Orientation-dependent Chord Length Distribution Research Articles

Orientation-Dependent Chord Length Distribution in a Convex Quadrilateral

This work contributes to the research devoted to the recognition of a convex body by probabilistic characteristics of its lower-dimensional sections. In this paper, for any convex quadrilateral, five orientation-dependent characteristics are introduced and explicitly evaluated per direction. In terms of these characteristics, simple explicit representations of the orientation-dependent chord length distribution function and the covariogram are obtained not only for an arbitrary convex quadrilateral but also for any right prism based on it.

Orientation-Dependent Chord Length Distribution Function for Right Prisms with Rectangular or Right Trapezoidal Bases

The paper continues the research to reconstruct a convex body in $${\mathbb{R}}^{n}$$ from the distribution of characteristics of its $$k$$ -dimensional sections ( $$k<n$$ ). In this paper we obtain explicit expressions for the covariogram and the orientation-dependent chord length distribution of right prisms with rectangular or right trapezoidal bases.

COVARIOGRAM OF A RIGHT PARALLELEPIPED

In this paper we obtain explicit expressions for the covariogram and the orientation-dependent chord length distribution of a right parallelepiped with square base.

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R n with a fixed direction u and length l lies entirely in the bounded convex body D ⊂ R n (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ⊂ D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ⊂ D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.

Orientation-dependent chord length distribution as a function of maximal chord

In [3] and [4] have been proved that covariogramand orientation-dependent chord length distribution function of a triangle and an ellipse depends on maximal chord tmax(u) in direction u. In this paper a necessary condition for orientation-dependent chord length distribution function as a function of maximal chord is obtained. A class of parallelograms is constructed for which this necessary condition is violated.

Orientation-dependent distribution of the length of a random segment and covariogram

The paper considers a random segment L(ω) in Rn with fixed direction and length that intersects a given bounded convex body D ⊂ Rn. We obtain a relationship between the covariogram of D and the distribution of the random variable |L| — the length of L ∩ D. A relationship between the distribution of |L| and the orientation-dependent chord length distribution is also obtained. As a consequence, we obtain a relationship between the chord length distribution function and the distribution of the random variable |L|.

Covariogram of a cylinder

In this paper we establish relationships between the covariogram and the orientation-dependent chord length distribution function of a cylinder and those of its base. Also, we obtain explicit expressions for the covariogram and the orientation-dependent chord length distribution function of a cylinder with cyclic, elliptical and triangular bases.

Covariogram of a parallelogram

In this paper we obtain the explicit forms of the covariogram and the orientation-dependent chord length distribution function for any parallelogram. The explicit form of the chord length distribution function for a parallelogram is also obtained.

English

The paper continues the investigations in (1). We consider the problem of recognition of convex bodies in n-dimensional Euclidean space by k-flats sections, in particular for n=3 by linear or planar sections. This problem is equivalent to recognition of convex bodies by orientation-dependent chord length distributions (for linear sections) and orientation-dependent area distributions (for planar sections). The main goal of the article is to enlarge the class of bodies for which the form of the orientation dependent chord length distribution function and the cross- section area distribution function are known.

Orientation-dependent section distributions for convex bodies

In the paper the orientation dependent chord length distribution functions for some bounded convex domains are obtained. In particular, formulae for the orientation dependent chord length distributions for a regular polygon and an ellipse are obtained. Explicit forms of the orientation dependent chord length distributions were known only in the cases of a disc and a triangle. We also obtain the cross-section area distribution functions for an ellipsoid and a cylinder. The cross-section area distribution function was known only in the case of a ball.

Recognition of triangles by covariogram

The paper considers the problem of recognition of triangles by orientation-dependent chord length distribution. The following results are obtained: 1. The explicit form of the covariogram and orientation-dependent chord length distribution function for a triangle. 2. The explicit form for the chord length distribution function for a triangle. 3. The length of the maximal chord of a triangle is continuous function on direction u ∈ S1 (S1 is the space of all directions in the plane). 4. If we have orientation-dependent chord length distribution function for an everywhere dense set of S1, then we can uniquely recognize the triangle with respect to reflections and translations. 5. For any finite subset A ⊂ S1, there are two non-congruent triangles with the same values of orientationdependent chord length distribution functions on A.

Generalized Pleijel identity

A family of identities primarily associated with isoperimetric inequalities for planar convex domains was discovered by Pleijel in 1956. We call these identities classical Pleijel identities. R. V. Ambartzumian gave combinatorial proof of these identities and pointed out that they can be applied to find chord length distribution functions for convex domains. In the classical Pleijel identities integration is over the measure in the space \( \mathbb{G} \) of lines which is invariant with respect to the all Euclidean motions. In the present paper they are considered for any locally-finite measure in the space \( \mathbb{G} \). These identities are applied to find the so-called orientation-dependent chord length distribution (or density) functions for bounded convex domains.

Orientation-dependent chord length distributions characterize convex polygons

It is shown that the convex polygons are uniquely determined (up to translation and reflection) by their covariograms. The covariogram can be represented by the ‘orientation-dependent chord length distribution', i.e. the distribution of the length of chords which are generated by random lines parallel to fixed directions. Thus the result contributes to answer Blaschke's question about the content of information comprised in chord length distributions.

Orientation-dependent chord length distributions characterize convex polygons

It is shown that the convex polygons are uniquely determined (up to translation and reflection) by their covariograms. The covariogram can be represented by the ‘orientation-dependent chord length distribution', i.e. the distribution of the length of chords which are generated by random lines parallel to fixed directions. Thus the result contributes to answer Blaschke's question about the content of information comprised in chord length distributions.