Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 34 (1993), No. 1, 31-44.

Inner Parallel Curves and Domains in the Equiaffine Geometry

Dedicated to W. Degen on occasion of his 60. birthday

Kurt Leichtweiß

Mathematisches Institut B, Universität Stuttgart Pfaffenwaldring 57, W-7000 Stuttgart 80

1. Introduction

In a previous paper, titled ”On inner parallel bodies in the equiaffine geometry” [7], we treated the definitions and properties of equiaffine inner parallel hypersurfaces resp. bodies in a n-dimensional affine space and compared them with the corresponding definitions and properties of euclidean inner parallel hypersurfaces resp. bodies. Clearly, these constructions become particularly obvious in the case of dimension 2 where many proofs strongly simplify and there exist instructive examples. It is the aim of this article to implement this program with the help of computer drawings together with an outlook on the inverse problem of determining outer equiaffine parallel curves, equivalent to the so called ”dual billiard problem”.

2. Equiaffine inner parallel curves

We begin with the following definition, proposed by G.Chakerian (see [7]),

Definition 2.1 Let C be a closed convex curve, piecewise of differentiability class C_{1}, which bounds a compact convex domain (with inner points) D in the affine plane. Then another closed convex curve C_{[δ]} of differentiability class C_{1} in the interior of D without singularities is called equiaffine inner parallel curve of C if all tangent lines of C_{[δ]} cut off from D segments of constant area δ > 0.

It is obvious that this definition is equiaffinely invariant. In contrast to the euclidean case the osculating point x_{[δ]} of C_{[δ]} with its tangent line T_{[δ]} depends on an open part of C containing the arc of C between the two points of the intersection C ∩ T_{[δ]}. This implies that the construction of C_{[δ]} as an envelope of the lines T_{[δ]} is not ”pointwise” like the corresponding euclidean construction x_{ϵ} = x + ϵξ (x = point of C with inner unit normal vector ξ, ϵ = const > 0). Nevertheless there are many analogies between the euclidean and equiaffine inner parallel curves C_{ϵ} and C_{[δ]} of C which will be described in the following.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

A basic tool for this purpose is

Proposition 2.2 a) The (only) osculating point x_{[δ]} of the (necessarily) strictly convex curve C_{[δ]} with T_{[δ]} is the midpoint of the chord x^{(1)}x^{(2)} := D ∩ T_{[δ]} of C.

b) The inner parallel curve C_{[δ]} has

R_{[δ]} = \frac{|x^{(1)}x^{(2)}|}{4}(cot β_{1} + cot β_{2})  (1)

as radius of curvature at the point x_{[δ]} where β_{1} resp. β_{2} are the angles between the chord

x^{(1)}x^{(2)} and the tangent halflines of the arc \overrightarrow{x^{(1)}x^{(2)}} at x^{(1)} resp. x^{(2)}. (Compare the proof of Satz 1 in [5] for the general dimension n and the subsequent remark given there.)

Proof. a) is obvious because of Definition 2.1 after applying a variation of the chord x^{(1)}x^{(2)} of C.

b) If we consider the chord x^{(1)}x^{(2)} of C and the infinitesimally neighbouring chord \overline{x}^{(1)}\overline{x}^{(2)} of C forming with x^{(1)}x^{(2)} the angle dα then according to a) we have for the points x_{[δ]} = \frac{1}{2}(x^{(1)} + x^{(2)}) and \overline{x}_{[δ]} = \frac{1}{2}(\overline{x}^{(1)} + \overline{x}^{(2)}) the relation

\overline{x}_{[δ]} − x_{[δ]} = dx_{[δ]} = \frac{1}{2}(\frac{|x^{(1)}x^{(2)}|}{2}dα(cot β_{1} + cot β_{2})) = R_{[δ]}dα  (2)

which yields (1) (see Fig. 1).

Figure 1

Corollary 2.3 C_{[δ]} is convex at x_{[δ]} iff

β_{1} + β_{2} ≤ π  (3)

holds for the angles β_{1} and β_{2} at the endpoints of the chord x^{(1)}x^{(2)} of C.

