Arrangements of lines and, in general, arrangements of hyperplanes are paramount data structures in computational geometry whose combinatorial properties have been extensively studied, partially motivated by the point-hyperplane duality. Pseudo-line arrangements are a combinatorial generalization of line arrangements. Defined by Levi in 1926 the full potential of working with these structures was first exploited by Goodman and Pollack.
While pseudo-lines can be considered either as combinatorial or geometric objects, they also lack certain geometric properties that may be needed in proofs. The following example motivated the research presented in this paper.
Consider a finite set of lines that are either red or blue, no two of them parallel and no three of them passing through the same point. Every such arrangement has a bichromatic triangle, i.e., an empty triangular cell bounded by red and blue lines. This can be shown using a distance argument similar to Kelly’s proof of the Sylvester-Gallai theorem (see, e.g., [2, p. 73]). We sketch another nice proof. Think of the arrangement as a union of two monochromatic arrangements in colors blue and red. Continuously translate the red arrangement in positive -direction while keeping the blue arrangement in place. Eventually the combinatorics of the union arrangement will change with a triangle flip, i.e., with a crossing passing a line. The area of monochromatic triangles is not affected by the motion. Therefore, the first triangle that flips is a bichromatic triangle in the original arrangement. See Figure 1 (left).
This argument does not generalize to pseudo-line arrangements. See Figure 1 (right). Actually the question whether all simple bichromatic pseudo-line arrangements have bichromatic triangles is by now open for several years. The crucial property of lines used in the above argument is that shifting a subset of the lines vertically again yields an arrangement, i.e., the shift does not introduce multiple crossings. We were wondering whether any pseudo-line arrangement can be drawn s.t. this property holds. In this paper, we show that this is not true and that arrangements where this is possible constitute an interesting class of pseudo-line arrangements.
Define an arrangement of pseudo-lines as a finite family of -monotone bi-infinite connected curves (called pseudo-lines) in the Euclidean plane s.t. each pair of pseudo-lines intersects in exactly one point, at which they cross. For simplicity, we consider the pseudo-lines to be indexed from to in top-bottom order at left infinity.111Pseudo-line arrangements are often studied in the real projective plane, with pseudo-lines being simple closed curves that do not separate the projective plane. All arrangements can be represented by -monotone arrangements . As -monotonicity is crucial for our setting and the line at infinity plays a special role, we use the above definition. A pseudo-line arrangement is simple if no three pseudo-lines meet in one point; if in addition no two pairs of pseudo-lines cross at the same -coordinate we call it -simple.
An arrangement of approaching pseudo-lines is an arrangement of pseudo-lines where each pseudo-line is represented by function-graph , defined for all , s.t., for any two pseudo-lines and with , the function is monotonically decreasing and surjective. This implies that the pseudo-lines approach each other until they cross, and then they move away from each other, and exactly captures our objective to vertically translate pseudo-lines in an arbitrary way while maintaining the invariant that the collection of curves is a valid pseudo-line arrangement (If is not surjective the crossing of pseudo-lines and may be lost upon vertical translations.) For most of our results, we consider the pseudo-lines to be strictly approaching, i.e., the function is strictly decreasing. For simplicity, we may sloppily call arrangements of approaching pseudo-lines approaching arrangements.
In this paper, we identify various notable properties of approaching arrangements. In Section 2, we show how to modify approaching arrangements and how to decide whether an arrangement is -isomorphic to an approaching arrangement in polynomial time. Then, we show a specialization of Levi’s enlargement lemma for approaching pseudo-lines and use it to show that arrangements of approaching pseudo-lines are dual to generalized configurations of points with an underlying arrangement of approaching pseudo-lines. In Section 5, we describe arrangements which have no realization as approaching arrangement. We also show that asymptotically there are as many approaching arrangements as pseudo-line arrangements. We conclude in Section 6
with a generalization of the notion of being approaching to three dimensions; it turns out that arrangements of approaching pseudo-planes are characterized by the combinatorial structure of the family of their normal vectors at all points.
Restricted representations of Euclidean pseudo-line arrangements have been considered already in early work about pseudo-line arrangements. Goodman  shows that every arrangement has a representation as a wiring diagram. More recently there have been results on drawing arrangements as convex polygonal chains with few bends  and on small grids . Goodman and Pollack  consider arrangements whose pseudo-lines are the function-graphs of polynomial functions with bounded degree. In particular, they give bounds on the degree necessary to represent all isomorphism classes of pseudo-line arrangements. Generalizing the setting to higher dimensions (by requiring that any pseudo-hyperplane can be translated vertically while maintaining that the family of hyperplanes is an arrangement) we found that such approaching arrangements are representations of Euclidean oriented matroids, which are studied in the context of pivot rules for oriented matroid programming (see [4, Chapter 10]).
2 Manipulating approaching arrangements
Lemma 1 shows that we can make the pseudo-lines of approaching arrangements piecewise linear. This is similar to the transformation of Euclidean pseudo-line arrangements to equivalent wiring diagrams. Before stating the lemma it is appropriate to briefly discuss notions of isomorphism for arrangements of pseudo-lines.
