MATEMATIQKI VESNIK UDK 514.752.2
originalni nauqni rad
49 (1997), 23{26
research pap er
PEDALS, AUTOROULETTES AND STEINER'S THEOREM
Momcilo 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 resp ect to the same generating p oint. This is an
equivalent to Steiner's theorem for p edals. Also, an extension of Steiner's theorem is asserted.
1. Pedals and Steiner's theorem
Definition 1. The p edal of a given curvewith resp ect to a point P is the
lo cus of the fo ot of the p erp endicular from P to a variable tangent line to the curve.
Steiner's theorem states as follows.
Theorem 1. [2], [3] When a closed curve rol ls on a straight line, the area
between the line and the roulette generated in a complete revolution by any point
on the rol ling curve is double the area of the pedal of the rol ling 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 b e 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 byapoint P in
xed p osition with resp ect to a rolling curve c which rolls without slipping along a
xed base curve l .
_ _
Let c = C C and l = L L be oriented piecewise regular curves with para-
1 2 1 2
metric equations r (t)=(x(t);y(t)) and (t)=(u(t);v(t)), a t b, resp ective-
1
ly. A parametric representation is regular if vector function r is of class C and
AMS Subject Classi cation : 53 A 04
Keywords and phrases : Pedal, autoroulettes, barycenter of curvatures
Communicated at the 11. Yugoslav Geometrical Seminar, Divcibare 1996.
24 M. Bjelica
r_ (t)=(_x (t); y_ (t)) 6= 0 for all a t b. Tangentvectors r_ can b e translated so that
T
they have same origin. The total curvature is the di erence in the values
1 0
c
of inclination of the tangenttothecurve at the end p oints of the curve. For the
T
closed curvewehave =2z, where z is an integer. The natural parameter is
c
R
the length of the arc s = jr_ (t)j dt, in which case tangentvectors have length 1. If
c is smo oth in some neighb orho o d of a p oint X , then the curvature (X )of c at X
is
d
= ; s ! 0; Y ! X; (X ) = lim
s ds
where is the angle b etween tangentvectors at X and Y , s is the length of
_
the arc XY . The classical formula for the curvature is
x_ (t)y (t) x(t)_y (t)
(X )= :
2 2 3=2
(_x (t) + y_ (t) )
At a singular p oint S 2 c the p ointcurvature is (S )=](_r (S ); r_ (S )), so that
+
_ _
(S ) . We take (S )= if lim \(_r (S ); SX )= , X ! S , x 2 SC .
2
_
If the whole arc SC is a straight segment opp osite to r_ (s), wecancho ose either
2
or . Mention that the curvature as a function of the length of the arc is the
natural equation of the curve.
Definition 3. The curvature measure on a curve c is de ned at the regular
p oints by the curvature functional (t), and at the singular p oints the p oint-measure
is equal to the p oint-curvature. The centroid B of curvature of a curve c is the
barycenter of c while c is by the curvature measure p ondered.
R
T T T
Note that = d. Also, the di erence is equal to the angle of
c c
l
c
rotation of c in tracing the roulette.
_ _
Let R(c; l ; P )= R R , and let l = L L where L and L are the rst and the
1 2 1 2 1 2
last contact p oints resp ectively. The area R(c; l ; P ) is the oriented area L L R R
1 2 2 1
_ _
b ounded by the base curve l = L L , the roulette R R , and straight segments
1 2 2 1
L R and R L .
2 2 1 1
Theorem 2. [1] Let B be the barycenter of a curve c, let c be pondered by
T
the di erence of curvaturemeasures , and let denote the total curvature.
c l
Then, the areabounded by the roulette and the base curve l is
Z
1
2
area R(c; l ; P ) = area cone Pc + d( ); X 2 c PX
c l
2
c
T T
c
2
l
BP +areaPC BC : = area R(c; l ; B )+
1 2
2
For the closedrol ling curve area cone Pc = area c and area PC BC =0. Further,
1 2
Z
length R(c; l ; P )= PX d( ); X 2 C:
c l c
Pedals, autoroulettes and Steiner's theorem 25
If c is a closed curve then p oints from the circle with center B trace roulettes
T T 2 T T
with equal areas. Note that ( )BP =2 = area sector (BP; ), where
c c
l l
the sector is circular, with radius BP , and which angle is equal to the angle of
rotation of c.
3. Autoroulettes
2
Recall that the area under a cycloid cusp is 3r , and the area b ounded bya
2
cardioid is 6r . This cardioid-cycloid prop ortion 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 (t)= (t), a t b.
l c
If c is a closed curve, then AR(c; P ) is closed also, and, obviously,
area AR(c; P )=areal + area R(c; l ; P ):
Theorem 3. Let c and l be closed symmetric curves (t)= (t), a t b,
l c
and let L beastraight 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 wehave
area AR(c; P )=areal +areaR(c; l ; P )
Z
1
2
=areal +areac + PX d(2 )
c
2
c
Z
1
2
=2 area c + PX d
c
2
c
=2 areaR(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
areapedal : areastraight-roulette : area autoroulette = 1 : 2 : 4
length pedal : length straight-roulette : length autoroulette = 1 : 1 : 2:
4. Extensions
Theorem 4. (a) Extendedstraight roulette | autoroulette proportion. Let c
be piecewise regular curve and L beastraight line, then
area AR(c; P ) area cone Pc =2[area R(c; L; P ) area cone Pc] :
26 M. Bjelica
(b) Extended Steiner's theorem. When a piecewise regular curve rol ls on a straight
line, the areabetween 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 C , the p edal denote by p = P P , and let AR(c; P )=A A .
1 2 1 2 1 2
Notice that the all areas are oriented.
1
[area AR(c; P ) + area cone Pc] area R(c; L; P )=
2
_
1
= area PC A A C P
1 1 2 2
2
h i
_
1
= area coneP (A A )+2PC P 2PC P
1 2 1 1 2 2
2
1
= [4 area cone Pp +2PC P 2PC P ]
1 1 2 2
2
=2areaconePp +PC P PC P :
1 1 2 2
If Area R(c; L; P ) denote the area b ounded by the roulette, the base straight line
and two p erp endiculars from the end p oints of the roulette to the line, then
area R(c; L; P ) = area R(c; L; P )+PC P PC P ;
1 1 2 2
so that
Area R(c; L; P ) = 2 area cone Pp;
what ends the pro of.
These two de nitions of the area of roulettes, by connecting the endoints of
the roulette by the endp oints of the base straight line, and taking p erp endiculars
from the end p oints to the base straight line, for closed curves give equal values.
REFERENCES
[1] M. Bjelica, Area and length for roulettes via curvature, Di erential geometry and applications,
Pro c. Conf. 1995, Brno, 1996, 245{248.
ie
[2] E. Goursat, Cours D'Analyse Mathematique, Tome I Gauthier-Villars et C ,Paris, pp. 240.
[3] M. S. Klamkin, Solution of problem 10254, Amer. Math. Monthly, 102, 4 (1995), 263{264.
(received 27.12.1996.)
UniversityofNovi Sad, \M. Pupin", Zrenjanin 23000, Yugoslavia