Under the assumption of Definition 2.1 it may occur that C has no equiaffine inner parallel curve C_{[δ]}. For example this is true for a triangle C because the condition (3) for the convexity of C_{[δ]} then is always violated for certain chords parallel to the edges of the triangle. Therefore we restrict ourselves for the moment to a (strictly) convex curve C of differentiability class C_{2}^{+} and show

Proposition 2.4 For a curve C of class C_{2}^{+} and minimal radius of curvature r_{0} there exist all equiaffine inner parallel curves C_{[δ]} with

0 < δ < δ_{0} := \frac{π}{2} (r_{0})^{2}.  (4)

Proof. Blaschke’s rolling theorem [1], p. 118, says that a circle of radius r_{0} rolls freely in the interior of C such that all chords D ∩ T_{[δ]} of C with F (D ∩ H_{[δ]}^{+}) = δ (H_{[δ]}^{+} denoting the exterior part of a closed halfplane bounded by T_{[δ]}) intersect C in acute angles β_{1}, β_{2} whence R_{[δ]} > 0 because of (1). (The example of a circle C of radius r_{0} shows that δ_{0} in condition (4) is sharp!).

Remark 2.5 Proposition 2.4 is an equiaffine counterpart of the fact that the (sharp) condition 0 < ϵ < r_{0} implies the existence of euclidean inner parallel curves C_{ϵ} of C.

Figure 2

For example see Fig. 2 with regular and singular euclidean inner parallel curves of an ellipse (the cusps of the latter lie on an astroide). It is very remarkable that Corollary 2.3 provides as an immediate consequence

Proposition 2.6 A centrally symmetric convex curve C of class C_{2}^{+} (with the origin o as center) possesses equiaffine inner parallels C_{[δ]} for all possible values of δ, i.e. for 0 < δ < \frac{1}{2}F (D). (Compare the much deeper result for dimension n in [8], Theorem 3, part Proof. Trivially, the tangent lines at the endpoints of a chord of C through the center o are parallel whence β_{1} + β_{2} = π for these chords. But this implies β_{1} + β_{2} < π for all the other chords and therefore R_{[δ]} > 0 because of (1).

Figure 3a  Figure 3b

For example see Fig. 3a with a curve C, represented in polar coordinates by r = 3 + 0.5 cos 2φ + 0.2 sin 2φ, and Fig. 3b with an algebraic curve C : ξ^{4} + η^{4} = 1 with respect to cartesian coordinates ξ, η. Conversely we note

Remark 2.7 If a convex curve C of class C_{2}^{+} has equiaffine inner parallel curves C_{[δ]} for all δ with 0 < δ < \frac{1}{2}F (D) then C is centrally symmetric. (Compare [8], Lemma 4.)

Proof. When δ tends to \frac{1}{2}F (D) the curves C_{[δ]} concentrate themselves to a point c which divides all chords of C through c into two parts of equal length (compare Proposition 2.2 a)). Therefore C must be centrally symmetric with the center c.

Further we should mention (without proof)

Remark 2.8 The procedure of forming an equiaffine inner parallel curve increases the differentiability class in general in the sense that the class of C_{[δ]} equals that of C plus 1.

For example we look at the inner parallels of a regular triangle (Fig. 4a), quadrangle (Fig. 4b), pentagon (Fig. 4c) and hexagon (Fig. 4d). They consist of arcs of hyperbolas fitting together at the midpoints of the chords going through a vertex of the polygon, smoothly in case of β_{1} + β_{2} < π or β_{1} + β_{2} > π (hereby with a cusp !) respectively with an angle in case of β_{1} + β_{2} = π. Cusps resp. angles may occur if the number of the vertices of the polygon is odd resp. even and δ is sufficiently near to \frac{1}{2}F (D). When cusps exist the inner parallel curve C_{[δ]} is not convex.

Figure 4a  Figure 4b

Figure 4c  Figure 4d

Figure 5

Moreover we assert Proposition 2.9 The procedure of forming an equiaffine inner parallel curve of a given curve is not associative (in contrast to the euclidean case), i.e. in general the equiaffine inner parallel curve (C_{[δ_{1}]})_{[δ_{2}]} of another equiaffine inner parallel curve C_{[δ_{1}]} is not an equiaffine inner parallel curve of C.