Since we have defined pseudo-lines as -monotone curves there are two faces of the arrangement containing the points at infinity of vertical lines. These two faces are the north-face and the south-face. A marked arrangement is an arrangement together with a distinguished unbounded face, the north-face. Pseudo-lines of marked arrangements are oriented such that the north-face is to the left of the pseudo-line. We think of pseudo-line arrangements and in particular of approaching arrangements as being marked arrangements.
Two pseudo-line arrangements are isomorphic iff there is an isomorphism of the induced cell complexes which maps north-face to north-face and respects the induced orientation of the pseudo-lines.
Two pseudo-line arrangements are -isomorphic iff a sweep with a vertical line meets the crossings in the same order.
Both notions can be described in terms of allowable sequences. An allowable sequence is a sequence of permutations tarting with the identity permutation in which (i) a permutation is obtained from the previous one by the reversal of one or more non-overlapping substrings, and (ii) each pair is reversed exactly once. An allowable sequence is simple if two adjacent permutations differ by the reversal of exactly two adjacent elements.
Note that the permutations in which a vertical sweep line intersects the pseudo-lines of an arrangement gives an allowable sequence. We refer to this as the allowable sequence of the arrangement and say that the arrangement realizes the allowable sequence. Clearly two arranements are -isomorphic if they realize the same allowable sequence.
Replacing the vertical line for the sweep by a moving curve (vertical pseudo-line) which joins north-face and south-face and intersects each pseudo-line of the arrangement exactly once we get a notion of pseudo-sweep. A pseudo-sweep typically has various options for making progress, i.e., for passing a crossing of the arrangement. Each pseudo-sweep also produces an allowable sequence. Two arrangements are isomorphic if their pseudo-sweeps yield the same collection of allowable sequences or equivalently if there are pseudo-sweeps on the two arrangements which produce the same allowable sequence.
For any arrangement of approaching pseudo-lines, there is an -isomorphic arrangement of approaching polygonal curves (starting and ending with a ray). If the allowable sequence of the arrangement is simple, then there exists such an arrangement without crossings at the bends of the polygonal curves.
Consider the approaching pseudo-lines and add a vertical ‘helper-line’ at every crossing. Connect the intersection points of each pseudo-line with adjacent helper-lines by segments. This results in an arrangement of polygonal curves between the leftmost and the rightmost helper-line. See Figure 2
. Since the original pseudo-lines were approaching, these curves are approaching as well; the signed distance between the intersection points with the vertical lines is decreasing, and this property is maintained by the linear interpolations between the points. To complete the construction, we add rays in negative-direction starting at the intersection points at the first-helper line; the slopes of the rays are to be chosen s.t. their order reflects the order of the original pseudo-lines at left infinity. After applying the analogous construction at the rightmost helper-line, we obtain the -isomorphic arrangement. If the allowable sequence of the arrangement is simple, we may choose the helper-lines between the crossings and use a corresponding construction. This avoids an incidence of a bend with a crossing. ∎
The construction used in the proof yields pseudo-lines being represented by polygonal curves with a quadratic number of bends. It might be interesting to consider the problem of minimizing bends in such polygonal representations of arrangements. Two simple operations which can help to reduce the number of bends are horizontal streching, i.e., a change of the -coordinates of the helper-lines which preserves their left-to-right order, and vertical shifts which can be applied a helper-line and all the points on it. Both operations preserve the -isomorphism class.
The two operations are crucial for our next result, where we show that the intersection points with the helper-lines can be obtained by a linear program. Asinowski defines a suballowable sequence as a sequence obtained from an allowable sequence by removing an arbitrary number of permutations from it. An arrangement thus realizes a suballowable sequence if we can obtain this suballowable sequence from its allowable sequence.
Given a suballowable sequence, we can decide in polynomial time whether there is an arrangement of approaching pseudo-lines with such a sequence.
We attempt to construct a polygonal pseudo-line arrangement for the given suballowable sequence. As discussed in the proof of Lemma 1, we only need to obtain the points in which the pseudo-lines intersect vertical helper-lines through crossings. The allowable sequence of the arrangement is exactly the description of the relative positions of these points. We can consider the -coordinates of pseudo-line at a vertical helper-line as a variable and by this encode the suballowable sequence as a set of linear inequalities on those variables, e.g., to express that is above at we use the inequality . Further, the curves are approaching iff for all and . These constraints yield a polyhedron (linear program) that is non-empty (feasible) iff there exists such an arrangement. Since the allowable sequence of an arrangement of pseudolines consists of permutations the linear program has inequalities in variables. Note that it is actually sufficient to have constraints only for neighboring points along the helper lines, this shows that inequalities are sufficient. ∎
Let us emphasize that deciding whether an allowable sequence is realizable by a line arrangement is an -hard problem , and thus not even known to be in NP. While we do not have a polynomial-time algorithm for deciding whether there is an isomorphic approaching arrangement for a given pseudo-line arrangement, Theorem 1 tells us that the problem is in NP, as we can give the order of the crossings encountered by a sweep as a certificate for a realization. The corresponding problem for lines is also -hard .
