A FOUR VERTEX THEOREM FOR POLYGONS

SERGE TABACHNIKOV

The classical 4-vertex theorem states that a closed smooth convex plane curve has at least 4 curvature extrema. Published in 1909 ([4]) this theorem and its various generalizations continue to attract interest – see [1] for a contemporary point of view on the subject.

One can formulate the 4-vertex theorem in terms of the 1-parameter family of normal lines to the curve; the envelope of these lines is called the caustic (or the evolute) of the curve. The caustic is not smooth: generically, it has semi-cubical cusp singularities, and these cusps correspond to the extrema of the curvature. One gives the smooth pieces of the caustic the orientation, induced by the inward orientation of the normals. As one traverses the caustic, the cusps are the points at which the orientation changes to the opposite – see figure 1.

Figure 1

The next two results, due to O. Musin and B. Wegner, respectively, extend the 4-vertex theorem from smooth curves to polygons. Let A_{1}, ..., A_{n} be consecutive vertices of a plane convex polygon P. Consider the circle, circumscribed around the triagle A_{i−1}A_{i}A_{i+1}, and let R_{i} be its radius. Likewise, let r_{i} be the radius of the circle touching the lines A_{i−1}A_{i}, A_{i}A_{i+1} and A_{i+1}A_{i+2} and lying on the same sides of these lines as P; in other words, the center of the circle is the intersection of the bisectors of the i-th and i + 1-st interior angles of P (here and elsewhere we understand the index i cyclically, that is, n + 1 = 1, etc). Then each of the two cyclic difference sequences

Research was supported in part by an NSF Grant.

(ΔR)_{i} = R_{i+1} − R_{i} and (Δr)_{i} = r_{i+1} − r_{i}, i = 1, ..., n

changes sign at least 4 times – see [5], [9]. In the limit n → ∞, when P approximates a smooth curve, both sequences R_{i} and r_{i} approximate the radius of curvature function, and the places where (ΔR)_{i} or (Δr)_{i} change signs become critical points of this function, i.e., vertices of the curve.

We propose the following generalization. Consider a convex polygon P with vertices A_{1}, ..., A_{n}, and let l_{i}, i = 1, ..., n be a line through the vertex A_{i} that intersect the interior of P and is oriented inwards. Denote by α_{i} and β_{i} the angles made by l_{i} with the sides A_{i}A_{i+1} and A_{i}A_{i−1}, respectively, and assume that α_{i} + β_{i+1} < π for all i.

The polygon P being an analog of a smooth curve, the lines l_{i} play the role of the normals. Let B_{i} = l_{i} ∩ l_{i+1}; the closed (possibly selfintersecting) polygon Q = (B_{1}, ..., B_{n}) will be the analog of the caustic. Each side of Q lies on a line l_{i} and gets orientation from it. Traverse Q and mark the vertices at which the orientations of the sides change to the opposite; in analogy with the smooth case, call them cusp vertices – on figure 2 the four cusp vertices are marked. Obviously, the number of cusp vertices is even.

Figure 2

Call a collection of lines l_{i} exact if Π_{1}^{n} sin α_{i} = Π_{1}^{n} sin β_{i}. For example, such is the collection of the bisectors of the interior angles of P. A collection of lines l_{i} is called generic if no three consecutive lines intersect at one point.

Theorem 1. Given an exact generic collection of lines l_{1}, ..., l_{n}, the polygon Q has at least 4 cusp vertices.

If l_{i} are the bisectors of the angles of P one obtains Wegner’s theorem. Note that without the exactness assumption one can easily do without cusp vertices at all – see figure 2 again.

Proof. Start with a geometrical interpretation of the exactness condition. Pick a point X_{1} ∈ l_{1}, draw a line through X_{1} parallel to A_{1}A_{2} until it intersects l_{2} at point X_{2}, draw a line through X_{2} parallel to A_{2}A_{3} until it intersects l_{3} at point X_{3},... – see figure 3.

Figure 3

We claim that the polygonal line X_{1}, X_{2}, ... closes, that is, that X_{n+1} = X_{1}, if and only if the collection of lines l_{i} is exact. Indeed, |X_{i+1}A_{i+1}|/|X_{i}A_{i}| = sin α_{i}/ sin β_{i+1}, therefore X_{n+1} = X_{1} if and only if Π sin α_{i}/Π sin β_{i} = 1.

