Abstract
In this paper we present a certain modification of the Holditch construction. This construction allows to consider a geometric family of pairs of ring domains. It is proved that the ratio of areas of ring domains of each pair belonging to this family is constant. Problems on extremal chords of constant length sliding around a given oval with both endpoints on it are also considered.
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1 Introduction
Hamnet Holditch, president of Caius College in Cambridge, published in [13] a remarkable theorem. Let \(C\;\) be a convex curve, and a chord h of length \(a+b\) be divided into parts of lengths a and b by a point A. Let \(C_{a,b}\) denote a curve traced out by the point A when the chord h slides around with both endpoints on C. Holditch proved that the area of a ring domain bounded by C and \(C_{a,b}\) is equal to \(\pi ab\), see Fig 1.
Arne Broman proved in [4] and [3] a much more general theorem and gave some kinematic applications. Further applications to mechanics were given in [11, 12], and [14]. Some additional remarks on Holditch’s theorem can be found also in [1, 8, 9], and [18], and recent related investigations are given in [15, 16], and [10].
In this paper we modify the Holditch construction in which one ring domain is considered. In our modification we deal with a family of pairs of ring domains and obtain a natural geometric generalization. As an application we derive some Crofton-type formula for a ring domain.
We denote by \({\mathcal {C}}^*\) the family of all closed strictly convex curves of class \(C^{1}\). Let \(C\in {\mathcal {C}}^*\) and let p denote a fixed support function of C. The parametric representation of the curve C has the form
where the dot denotes differentiation with respect to t, see [2] and [17]. We denote by \({\mathcal {C}}\) a subfamily of \({\mathcal {C}}^*\) defined as follows: a curve \(C\in {\mathcal {C}}^*\) belongs to \({\mathcal {C}}\) if and only if the function \(R=p+\ddot{p}\) satisfies the inequality
Note that the function R is the curvature radius of C if the curve is of class \(C^2\).
We fix \(\alpha \in \left( 0,\pi \right) \), and we denote by \(z_{\alpha }\left( t\right) \) the intersection point of the tangent lines at \(z\left( t\right) \) and \(z\left( t+\alpha \right) \). A curve \(C_\alpha :t\rightarrow z_{\alpha }\left( t\right) \) is called an \(\alpha \)-isoptic, see Fig. 2.
We will use the notations introduced in [6] and [7], namely
where
Moreover, if
then we have
see [7]. We denote by \(\xi _{\alpha }\left( t\right) \) the intersection point of the normal lines at \(z\left( t\right) \) and \(z\left( t+\alpha \right) \). We have
see Fig. 3.
Simple calculations lead us to the formulas
Comparing (1.6) and (1.13), we get
Moreover, it is easy to verify that
2 Extremal chords
Let us fix a curve \(C\in {\mathcal {C}}\). We consider lengths of chords joining a fixed point \(z\left( t\right) \) and \(z\left( t+\alpha \right) \) for \(\alpha \in \left( 0, 2\pi \right) \). Let
We denote by \(\left\langle -,-\right\rangle \) the Euclidean scalar product. Differentiating the function \(H_{t}\) given by the formula (2.1) and making use of (1.8) and (1.15), we obtain
and
The above formula implies immediately the following statement.
If the chord joining a fixed point \(z\left( t\right) \)and a point \(z\left( t+\alpha \right) \)for some \(\alpha \in \left( 0, 2\pi \right) \)has maximal length, then the normal line at \(z\left( t+\alpha \right) \)intersects C at \(z\left( t\right) \).
Let
Now we consider the particular but important case of ellipses.
Proposition 2.1
Let us fix an ellipse E, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a>b>0\). Then the maximal length of chords is given by the formula
Proof
For a given point P(r, s) of E we consider the normal line to E at P. This normal line intersects E at the second point \({\tilde{P}}({\tilde{r}}, {\tilde{s}})\), where
Hence the distance between the points P and \({\tilde{P}}\) is \(d(P,{\tilde{P}})=2 (b^4r^2+a^4s^2)^\frac{3}{2}(b^6r^2+a^6s^2)^{-1}.\) Since \(a^2s^2=a^2b^2-b^2r^2\), it suffices to find the minimum of the function
\(\square \)
Example
Let us fix an ellipse \(E, \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). Each chord of the length m which slides around the curve E with endpoints on E determines some curve \(E_m\). We find the equation of the curve \(E_m\). For this aim we solve the system of equations
where the second equation represents the line dual to an exterior point (u, v) with respect to E. Then its intersection points with E are
Since \(m^2=(r-{\tilde{r}})^2+(s-{\tilde{s}})^2\), we get
where we substituted x, y instead of u, v, respectively.