The following observation is the main property that makes approaching pseudo-lines interesting.
Given an arrangement of strictly approaching pseudo-lines and a pseudo-line , any vertical translation of in results again in an arrangement of strictly approaching pseudo-lines.
Doing an arbitrary translation, we may run into trouble when the pseudo-lines are not strictly approaching. In this case it can happen that two pseudo-lines share an infinite number of points. The following lemma allows us to ignore non-strictly approaching arrangements for many aspects.
Any simple arrangement of approaching pseudo-lines is homeomorphic to an -isomorphic arrangement of strictly approaching pseudo-lines.
Given an arrangement , construct a polygonal arrangement as described for Lemma 1. If the resulting pseudo-lines are strictly approaching, we are done. Otherwise, consider the rays that emanate to the left. We may change their slopes s.t. all the slopes are different and their relative order remains the same. Consider the first vertical slab defined by two neighboring vertical lines and that contains two segments that are parallel (if there are none, the arrangement is strictly approaching). Choose a vertical line slightly to the left of the slab and use and as helper-lines to redraw the pseudo-lines in the slab. Since the arrangement is simple the resulting arrangement is -isomorphic and it has fewer parallel segments. Iterating this process yields the desired result. ∎
If is an approaching arrangement with a non-simple allowable sequence, then there exists an approaching arrangement whose allowable sequence is a refinement of the allowable sequence of , i.e., the sequence of may have additional permutations between consecutive pairs in the sequence of .
Since its allowable sequence is non-simple arrangement has a crossing point where more than two psudo-lines cross or has several crossings with the same -coordinate. Let be a pseudo-line participating in such a degeneracy. Translating slightly in vertical direction a degeneracy is removed and the allowable sequence is refined. ∎
Ringel’s homotopy theorem [4, Thm. 6.4.1] tells us that given a pair , of pseudo-line arrangements, can be transformed to by homeomorphisms of the plane and so-called triangle flips, where a pseudo-line is moved over a crossing. Within the subset of arrangements of approaching pseudo-lines, the result still holds. We first show a specialization of Ringel’s isotopy result [4, Prop. 6.4.2]:
Two -isomorphic arrangements of approaching pseudo-lines can be transformed into each other by a homeomorphism of the plane s.t. all intermediate arrangements are -isomorphic and approaching.
Given an arrangement ‘ of approaching pseudo-lines, we construct a corresponding polygonal arrangement
. Linearly transforming a pointon a pseudo-line in to the point on the corresponding line in gives a homeomorphism from to which can be extended to the plane. Given two -isomorphic arrangements and of polygonal approaching pseudo-lines, we may shift helper-lines horizontally, so that the helper-lines of the two arrangements become adjusted, i.e., are at the same -coordinates; again there is a corresponding homeomorphism of the plane. Now recall that these arrangements can be obtained from solutions of linear programs. Since and have the same combinatorial structure, their defining inequalities are the same. Thus, a convex combination of the variables defining the two arrangements is also in the solution space, which continuously takes us from to and thus completes the proof. ∎
Given two simple arrangements of approaching pseudo-lines, one can be transformed to the other by homeomorphisms of the plane and triangle flips s.t. all intermediate arrangement are approaching.
Let be a fixed simple arrangement of lines. We show that any approaching arrangement can be transformed into with the given operations. Since the operations are invertible this is enough to prove that we can get from to .
Consider a vertical line in such that all the crossings of are to the right of and replace the part of the pseudo-lines of left of by rays with the slopes of the corresponding lines of . This replacement is covered by Lemma 4. Let be a vertical line in which has all the crossings of to the left. Now we vertically shift the pseudo-lines of to make their intersections with an identical copy of their intersections with . During the shifting we have a continuous family of approaching arrangements which can be described by homeomorphisms of the plane and triangle flips. At the end the order of the intersections on is completely reversed, all the crossings are left of where the pseudo-lines are straight and use the slopes of . It remains to replace the part of the pseudo-lines of to the right of by rays with the slopes of the corresponding ∎
Note that the proof requires the arrangement to be simple. Vertical translations of pseudo-lines now allows us to prove a restriction of our motivating question.
An arrangement of approaching red and blue pseudo-lines contains a triangular cell that is bounded by both a red and a blue pseudo-line unless it is a pencil, i.e., all the pseudo-lines cross in a single point.
By symmetry in color and direction we may assume that there is a crossing of two blue pseudo-lines above a red pseudo-line. Translate all the red pseudo-lines upwards with the same speed. Consider the first momentwhen the isomorphism class changes. This happens when a red pseudo-line moves over a blue crossing, or a red crossing is moved over a blue pseudo-line. In both cases the three pseudo-lines have determined a bichromatic triangular cell of the original arrangement.