For example, see Fig. 5 with equiaffine inner parallel curves of a regular pentagon. Very important seems to us the subsequent variation property of equiaffine inner parallels (being similar to the corresponding property in the euclidean case).

This is formulated in

Proposition 2.10 Let C be a closed convex curve of class C_{3}^{+}. Then the variation vector

δ_{aff}x := lim_{δ→0} \frac{x_{[δ]} − x}{δ^{\frac{2}{3}}}  (5)

of the ”normal variation” of the equiaffine inner parallel curves C_{[δ]} (with parallel tangent lines at corresponding points) and the affine normal vector y of C at x coincide up to a constant factor:

c · δ_{aff}x = y  (6)

with

c := 2(\frac{2}{3})^{\frac{2}{3}} .  (7)

(See also [6].)

Proof. We consider a special point x = (0, 0) of C with tangent line T: η = 0 bounding the interior halfplane H^{−} given by η ≥ 0 with respect to suitably chosen affine coordinates ξ, η such that C may be represented by

η = \frac{1}{2}aξ^{2} + \frac{1}{6}bξ^{3} + o(ξ^{3})  (a > 0)  (8)

in a neighbourhood of x. Introducing ζ ≥ 0 by

ζ = η^{\frac{1}{2}}  (9)

we get

ζ = signξ · ((\frac{a}{2})^{\frac{1}{2}} ξ + \frac{b}{6(2a)^{\frac{1}{2}}} ξ^{2} + o(ξ^{2}))

(see also [4], p. 50) or - conversely -

ξ = ±(\frac{2}{a})^{\frac{1}{2}} ζ − \frac{b}{3a^{2}} ζ^{2} + o(ζ^{2})  (10)

whence

x_{[δ]} − x = (−\frac{b}{3a^{2}} ζ^{2} + o(ζ^{2}), ζ^{2})  (11)

K. Leichtweiß: Inner Parallel Curves and Domains in the Equaffine Geometry 37 (see Proposition 2.2 a) !). Furthermore the integration of (10) yields

δ = ∫_{0}^{ζ} (2(\frac{2}{a})^{\frac{1}{2}} τ + o(τ^{2}))d(τ^{2}) = \frac{4}{3}(\frac{2}{a})^{\frac{1}{2}} ζ^{3} + o(ζ^{3})  (12)

whence

δ^{\frac{2}{3}} = c · a^{− \frac{1}{3}} · ξ^{2} + o(ξ^{2})  (13)

(see also (7)). The combination of (11) and (13) then provides

c · δ_{aff}x = a^{\frac{1}{3}} (−\frac{b}{3a^{2}} , 1).  (14)

On the other side, after introducing the affine arc length parameter s by

s = ∫ (\frac{dx}{dt}, \frac{d^{2}x}{dt^{2}})^{\frac{1}{3}} dt,

we may deduce for the affine normal vector y of C

y := \frac{d^{2}x}{ds^{2}} = (\frac{dx}{dt}, \frac{d^{2}x}{dt^{2}})^{− \frac{5}{3}} ((\frac{dx}{dt}, \frac{d^{2}x}{dt^{2}})\frac{d^{2}x}{dt^{2}} − \frac{1}{3}(\frac{dx}{dt}, \frac{d^{3}x}{dt^{3}})\frac{dx}{dt})  (15)

(see [2], §5). Then the specialization t = ξ and application of the representation (8) yield x(ξ) = (ξ, \frac{1}{2}aξ^{2} + \frac{1}{6}bξ^{3} + o(ξ^{3})) such that

y = a^{− \frac{5}{3}} ((0, a^{2}) − (\frac{b}{3}, 0)) = a^{\frac{1}{3}} (−\frac{b}{3a^{2}} , 1)  (16)

holds. (14) and (16) show the validity of proposition 2.10.