Translate the vectors A_{i}X_{i} to an origin O to obtain a polygon V_{1}...V_{n} – see figure 3. It follows from the previous paragraph that the sides of this polygon are parallel to those of P. For every i the triangle B_{i}A_{i}A_{i+1} is similar to OV_{i}V_{i+1}; denote by t_{i} the similarity coefficient. As long as t_{i} monotonically increase or decrease the orientation of the sides of the ”caustic” Q do not change, and the cusp vertices of Q correspond to the change of sign of the difference sequence (Δt)_{i} = t_{i+1} − t_{i}; more precisely, B_{i} is a cusp vertex if and only if (Δt)_{i−1} and (Δt)_{i} have opposite signs.

It remains to show that the cyclic difference sequence (Δt)_{i} changes sign at least 4 times. One has: t_{i}(A_{i+1} − A_{i}) = V_{i+1} − V_{i}, therefore

∑_{1}^{n}(Δt)_{i}A_{i} = − ∑_{1}^{n} t_{i}(A_{i+1} − A_{i}) = − ∑_{1}^{n}(V_{i+1} − V_{i}) = 0,

(the first equality is ”integration by parts”). Since ∑(Δt)_{i} = 0, the cyclic sequence (Δt)_{i} changes sign at least twice. Assume that it happens exactly twice: say, (Δt)_{i} > 0 for i = 1, ..., k and (Δt)_{i} < 0 for i = k + 1, ..., n. Choose an origin on a line m that intersects the segments A_{k}A_{k+1} and A_{n}A_{1}. With this choice, all vectors (Δt)_{i}A_{i} lie on the same side of m, and ∑_{1}^{n}(Δt)_{i}A_{i} cannot vanish. This contradiction proves the theorem.

Remarks. 1. The continuous case n → ∞ of the above theorem is discussed in [7], [8]. For another aspect of exact systems of lines see [6].

2. The argument in the proof is a particular case of a discrete version of the following theorem (Sturm, Hurwitz, Kellog,...): if a function on a circle is L_{2}-orthogonal to a k-dimensional Chebyshev system then this function has at least k + 1 distinct zeroes – see [1] for a continuous and [3] for a discrete version.

3. One may argue that the oldest 4-vertex type result is the following Cauchy lemma (1813) that plays a crucial role in the proof of convex polyhedra rigidity: given two convex n-gons whose i-th sides have equal lenghts for all i, the cyclic sequence of the differences of the corresponding angles of the polygons changes sign at least 4 times. Our theorem concerns a dual configuration: two polygons whose corresponding angles are equal – figure 3.

We conclude with a curious and not immediately obvious property of exact systems of lines.

Lemma 2. Exactness is invariant under projective transformations of the plane.

Proof. Choose a point O inside the polygon P, let l′_{i} be the line A_{i}O and denote by α′_{i} and β′_{i} the angles made by l′_{i} with sides A_{i}A_{i+1} and A_{i}A_{i−1}, respectively. The new lines l′_{i} form an exact system: consider the Sine Rule for the triangle A_{i}A_{i+1}O and take the product over all i. The cross-ratio of the four lines through vertex A_{i}, namely, the two adjacent sides of P and the lines l_{i} and l′_{i}, equals (sin α′_{i}/ sin α_{i})/(sin β′_{i}/ sin β_{i}), and the product of these ratios is equal to one.

Apply a projective transformation F to this configuration of lines and points. The lines F (l′_{i}) still make an exact system since they pass through a point F (O). The cross-ratios of each quadruple of lines remains the same, and so their product still equals one. It follows that F (l_{i}) form an exact collection of lines as well.

Acknowledgments. This work was done during the author’s visit at Max-Planck-Institut in Bonn; it is a pleasure to thank the Institute for its hospitality.

References

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[2] R. Connelly. Rigidity. Handbook of Convex Geometry, pp. 223-272. North-Holland, 1993.

[3] F. Gantmakher, M. Krein. Sur les matrices complètement non négative et oscillatoires. Compositio Math., 4 (1937), 445-476.

[4] S. Mukhopadhyaya. New methods in the geometry of a plane arc. Bull. Calcutta Math. Soc., 1 (1909), 31-37.

[5] V. Sedykh. Theorem of four support vertices of a polygonal line. Funct. Anal. Appl., 30, No 3 (1996), 216-218.

[6] Problem No 10724 , Amer. Math. Monthly, 106, No 3 (1999).

[7] S. Tabachnikov. The four vertex theorem revisited – two variations on the old theme. Amer. Math. Monthly, 102 (1995), 912-916.

[8] S. Tabachnikov. Parametrized plane curves, Minkowski caustics, Minkowski vertices and conservative line fields. l’Enseign. Math., 43 (1997), 3-26.

[9] B. Wegner. On the evolutes of piecewise linear curves in the plane. Rad. Hrvat. Acad. Znan. Um., 467 (1994), 1-16.

Department of Mathematics, University of Arkansas, Fayetteville, AR 72701

 E-mail address: serge@comp.uark.edu