Now consider the case that at time parallel segments of different color are concurrent. In this case we argue as follows. Consider the situation at time right after the start of the motion. Now every multiple crossing is monochromatic and we can use an argument as in the proof of Lemma 2 to get rid of parallel segments of different colors. Continuing the translation after the modification reveals a bichromatic triangle as before. ∎
3 Levi’s lemma for approaching arrangements
Proofs for showing that well-known properties of line arrangements generalize to pseudo-line arrangements often use Levi’s enlargement lemma. (For example, Goodman and Pollack  give generalizations of Radon’s theorem, Helly’s theorem, etc.) Levi’s lemma states that a pseudo-line arrangement can be augmented by a pseudo-line through any pair of points. In this section, we show that we can add a pseudo-line while maintaining the property that all pseudo-lines of the arrangement are approaching.
Given an arrangement of approaching pseudo-lines containing two pseudo-lines and (each a function ), consider , for some . The arrangement augmented by is still an arrangement of approaching pseudo-lines.
Consider any pseudo-line of the arrangement, . We know that for , , whence . Similarly, we have . Adding these two inequalities, we get
The analogous holds for any . ∎
The lemma gives us a means of producing a convex combination of two approaching pseudo-lines with adjacent slopes. Note that the adjacency of the slopes was necessary in the above proof.
Given an arrangement of approaching pseudo-lines, we can add a pseudo-line for any and still have an approaching arrangement.
With we also get for all . ∎
Given an arrangement of strictly approaching pseudo-lines and two points and with different -coordinates, the arrangement can be augmented by a pseudo-line containing and to an arrangement of approaching pseudo-lines. Further, if and do not have the same vertical distance to a pseudo-line of the initial arrangement, then the resulting arrangement is strictly approaching.
Let have smaller -coordinate than . Vertically translate all pseudo-lines such that they pass through (the pseudo-lines remain strictly approaching, forming a pencil through ). If there is a pseudo-line that also passes through , we add a copy of it. If is between and , then we find some such that contains and . By Lemma 5 we can add to the arrangement. If is above or below all pseudo-lines in the arrangement, we can use Lemma 6 to add a pseudo-line; we choose large enough such that the new pseudo-line contains . Finally translate all pseudo-lines back to their initial position. This yields an approaching extension of the original arrangement with a pseudo-line containing and . Observe that the arrangement is strictly approaching unless the new pseudo-line was chosen as copy of . ∎
Following Goodman et al. , a spread of pseudo-lines in the Euclidean plane is an infinite family of simple curves such that
each curve is asymptotic to some line at both ends,
every two curves intersect at one point, at which they cross, and
there is a bijection from the unit circle to the family of curves such that is a continuous function (under the Hausdorff metric) of .
It is known that every projective arrangement of pseudolines can be extended to a spread  (see also ). For Euclidean arrangements this is not true because condition 1 may fail (for an example take the parabolas as pseudo-lines). However, given an Euclidean arrangement we can choose two vertical lines and such that all the crossings are between and and replace the extensions beyond the vertical lines by appropriate rays. The reult of this procedure is called the truncation of . Note that the truncation of and are -isomorphic and if is approaching then so is the truncation. We use Lemma 5 to show the following.
The truncation of every approaching arrangement of pseudo-lines can be extended to a spread of pseudo-lines and a single vertical line such that the non-vertical pseudo-lines of that spread are approaching.
Let be the pseudo-lines of the truncation of an approaching arrangement. Add two almost vertical straight lines and such that the slope of the line connecting two points on a pseudoline is between the slopes of and . The arrangement with pseudo-lines is still approaching. Initialize with these pseudolines. For each and each add the pseudo-line to . The proof of Lemma 5 implies that any two pseudo-lines in are approaching. Finally, let be the intersection point of and and add all the lines containing and some point above these two lines to . This completes the construction of the spread . ∎
4 Approaching generalized configurations
Levi’s lemma is the workhorse in the proofs of many properties of pseudo-line arrangements. Among these, there is the so-called double dualization by Goodman and Pollack  that creates, for any arrangement of pseudo-lines, a corresponding primal generalized configuration of points.
A generalized configuration of points is an arrangement of pseudo-lines with a specified set of vertices, called points, such that any pseudo-line passes through two points, and, at each point, pseudo-lines cross. We assume for simplicity that there are no other vertices in which more than two pseudo-lines of the arrangement cross.
Let be a generalized configuration of points consisting of an approaching arrangement , and a set of points , which are labeled by increasing -coordinate. We denote the pseudo-line of connecting points by .
Consider a point moving from top to bottom at left infinity. This point traverses all the pseudo-lines of in some order. We claim that if we start at the top with the identity permutation , then, when passing we can apply the (adjacent) transposition to . Moreover, by recording all the permutations generated during the move of the point we obtain an allowable sequence .
Consider the complete graph on the set . Let be an unbounded cell of the arrangement , when choosing as the north-face of we get a left to right orientation on each . Let this induce the orientation of the edge of . These orientations constitute a tournament on . It is easy to verify that this tournament is acyclic, i.e., it induces a permutation on .