Figure 6  Figure 7

For example see Fig. 6 with a computed affine normal vector y. Now we are able to formulate as main result of this chapter (analogous to the euclidean case):

38 K. Leichtweiß: Inner Parallel Curves and Domains in the Equaffine Geometry Theorem 2.11 Let C be a closed convex curve of class C_{3}^{+}. Then the limit of the term \frac{F (D) − F (D_{[δ]})}{δ^{\frac{2}{3}}} exists and equals the affine perimeter L_{aff}(C) of C up to the factor c in (7):

c · lim_{δ→0} \frac{F (D) − F (D_{[δ]})}{δ^{\frac{2}{3}}} = ∮ (dx, y) = ∮ (\frac{dx}{dt}, \frac{d^{2}x}{dt^{2}})^{\frac{1}{3}} dt =: L_{aff}(C),  (17)

D_{[δ]} denoting the compact domain bounded by C_{[δ]}. (See also the deeper result in the C_{2}^{+}-case for dimension n in [5], Satz 2). Proof. We first show the validity of Lemma 2.12 If C is a closed convex curve of class C_{3}^{+} with the equiaffine inner parallel curves C_{[δ]} and if R and R_{[δ]} are the radii of curvature of C and C_{[δ]} at the corresponding points x and x_{[δ]} (with parallel tangent lines) then

lim_{δ→0} R_{[δ]} = R  (18)

holds. Proof of Lemma 2.12. The formula (1) may geometrically be interpreted by

R_{[δ]} = \frac{1}{2}(|x_{[δ]}z_{[δ]}^{(1)}| + |x_{[δ]}z_{[δ]}^{(2)}|)  (19)

(see Fig. 7 !). Now we also may derive from Fig. 7

0 ≤ ||xz^{(k)}| − |x_{[δ]}z_{[δ]}^{(k)}|| ≤ |xw| + |x_{[δ]}w| cot β_{k}
≤ |xw| + 2\frac{|x_{[δ]}w|}{|xw|} · \frac{1}{2}|x^{(1)}x^{(2)}| = O(|xw|^{\frac{1}{2}} ) (k = 1, 2)  (20)

as

2\frac{|x_{[δ]}w|}{|xw|} = O(1)

because of (5) and (6) and

\frac{1}{2}|x^{(1)}x^{(2)}| ≤ ((R_{max})^{2} − (R_{max} − |xw|)^{2})^{\frac{1}{2}} = O(|xw|^{\frac{1}{2}} )

after Blaschke’s rolling theorem. But (20) implies

lim_{δ→0} |x_{[δ]}z_{[δ]}^{(k)}| = lim_{δ→0} |xz^{(k)}| = R (k = 1, 2)  (21)

whence (18) follows because of (19).

Proof of Theorem 2.11 (continuation): The difference F (D) − F (D_{[δ]}) may be transformed into

F (D) − F (D_{[δ]}) := \frac{1}{2} ∮ (x, dx) − \frac{1}{2} ∮ (x_{[δ]}, dx_{[δ]}) = \frac{1}{2} ∮ (x − x_{[δ]}, dx + dx_{[δ]})  (22)

as the application of Stokes’ formula yields

∮ ((dx, x_{[δ]}) + (x, dx_{[δ]})) = 0.  (23)

From (22) we deduce

c · \frac{F (D) − F (D_{[δ]}}{δ^{\frac{2}{3}}} = \frac{1}{2} ∫_{0}^{2π} (c\frac{x − x_{[δ]}}{δ^{\frac{2}{3}}} , t)(R + R_{[δ]})dα,  (24)

t denoting the unit tangent vector of C resp. C_{[δ]}. But now Lebesgue’s convergence theorem

(applicable because of the uniform boundedness of (\frac{x − x_{[δ]}}{δ^{\frac{2}{3}}} , t) =< \frac{x − x_{[δ]}}{δ^{\frac{2}{3}}} , −ξ > after

Blaschke’s rolling theorem (see [5], Hilfssatz 2) and of R_{[δ]} after the proof of Lemma 2.12 using a compactness argument) together with Proposition 2.10 and Lemma 2.12 provides indeed

c · lim_{δ→0} \frac{F (D) − F (D_{[δ]})}{δ^{\frac{2}{3}}} = \frac{1}{2} ∫_{0}^{2π}(−y, t) · 2Rdα = ∮ (dx, y) = ∮ (\frac{dx}{dt}, \frac{d^{2}x}{dt^{2}})^{\frac{1}{3}} dt (25)

after insertion of (15), being the assertion of Theorem 2.11.

For illustration of Theorem 2.11 see Fig. 8.