3 Sliding a chord around a curve
We denote by \({\mathcal {N}}\) a family of functions \(\nu :[0, +\infty )\rightarrow {\mathbb {R}}\) of the class \(C^1(0, +\infty )\) satisfying the following conditions:
Let C be a curve \(t\rightarrow z(t)\) given by (1.1), and \(\nu \in {\mathcal {N}}\) be a function. We associate with C a vector field Q along the curve C, defined as follows:
In view of (1.8) we have
Differentiating (3.5) and using the formulas
given in [7], we obtain
We note that
Hence we have equivalence of the following conditions:
With respect to (3.9) we consider the implicit equation
Differentiating the above equation and using the formulas (1.2) and (1.4), we get
and therefore
If the maximal width is attained at \(t=t_0\), then \(\Gamma (t_0)=\pi \). We note that \(\Gamma (t)>\frac{\pi }{2}\), since for the orthoptic curve we have \(f=-\mu \not =0\) and the considered function \(\Gamma \) is differentiable.
We associate with the curve C and \(\nu \in {\mathcal {N}}\) a curve \(C_\nu \) defined by
Theorem 3.1
The integral formula
holds, where \([a+bi, c+di]=ad-bc.\)
Proof
Let \(\alpha \left( t\right) =\nu \left( t\right) -t\). Using (1.10), (1.7), and (1.8), we obtain
\(\square \)
4 The main theorem
Now we assume that a chord of constant length m slides around with both endpoints on C which is given by formula (1.1). The endpoints of the sliding chord determine an increasing function \(\nu \in {\mathcal {N}}\). We assume that \( |z(0)-z(t_0)|=m \) for some \(t_0\in (0,2\pi ).\) Thus the function \(\nu \) satisfies the differential equation (3.9) with the initial condition \(\nu (0)=t_0\). For a fixed \(\xi \in [0,1]\) we consider a curve \(C(m,\xi )\) given by the formula
Obviously, we have \(|Q|\equiv m\). We note that
Hence we get immediately
We note that the graphs of the curves C(m, 0) and C(m, 1) coincide with the graph of C. Thus for \(\xi =1\) from (4.2) it follows immediately that \(\int \limits _0^{2\pi }[{\dot{z}}, Q]dt=-\pi m^2.\) Now formula (4.2) can be rewritten in the form
Letting \(m=a+b\) and \(\xi =\frac{a}{a+b}\), we get the well-known Holditch formula
Now we associate with C and \(C_{\nu }\) a certain curve \(D_{\nu ,\gamma }\) defined as follows:
where \(\gamma \) is a nonnegative constant.
Let us fix \(\xi \in (0,1)\). We consider a curve \(C_{\nu , \xi }\), \(t\rightarrow v_{\nu , \xi }(t)=(1-\xi )z(t)+\xi z(\nu (t))\) for \( t\in [0, 2\pi ]\), see Fig. 4.
Theorem 4.1
If \(C\in {\mathcal {C}}\), \(\nu \in {\mathcal {N}}\), \(\xi \in (0,1)\) and \(\gamma \not =0\), then the following formula holds:
Proof
We have
Applying Theorem 3.1, we obtain
It was proved in [5] that
Comparing the formulas (4.6) and (4.7), we get (4.5). \(\square \)
As corollaries of Theorem 4.1 we have the following Holditch-type formulas.
Corollary 4.2
If \(a+b<ww(C)\), then
Corollary 4.3
If \(m<ww\left( C\right) \), then
5 Crofton-type integral formulas
In this section we provide an interesting application of the developed theory and derive a new, geometrically justified Crofton-type formula.
Let us fix \(C\in {\mathcal {C}}\) and \(r<ww\left( C\right) \). The function \(\nu \left( t,m\right) \) determined by \(m\in \left( 0,r\right) \) satisfies the condition (3.4). Differentiating (3.4) with respect to m, we obtain
Let \(\alpha \left( t,m\right) =\nu \left( t,m\right) -t.\) With respect to (3.5) and (1.15) we have
The above calculations imply that
Now, we consider the ring domain \(CC_{r\text { }}\), and we introduce the notations as in Fig. 6, maintaining at the same time the notations of Santaló from [17].
Let \(R_{1}(x,y)\), \(R_{2}(x,y)\) denote the radii of curvature of C at the tangent points \(A_{1}\), \(A_{2}\), respectively.
Crofton proved the integral formula
where \({\text {ext}}C\) denotes the exterior of C, see [13]. We will prove some Crofton-type theorem, namely
Theorem 5.1
If \(C\in {\mathcal {C}}\) and \(r<ww\left( C\right) \), then the following integral formula holds:
Proof
We consider a mapping \(T:\left( 0,2\pi \right) \times \left( 0,r\right) \rightarrow \) interior of \(CC_{r\text { }}\backslash \){some segment} defined by the formula
We note that T is a bijection and
where h is some function. Thus the Jacobian JT of T at \(\left( t,m\right) \) is given by the formula
On the other hand, since \( \alpha \left( t,m\right) =\nu \left( t,m\right) -t\), so \(\frac{\partial \nu }{\partial m}=\frac{\partial \alpha }{\partial m}\) and
Thus the Jacobian of T at \(\left( t,m\right) \) has the form
Now we have
\(\square \)
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Cieślak, W., Martini, H. & Mozgawa, W. On Holditch’s theorem. J. Geom. 111, 24 (2020). https://doi.org/10.1007/s00022-020-00536-5
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DOI: https://doi.org/10.1007/s00022-020-00536-5