The order corresponding to the top cell equals the left-to-right order on . Since we have labeled the points by increasing -coordinate this is the identity.
When traversing to get from a cell to an adjacent cell the two orientations of the complete graph only differ in the orientation of the edge . Hence, and are related by the adjacent transposition .
The allowable sequence and the allowable sequence of are different objects, they differ even in the length of the permutations.
We say that an arrangement of pseudo-lines is dual to a (primal) generalized configuration of points if they have the same allowable sequence. Goodman and Pollack  showed that for every pseudo-line arrangement there is a primal generalized configuration of points, and vice versa. We prove the same for the sub-class of approaching arrangements.
For every generalized configuration of points on an approaching arrangement , there is an approaching arrangement with allowable sequence .
Let . We call the adjacent transposition at when . To produce a polygonal approaching arrangement we define the -coordinates of the pseudo-lines at -coordinates . Let be the transposition at . Consider the pseudo-line of . Since is -monotone we can evaluate . The -coordinate of the pseudo-line dual to the point at is obtained as .
We argue that the resulting pseudo-line arrangement is approaching. Let and be transpositions at and , respectively, and assume . We have to show that , for all . From it follows that is left of , i.e., . and are approaching, we get , i.e., , which translates to . This completes the proof. ∎
Goodman and Pollack use the so-called double dualization to show how to obtain a primal generalized configuration of points for a given arrangement of pseudo-lines. In this process, they add a pseudo-line through each pair of crossings in , using Levi’s enlargement lemma. This results in a generalized configuration of points, where the points are the crossings of . From this, they produce the dual pseudo-line arrangement . Then, they repeat the previous process for (that is, adding a line through all pairs of crossings of ). The result is a generalized configuration of points, which they show being the primal generalized configuration of . With Theorem 4 and Lemma 7, we know that both the augmentation through pairs of crossings and the dualization process can be done such that we again have approaching arrangements, yielding the following result.
For every arrangement of approaching pseudo-lines, there is a primal generalized configuration of points whose arrangement is also approaching.
An allowable sequence is the allowable sequence of an approaching generalized configuration of points if and only if it is the allowable sequence of an approaching arrangement.
5 Realizability and counting
Considering the freedom one has in constructing approaching arrangements, one may wonder whether actually all pseudo-line arrangements are -isomorphic to approaching arrangements. As we will see in this section, this is not the case. We use the following lemma, that can easily be shown using the construction for Lemma 1.
Given a simple suballowable sequence of permutations , where is the identity permutation, the suballowable sequence is realizable with an arrangement of approaching pseudo-lines if and only if it is realizable as a line arrangement.
Consider any realization of the simple suballowable sequence with an arrangement of approaching pseudo-lines. Since the arrangement is simple, we can consider the pseudo-lines as being strictly approaching, due to Lemma 2. There exist two vertical lines and s.t. the order of intersections of the pseudo-lines with them corresponds to and , respectively. We claim that replacing pseudo-line by the line connecting the points and we obtain a line arrangement representing the suballowable sequence .
To prove the claim we verify that for the slope of is less than the slope of . Since is approaching we have , i.e., . The slopes of and are obtained by dividing both sides of this inequality by , which is negative. ∎
Asinowski  identified a suballowable sequence , with permutations of six elements which is not realizable with an arrangement of lines.
There exist simple suballowable sequences that are not realizable by arrangements of approaching pseudo-lines.
With the modification of Asinowski’s example shown in Figure 3, we obtain an arrangement not having an isomorphic approaching arrangement. The modification adds two almost-vertical lines crossing in the north-cell s.t. they form a wedge crossed by the lines of Asinowski’s example in the order of . We do the same for . The resulting object is a simple pseudo-line arrangement, and each isomorphic arrangement contains Asinowski’s sequence.
There are pseudo-line arrangements for which there exists no isomorphic arrangement of approaching pseudo-lines.
Aichholzer et al.  construct a suballowable sequence on lines s.t. all line arrangements realizing them require slope values that are exponential in the number of lines. Thus, also vertex coordinates in a polygonal representation as an approaching arrangement are exponential in .
Ringel’s Non-Pappus arrangement  shows that there are allowable sequences that are not realizable by straight lines. It is not hard to show that the Non-Pappus arrangement has a realization with approaching pseudo-lines. We will show that in fact the number of approaching arrangements, is asymptotically larger than the number of arrangements of lines.
There exist isomorphism classes of simple arrangements of approaching pseudo-lines.
