (1995), 263–264.

(received 27.12.1996.)

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