Abstract
Can one determine a centrally symmetric lattice polygon by its projections? In 2005, Gardner et al. proposed the above discrete version of Aleksandrov’s projection theorem. In this paper, we define a coordinate matrix for a centrally symmetric convex lattice set and suggest an algorithm to study this problem.
1. Introduction
According to Gardner [1], geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross sections by planes. When the geometric object is a convex body, there are many results from convex geometry. A famous example is the celebrated Aleksandrov’s projection theorem, that is, for an origin symmetric convex body (compact, convex subset with nonempty interior in the -dimensional Euclidean space ); it can be determined by its projections. For more details, see [1]. When the object is a convex lattice set, it models the atoms in a crystal which will lead to a range of topics concerned with electron microscopy in discrete tomography [2, 3].
If is a set in , we denote the convex hull of by . The dimension of is the dimension of its affine hull and is denoted by . The notation for the usual orthogonal projection of on a subspace is . A convex lattice set is a finite subset of the -dimensional integer lattice such that and we denote the cardinality of by . Let denote the set of all centrally symmetric convex lattice sets about origin of .
In 2005, Gardner et al. [4] proposed the following discrete version of Aleksandrov’s projection theorem.
Question 1. Let and belong to with . If holds for every hyperplane with a norm , do and coincide up to a translation?
In [4], Gardner et al. discovered a counterexample to this problem in with cardinality (Figure 1). In [5], Zhou proved that the counterexample discovered by Gardner et al. was the only counterexample in , up to unimodular transformations and with cardinality not larger than 17. In [6], Xiong proved that an origin symmetric convex lattice set can be uniquely determined by its lattice projection counts if -coordinate’s absolute value is not bigger than 2 for every point in this origin symmetric convex lattice set and its cardinality is not 11.
All the results above [4–6] show that the discrete Aleksandrov’s projection theorem must have additional hypotheses. Motivated by the work of Gardner, Gronchi, Zong, Zhou, and Xiong, in the present paper, we introduced the coordinate matrix for convex lattice set and present some results about coordinate matrix. Considering that counting of projection of a convex lattice set in the discrete Aleksandrov’s projection theorem is important, we propose an algorithm based on the coordinate matrix and its properties.
2. Coordinate Matrix
Let belong to . If has points on the line , where and , then we denote by . Clearly, since is an origin symmetric convex lattice set, we have that , .
Definition 1. For , denote the minimum -coordinate by and the maximum -coordinate by on the line , respectively. Define the coordinate matrix of by the following:
Obviously, a convex lattice set and its coordinate matrix can be uniquely determined by each other.
Example 2. Considering the coordinate matrix of in Figure 1, we get
We summarize some basic properties of coordinate matrix as the following propositions.
Theorem 3. Letbe its coordinate matrix. Then has the following properties:(1).(2).(3).
Proof. Letbe its coordinate matrix.(1)By the definition of the coordinate matrix and the symmetry of , we have (2)From the the definition of the coordinate matrix , we can get it.(3)Note that is symmetric about the origin.
For and its coordinate matrixwe can denote by , , and any three points in , where , and , , .
Remark 4. Here , , and denote one of the following cases: for , , , and .
Ifwe know that , , and can be regarded as three vertexes of a nondegenerate triangle.
Then, with and , we have that is a convex combination of , , and . If , by the convexity of , belongs to . Thus we obtain the following theorem.
Theorem 5. Letwith and , . Then is the coordinate matrix of some if and only if for any with and there exist in such that
3. Algorithm for Counting of Projection of a Convex Lattice Set
Let . The primitive directions of , is defined as Here, gcd represents the greatest common divisor.
Letbe the coordinate matrix of , and . Based on the line segment intersection algorithm [7], the computation of planar convex hulls algorithm [7], and the coordinate matrix of , an algorithm named as plane sweep algorithm about is given in Algorithm 1.
| ||||||||||||||||||||||||||||||||||||||||
Remark 6. Moving a horizontal sweep line downwards over the plane, the sweep line halts at certain event points. If sweep line reaches an event point, then a lattice point must be added to event queue . This point is tested against the ones already have been reached. If two points have the same -coordinate, then the one with smaller -coordinate will be added to event queue first.
Remark 7. Based on Theorem 3, is a matrix and it presents points. We denote by . Thus could be recorded as .
Remark 8. , , and are obtained after the While-loop is completed. At the end of Algorithm 2, and represent the number of points in and , respectively. is defined in Algorithm 3.
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ||||||||||||||||
Remark 9. The points on the -axis should be considered separately because their symmetric points about the origin are still on the -axis. The If-loop at the end of Algorithm 1 is used to do this. represents the number of points whose -coordinates are zero in .
The procedures for finding the lattice points of the projection overlapped as follows: we simply test if the direction of a straight line that is determined by two lattice points is parallel to the primitive directions. We could delete the lattice points which conform to the conditions in order to simplify the computation. Hence, the procedure HANDLEEVENTPOINT is defined as in Algorithm 2.
By the origin symmetry, we have less calculation of nearly half of the points. The next procedure ORIGINSYMMETRY, defined as in Algorithm 3, is used to handle the missing points which are caused by the origin symmetry. In this procedure the way for process the lattice points whose projection will be overlap is the same as the procedure HANDLEEVENTPOINT.
Remark 10. If we do not consider the origin symmetry, we should calculate points. This will take time.
The time required for the parts of HANDLEEVENTPOINT and ORIGINSYMMETRY dominates the total running time. In HANDLEEVENTPOINT, considering the worst case, we check every point with other points whether they are in the same straight line with primitive direction. This will take time. In ORIGINSYMMETRY, we check points with other points. This will take time.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by the Young Talent Program of Beijing (YETP0770), the State Forestry Administration 948 Project (no. 2013-4-66), the Fundamental Research Funds for the Central Universities (no. 2015ZCQ-LY-01), and the National Natural Science Foundation of China (nos. 11371224 and 11001014).