The upper bound follows from the number of non-isomorphic arrangements of pseudo-lines. Our lower-bound construction is an adaptation of the construction presented by Matoušek [16, p. 134] for general pseudo-line arrangements. See the left part Figure 4 for a sketch of the construction. We start with a construction containing parallel lines that we will later perturb. Consider a set of vertical lines , for . Add horizontal pseudo-lines , for . Finally, add parabolic curves , defined for , some , and (we will add the missing part towards left infinity later). Now, passes slightly below the crossing of and at . See the left part Figure 4 for a sketch of the construction. We may modify to pass above the crossing at by replacing a piece of the curve near this point by a line segment with slope ; see the right part of Figure 4. Since the derivatives of the parabolas are increasing and the derivatives of at and of at are both the vertical distances from the modified to and remain increasing, i.e., the arrangement remains approaching.
For each crossing , we may now independently decide whether we want to pass above or below the crossing. The resulting arrangement contains parallel and vertical lines, but no three points pass through a crossing. This means that we can slightly perturb the horizontal and vertical lines s.t. the crossings of a horizontal and a vertical remain in the vicinity of the original crossings, but no two lines are parallel, and no line is vertical. To finish the construction, we add rays from the points on with , each having the slope of at . Each arrangement of the resulting class of arrangements is approaching. We have crossings for which we make independent binary decisions. Hence the class consists of approaching arrangements of pseudo-lines. ∎
As there are only isomorphism classes of simple line arrangements , we see that we have way more arrangements of approaching pseudo-lines.
The number of allowable sequences is . We show next that despite of the existence of nonrealizable suballowable sequences (Corollary 1), the number of allowable sequences for approaching arrangements, i.e., the number of -isomorphism classes of these arrangements, is asymptotically the same as the number of all allowable sequences.
There are allowable sequences realizable as arrangements of approaching pseudo-lines.
The upper bound follows from the number of allowable sequences. For the lower bound, we use the construction in the proof of Theorem 7, but omit the vertical lines. Hence, we have the horizontal pseudo-lines and the paraboloid curves , defined for and . For a parabolic curve and a horizontal line , consider the neighborhood of the point . Given a small value we can replace a piece of by the appropriate line segment of slope such that the crosssing of and the modified has -coordinate .
For fixed and any permutation of we can define values for such that . Choosing the offset values according to different permutations yields different vertical permutations in the neighborhood of , i.e., the allowable sequences of the arrangements differ. Hence, the number allowable sequences of approaching arrangements is at least the superfactorial , which is in . ∎
We have seen that some properties of arrangements of lines are inherited by approaching arrangements. It is known that every simple arrangement of pseudo-lines has triangles, the same is true for non-simple non-trivial arrangements of lines, however, there are non-simple non-trivial arrangements of pseudo-lines with fewer triangles, see . We conjecture that in this context approaching arrangements behave like line arrangements.
Every non-trivial arrangement of approaching pseudo-lines has at least triangles.
6 Higher dimensions
An arrangement of pseudo-hyperplanes in is a finite set of hypersurfaces, each homeomorphic to , with the property that any of them intersect as hyperplanes (no two of them parallel) do. More formally, for any , , the cell complex induced by is isomorphic to the cell complex of where is the hyperplane whose normal is the th vector of the standard basis.
We focus on arrangements of pseudo-planes in . We define arrangements of approaching pseudo-planes via one of the key properties observed for arrangements of approaching pseudo-lines (Observation 1).
An arrangement of approaching pseudo-planes in is an arrangement of pseudo-planes where each pseudo-plane is the graph of a continuously differentiable function such that for any , the graphs of form a valid arrangement of pseudo-planes. This means that we can move the pseudo-planes up and down along the -axis while maintaining the properties of a pseudo-plane arrangement. Clearly, arrangements of planes (without parallels) are approaching.
Consider an arrangement of approaching pseudo-lines with pseudo-lines given by continuously differentiable functions. The condition that is strictly monotonically decreasing implies that for any the slope of is at most the slope of , where at some single points, they might be equal, e.g. at for and . In other words for each the identity permutation is the sorted order of the slopes of the tangents at . Note that we may think of permutations as labeled Euclidean order types in one dimension.
In this section we show an analogous characterization of approaching arrangements of pseudo-planes: The two-dimensional order type associated to the tangent planes above a point is the same except for a sparse set of exceptional points where the order type may degenerate.
Let be a collection of graphs of continuously differentiable functions . For any point in , let be the upwards normal vector of the tangent plane of above . We consider the vectors as points in the plane with homogeneous coordinates. (That is, for each vector we consider the intersection of its ray with the plane .) We call a characteristic point and let be the set of characteristic points. The Euclidean order type of the point multiset is the characteristic order type of at , it is denoted .
We denote by the set of characteristic order types of on the whole plane, that is, . We say that is admissible if the following conditions hold:
for any two points and in the plane, we have that if an ordered triple of characteristic points in is positively oriented, then the corresponding triple in is either positively oriented or collinear;
for any triple of characteristic points, the set of points in the plane for which are collinear is either the whole plane or a discrete set of points (i.e, for each in this set there is some such that the -disc around contains no further point of the set);
for any pair of characteristic points, the set of points in the plane for which has dimension 0 or 1 (this implies that for each in this set and each the -disc around contains points which are not in the set).