Figure 8  Figure 9

3. Equiaffine inner parallel domains

As in the euclidean case one may avoid the fact that not all equiaffine inner parallel curves are convex by a modified notion introduced by C. Schütt and E. Werner (see [10]), namely Definition 3.1 Let D be a compact convex domain in the affine plane with inner points. Then the intersection D_{(δ)} of all closed halfplanes H_{[δ]}^{−} with F (D ∩ H_{[δ]}^{−}) = F (D) − δ is called equiaffine inner parallel domain of D (0 < δ ≤ \frac{1}{2}F (D)).

For example see an array of equiaffine inner parallel domains of a regular solid triangle D in Fig. 9. It is created by truncation of the equiaffine inner parallel curves of the boundary of D in Fig. 4a.

Clearly the equiaffinely invariant domains D_{(δ)} are convex and not empty for all δ with

0 < δ ≤ δ_{1} := \frac{π}{2} (r_{1})^{2}  (26)

where r_{1} is the inradius of D (because they contain the midpoint of the incircle of D). Moreover, in the case of the existence of the equiaffine inner parallel curve C_{[δ]} for a domain D with smooth boundary C and a certain δ, C_{[δ]} is the boundary of D_{(δ)}.

We should still make two remarks:

Remark 3.2 (see C. Schütt and E. Werner for dimension n [10]): Let D be an arbitrary

compact convex domain with inner points. Then the limit of the term c · \frac{F (D) − F (D_{(δ)})}{δ^{\frac{2}{3}}}

exists and equals the Lebesgue integral on the boundary ∂D of D with the (euclidean) arc length σ over the third root of the (almost everywhere existing) curvature κ of ∂D:

c · lim_{δ→0} \frac{F (D) − F (D_{(δ)})}{δ^{\frac{2}{3}}} = ∫_{∂D} κ^{\frac{1}{3}} dσ.  (27)

(Therefore this integral may be used for the definition of an affine perimeter of ∂D corresponding to (17) in the C_{2}^{+}-case.)

The proof of (27) uses Aleksandrov’s theorem of the twice differentiability of a convex function almost everywhere and some other techniques. In this context we state as a new characterization of an ellipse

Remark 3.3 (see C. Schütt and E. Werner for dimenion n [11]): If there is a sequence {δ_{k}}_{k∈N} converging to 0 such that all equiaffine inner parallel domains D_{(δ_{k})} are homothetic to D then D is a (solid) ellipse.

It is remarkable that this fact has an euclidean analogon: The tangential domains D of a circular disc (see [3], p. 18) are characterized by the property that a euclidean inner parallel domain D − ϵB (B denoting the unit circular disc about the origin) is homothtic to D (personal communication of the author to Mr. Melissen, Eindhoven, 1992).

4. Equiaffine outer parallel curves

In this last section we consider the following inverse Problem 4.1 What are the conditions such that a given (strictly) convex C_{2}^{+}-curve C′ is the equiaffine inner parallel curve C_{[δ]} for a suitable convex curve C - or in other words - that the given curve C′ possesses a convex ”equiaffine outer parallel curve” C?

With regard to proposition 2.2 a) Problem 4.1 is equivalent to that of the existence of a convex curve C as a solution of the so called ”dual billiard problem” for a given curve C′ as follows: We consider the (area preserving) map τ_{D′} of R^{2} \ D′ into itself (D′ denoting a compact convex domain with boundary C′), defined by the claim that all the segments pτ_{D′}(p) (p ∈ R^{2} \ D′) touch C′ at their midpoints in accordance with the orientation of C′. Then a convex curve C is called a solution of the dual billiard problem for C′ iff C is invariant under τ_{D′}:

τ_{D′}(C) = C.  (28)

Figure 10a  Figure 10b

Remark 4.2. It is rather trivial that for a given ellipse C′ the dilatated ellipses C = α·C′ (α > 1) are the only solutions of (28) (see [9], p. 11 or [7], Proposition 4.2).

But in the other (nontrivial) cases the question of the existence of convex solutions C for the dual billiard problem, i.e. the existence of convex equiaffine outer parallel curves C for a given C_{2}^{+}-curve C′ is open. J. Moser obtained as the most far going result in this direction the

Theorem 4.3 If C′ is a convex curve of class C_{6}^{+} then there exists an infinite number of (not necessarily convex) C_{1}-curves which are solutions of (28) (see [9], p. 11).