From the above conditions, we deduce another technical but useful property of admissible characteristic order types.
Let be an admissible order type and . For any pair and for every point in the plane for which there is a neighborhood such that for , the positive hull of contains no line.
Choose such that . In a small neighborhood of point will stay away from the line spanned by and (continuity). If in the positive hull of contains a line, then the orientation of changes from positive to negative in , this contradicts condition (1) of admissible characteristic order types. ∎
Let be a collection of graphs of continuously differentiable functions . Then is an arrangement of approaching pseudo-planes if and only if is admissible and all the differences between two functions are surjective.
Note that being surjective is a necessary condition for the difference of two functions, as otherwise we can translate them until they do not intersect. Thus, in the following, we will assume that all the differences between two functions are surjective. We first show that if is admissible then is an arrangement of approaching pseudo-planes. Suppose is not an arrangement of approaching pseudo-planes. Suppose first that there are two functions and in whose graphs do not intersect in a single pseudo-line. Assume without loss of generality that , i.e., is the constant zero function. Let denote the intersection of the graphs of and . If the intersection has a two-dimensional component, the normal vectors of the two functions are the same for any point in the relative interior of this component, which contradicts condition (3), so from now on, we assume that is at most one-dimensional. Also, note that due to the surjectivity of , the intersection is not empty. Note that if is a single pseudo-line then for every there exists a neighborhood in such that is a pseudo-segment. Further, on one side of the pseudo-segment, is below , and above on the other, as otherwise we would get a contradiction to Lemma 10. In the next two paragraphs we argue that indeed is a single pseudo-line. In paragraph (a) we show that for every the intersection locally is a pseudo-segment; in (b) we show that contains no cycle and that has a single connected component.
(a) Suppose for the sake of contradiction that contains a point such that for every neighborhood of in we have that is not a pseudo-segment. For let be the -disc around . Consider small enough such that consists of a single connected component. Further, let be small enough such that whenever we walk away from in a component where is above (below) , the difference is monotonically increasing (decreasing). The existence of such an follows from the fact that and are graphs of continuously differentiable functions. Then partitions into several connected components , ordered in clockwise order around . In each of these components, is either above or below , and this sidedness is different for any two neighboring components. In particular, the number of components is even, that is, , for some natural number . We will distinguish the cases where
is even and odd, and in both cases we will first show that atwe have and then apply Lemma 10.
We start with the case where is even. Consider a differentiable path starting in , passing through and ending in . As is even, is above in if and only if is also above in . In particular, the directional derivative of for at is . This holds for every choice of and , thus at all directional derivatives of vanish. This implies that at the normal vectors of and , coincide, hence . Now, consider the boundary of . Walking along this boundary, is the constant zero function, and thus the directional derivatives vanish. Hence, at any point on this boundary, must be orthogonal to the boundary, pointing away from if is above in , and into otherwise. Let now and be the intersections of the boundary of with the boundary of . The above argument gives us two directions of vectors, and , and a set of possible directions of vectors , , between them. By continuity, all of these directions must be taken somewhere in (see Figure 5 for an illustration). Let now be the set of all components where is above , and let be the set of all directions of vectors , . Further, let be the set of rays emanating from which are completely contained in . By continuity, for every small enough , there are two rays in which together span a line. It now follows from the above arguments, that for these , the directions in also positively span a line. This is a contradiction to Lemma 10.
Let us now consider the case where is odd. Consider the boundary between and and denote it by . Similarly, let be the boundary between and . Let now be the path defined by the union of and and consider the vectors when walking along . Assume without loss of generality that , and thus and . Analogous to the arguments in the above case, along the vectors are orthogonal to , pointing from into . In particular, they always point to the same side of . However, at the path is also incident to and to . The same argument now shows that at , the vector must point from into , that is, into the other side of . This is only possible if , and thus, as claimed, we again have at . We can now again consider the set of directions , and this time, for every small enough , the set is the set of all possible directions (see Figure 6 for an illustration), which is again a contradiction to Lemma 10. This concludes the proof of claim (a).
(b) Suppose that the intersection contains a cycle. In the interior of the cycle, one function is above the other, so we can vertically translate it until the cycle contracts to a point, which again leads to a contradiction to Lemma 10. Now suppose that the intersection contains two disjoint pseudo-lines. Between the pseudo-lines, one function is above the other, so we can vertically translate it until the pseudo-lines cross or coincide. If they cross, we are again in the case discussed in (a) and get a contradiction to Lemma 10. If they coincide, has the same sign on both sides of the resulting pseudo-line which again leads to a contradiction to Lemma 10.
Thus, we have shown that if is admissible then any two pseudo-planes in intersect in a single pseudo-line.
Now consider three functions such that any two intersect in a pseudo-line but the three do not form a pseudo-hyperplane arrangement. Then in one of the three functions, say , the two pseudo-lines defined by the intersections with the other two functions do not form an arrangement of two pseudo-lines; after translation, we can assume that they touch at a point or intersect in an interval. First assume that they touch at a point. At this touching point, one normal vector of tangent planes is the linear combination of the other two: assume again without loss of generality that . Further assume without loss of generality that the curves and touch at the point and that the -axis is tangent to at this point. Then, as the two curves touch, the -axis is also tangent to . In particular, the normal vectors to both and lie in the --plane. As the normal vector to lies on the -axis, the three normal vectors are indeed linearly dependent. For the order type, this now means that one vector is the affine combination of the other two, i.e., the three vectors are collinear. Further, on one side of the point the three vectors are positively oriented, on the other side they are negatively oriented, which is a contradiction to condition (1). On the other hand, if they intersect in an interval, then the set of points where the vectors are collinear has dimension greater than 0 but is not the whole plane, which is a contradiction to condition (2).
This concludes the proof that if is admissible then is an arrangement of approaching pseudo-planes.
For the other direction consider an approaching arrangement of pseudo-planes and assume that is not admissible. First, assume that condition (1) is violated, that is, there are three pseudo-planes whose characteristic points change their orientation from positive to negative. In particular, they are collinear at some point. Assume without loss of generality that and are planes containing the origin whose characteristic points are thus constant, and assume without loss of generality that they are and . In particular, the intersection of and is the -axis in . Consider now a -disc around the origin in and let , and be the subsets of with , and , respectively. Assume without loss of generality that in the characteristic point is to the left of the -axis in , to the right in , and on the -axis in . Also, assume that contains the origin in . But then, is below the -plane everywhere in . In particular, touches in a single point, namely the origin. Hence, and is not an arrangement of two pseudo-lines in .
Similar arguments show that
if condition (2) is violated, then after some translation the intersection of some two pseudo-planes in a third one is an interval,
if condition (3) is violated, then after some translation the intersection of some two pseudo-planes has a two-dimensional component,
On the other hand, from the above it does not follow to what extent an arrangement of approaching pseudo-planes is determined by its admissible family of characteristic order types. In particular, we would like to understand which admissible families of order types correspond to families of characteristic order types. To that end, note that for every graph in an arrangement of approaching pseudo-planes, the characteristic points define a vector field , namely its gradient vector field (a normal vector can be written as .) In particular, the set of all graphs defines a map with the property that and the order type of is . We call the family of vector fields obtained by this map the characteristic field of . A classic result from vector analysis states that a vector field is a gradient vector field of a scalar function if and only if it has no curl. We thus get the following result:
Let be a family of vector fields. Then is the characteristic field of an arrangement of approaching pseudo-planes if and only if each is curl-free and for each , the set of order types defined by is admissible.
Let now be an arrangement of approaching pseudo-planes. A natural question is, whether can be extended, that is, whether we can find a pseudo-plane such that is again an arrangement of approaching pseudo-planes. Consider the realization of for some . Any two points in this realization define a line. Let be the line arrangement defined by all of these lines. Note that even if is the same order type for every , the realization might be different and thus there might be a point such that is not isomorphic to . For an illustration of this issue, see Figure 7. (This issue also comes up in the problem of extension of order types, e.g. in , where the authors count the number of order types with exactly one point in the interior of the convex hull.)
We call a cell of admissible, if its closure is not empty in for every . Clearly, if we can extend with a pseudo-plane , then characteristic point of the normal vector must lie in an admissible cell . On the other hand, as is admissible, it is possible to move continuously in , and if all the vector fields are curl-free, then so is the vector field obtained this way. Thus, is the vector field of a differentiable function and by Corollary 3, its graph extends . In particular, can be extended if and only if contains an admissible cell. As the cells incident to a characteristic point are always admissible, we get that every arrangement of approaching pseudo-planes can be extended. Furthermore, by the properties of approaching pseudo-planes, can be chosen to go through any given point in . In conclusion, we get the following:
Let be an arrangement of approaching pseudo-planes and let be a point in . Then there exists a pseudo-plane such that is an arrangement of approaching pseudo-planes and lies on .
On the other hand, it could possible that no cell but the ones incident to a characteristic point are admissible, heavily restricting the choices for . In this case, every pseudo-plane that extends is essentially a copy of one of the pseudo-planes of . For some order types, there are cells that are not incident to a characteristic point but still appear in every possible realization, e.g. the unique -gon defined by points in convex position. It is an interesting open problem to characterize the cells which appear in every realization of an order type.
In this paper, we introduced a type of pseudo-line arrangements that generalize line arrangements, but still retain certain geometric properties. One of the main algorithmic open problems is deciding the realizability of a pseudo-line arrangement as a isomorphic approaching arrangement. Further, we do not know how projective transformations influence this realizability. The concept can be generalized to higher dimensions. Apart from the properties we already mentioned in the introduction, we are not aware of further non-trivial observations. Eventually, we hope for this concept to shed more light on the differences between pseudo-line arrangements and line arrangements. For higher dimensions, we gave some insight into the structure of approaching hyperplane arrangements via the order type defined by their normal vectors. It would be interesting to obtain further properties of this setting.
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