As a consequence of this theorem all the orbits

M_{D′}(p) := ⋃_{ν=−∞}^{∞}(τ_{D′})^{ν}(p) (p ∈ R^{2} \ D′)  (29)

of the dual billiard tranformation τ_{D′} for a C_{6}^{+}-curve are bounded. We have computed these orbits for a curve C′ near to an ellipse represented in polar coordinates by r = 3 + 0.2 cos 2φ + 0.2 sin 2φ with p = (4, 1) where the orbit lies on a convex curve C (Fig. 10a), for another such curve C′ represented by r = 2.5 − 0.3 cos(2φ + π) + 0.35 sin(2φ + π) with p = (4, 1) where the orbit lies on a simply closed nonconvex curve C (Fig. 10b), and for the algebraic curve ξ^{4} + η^{4} = 1 with p = (2, 0.6) where the orbit lies on four convex closed curves (Fig. 10c) resp. with p = (1.8, 1.03) where the orbit does not lie on a simply closed curve (Fig. 10d). Finally, if in the extreme case C′ is a quadrangle |ξ| + |η| = 1, then the orbits are finite, i.e. periodic with the period 4 for all initial points p belonging to the interior of the four squares in R^{2} \ D′ adjacent to D′ at an edge of D′ for a suitably chosen well defined map τ_{D′} (Fig. 10e). For more information see the Diplomarbeit of B. Bunse, Freiburg Br., 1991.

Figure 10c  Figure 10d

Figure 10e

We want to finish this paper with

Remark 4.4 There cannot exist an ordinary differential equation of first order fulfilling Lipschitz conditions with the property that each strictly convex C_{1}-curve with a proper convex equiaffine outer parallel curve C is the solution of this differential equation.

Proof. We construct two strictly convex C_{1}-curves C′_{1}, C′_{2}, coinciding for a certain part without being identical, having proper convex equiaffine outer parallels C_{1}, C_{2}. This will give a contradiction to the existence of the mentioned differential equation. For this purpose we consider an equilateral triangle Δ with inscribed circle C′_{1} and circumscribed circle C_{1}. Trivially C_{1} is the equiaffine outer parallel of C′_{1} for a certain δ_{1}. The three vertices p_{1}, p_{2}, p_{3} of Δ divide C_{1} into the three arcs C_{1}^{(1)} = \overrightarrow{p_{2}p_{3}}, C_{1}^{(2)} = \overrightarrow{p_{3}p_{1}}, C_{1}^{(3)} = \overrightarrow{p_{1}p_{2}} and the points of contact q_{1}, q_{2}, q_{3} of Δ with C′_{1} divide C′_{1} into the three arcs C_{1}^{′(1)} = \overrightarrow{q_{2}q_{3}}, C_{1}^{′(2)} = \overrightarrow{q_{3}q_{1}}, C_{1}^{′(3)} = \overrightarrow{q_{1}q_{2}} (see Fig. 11). Now we deform the arc C_{1}^{(1)} a little in a way such that the changed arc C_{2}^{(1)} remains convex and touches C_{1}^{(1)} at p_{2} and p_{3}. If this change is not too big we obtain C_{2}^{(2)} := τ_{D′_{1}}(C_{2}^{(1)}) and C_{2}^{(3)} := τ_{D′_{1}}(C_{2}^{(2)}) = (τ_{D′_{1}})^{2}(C_{2}^{(1)}) have the analogous property such that C_{2}^{(1)}, C_{2}^{(2)}, C_{2}^{(3)} fit together to a convex C_{1}-curve C_{2}. But because of this construction, after the choice of a suitable δ_{2} > 0, we find an equiaffine inner parallel C′_{2} = (C_{2})_{[δ_{2}]} of C_{2} which is different from C′_{1} because of Remark 4.2 but a part of which coincides with C_{1}^{′(1)} ∪ C_{1}^{′(3)}. This fact completes the proof of Remark 4.4.

Figure 11

Acknowledgement: I am very grateful to Mr. Uwe Schlenther who kindly prepared the figures to this article.

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