On the Complexity of the k-Level in Arrangements of Pseudoplanes

1 Introduction

Let $Λ$ be a set of $n$ non-vertical planes (resp., pseudoplanes, as will be formally defined shortly) in $R^{3}$ , in general position. We say that a point $p$ lies at level $k$ of the arrangement $A (Λ)$ , and write $λ (p) = k$ , if exactly $k$ planes (resp., pseudoplanes) of $Λ$ pass below $p$ . The $k$ -level of $A (Λ)$ is the closure of the set of points that lie on the planes of $Λ$ and are at level $k$ . Our goal is to obtain an upper bound on the complexity of the $k$ -level of $A (Λ)$ , which is measured by the number of vertices of $A (Λ)$ that lie at level $k$ . (The level may also contain vertices at level $k - 1$ or $k - 2$ , but we ignore this issue – it does not effect the worst-case asymptotic bound that we are after.) Using a standard duality transform that preserves the above/below relationship (see, e.g., [10]), the case of planes is the dual version of the following variant of the $k$ -set problem: given a set of $n$ points in $R^{3}$ in general position, how many triangles spanned by $P$ are such that the plane supporting the triangle has exactly $k$ points of $P$ below it? We refer to these triangles as k-triangles. This has been studied by Dey and Edelsbrunner [9], in 1994, for the case of halving triangles, namely $k$ -triangles with $k = (n - 3) / 2$ (and $n$ odd). They have shown that the number of halving triangles is $O (n^{8 / 3})$ . In 1998, Agarwal et al. [1] generalized this result for $k$ -triangles, for arbitrary $k$ , showing that their number is $O (n k^{5 / 3})$ , using a probabilistic argument. In 1999, Sharir, Smorodinsky and Tardos [15] improved the upper bound for the number of $k$ -triangles in $S$ to $O (n k^{3 / 2})$ .

The three-dimensional case extends the more extensively studied planar case. In its primal setting, we have a set $S$ of $n$ points in the plane in general position, and a parameter $k < n$ , and we seek bounds on the maximum number of $k$ -edges, which are segments spanned by pairs of points of $S$ so that one of the halfplanes bounded by the line supporting the segment, say the lower halfplane, contains exactly $k$ points of $S$ . In the dual version, we seek bounds on the maximum number of vertices of an arrangement of $n$ nonvertical lines in general position that lie at level $k$ . The best known upper bound for this quantity, due to Dey [8], is $O (n k^{1 / 3})$ , and the best known lower bound, due to Tóth [18] is $n e^{Ω (\sqrt{ln k})}$ (Nivasch [13] has slightly improved this bound for the case of halving edges).

In this paper we consider the dual version of the problem in three dimensions, where the points are mapped to planes, and the $k$ -triangles are mapped to vertices of the arrangement of these planes at level $k$ . We translate parts of the machinery developed in [15] to the dual setting, and then extend it to handle the case of pseudoplanes. In the primal setting, we have a set $S$ of $n$ points in $R^{3}$ in general position, and the set $T$ of $k$ -triangles spanned by $S$ . We say that triangle $Δ_{1}$ crosses another triangle $Δ_{2}$ if the triangles share exactly one vertex, and the edge opposite to that vertex in $Δ_{1}$ intersects the interior of $Δ_{2}$

. Denote the number of ordered pairs of crossing

k

-triangles by

X^{k}

. The general technique in [15] is to establish an upper bound and a lower bound on

X^{k}

, and to combine these two bounds to derive an upper bound for the number of

k

-triangles in

S

The upper bound in [15] is based on the 3-dimensional version of the Lovász Lemma, as in [5]: Any line crosses at most $O (n^{2})$ interiors of $k$ -triangles. The lemma follows from the main property of the set $T$ , which is its antipodality. Informally, the property asserts that for each pair of points $a, b \in S$ , the $k$ -triangles having $a b$ as an edge form an antipodal system, in the sense that for any pair $Δ_{a b c}, Δ_{a b d}$ of such triangles that are consecutive in the circular order around $a b$ , the dihedral wedge that is formed by the two halfplanes that contain $Δ_{a b c}, Δ_{a b d}$ , and are bounded by the line through $a b$ , has the property that its antipodal wedge, formed by the two complementary halfplanes within the planes supporting $Δ_{a b c}, Δ_{a b d}$ , contains a point $e \in S$ such that $Δ_{a b e}$ is also a $k$ -triangle; See Figure 1.

Figure 1: The antipodality property of $k$ -triangles: the edge $a b$ is drawn from a side-view as a point. The triangles $Δ_{a b c}, Δ_{a b d}$ and $Δ_{a b e}$ , drawn as segments in the figure, are all $k$ -triangles, where $e$ is contained in the antipodal wedge formed by the two complementary halfplanes within the planes supporting $Δ_{a b c}, Δ_{a b d}$ .

To obtain a lower bound on $X^{k}$ , the technique in [15] defines, for each $a \in S$ , a graph $G_{a} = (V_{a}, E_{a})$ drawn in a horizontal plane $h_{a}^{+}$ slightly above $a$ , whose edges are, roughly, the cross-sections of the $k$ -triangles incident to $a$ with the plane. See Figure 2 for an illustration. The analysis in [15] shows that $G_{a}$ inherits the antipodality property of the $k$ -triangles, and uses this fact to decompose $G_{a}$

into a collection of convex chains, and to estimate the number of crossings between the chains. Summing these bounds over all

a \in S

, the lower bound on

X^{k}

follows.

Figure 2: (1) The graph $G_{a}$ on $h_{a}^{+}$ . $Δ_{a u w}, Δ_{a u v}$ are halving triangles. $u v$ is mapped to the segment $u^{*} v^{*}$ , and $u w$ is mapped to a ray emanating from $u^{*}$ on $h_{a}^{+}$ . (2) The antipodality property in $G_{a}$ .

We omit further details of the way in which these lower bounds are derived in [15], because, in the dual version that we present here, we use a weaker lower bound, which is based on a dual version of the Crossing Lemma (see [4]), and which is easier to extend to the case of pseudoplanes. Let $G = (V, E)$ be a simple graph, and define the crossing number of $G$ as the minimum number of intersecting pairs of edges in any drawing of $G$ in the plane. In the primal setting, the Crossing Lemma asserts that any simple graph $G = (V, E)$ drawn in the plane, with $| E | > 4 | V |$ , has crossing number at least¹¹1The constant of proportionality has been improved in subsequent works, but we will stick to this bound. $\frac{| E |^{3}}{64 | V |^{2}}$ . Using this technique for deriving a lower bound on $X^{k}$ , instead of the refined technique in [15], one can show that the number of $k$ -triangles is $O (n^{8 / 3})$ , or, with the additional technique of [1], $O (n k^{5 / 3})$ .

We now present the dual setting for the problem, where the input is a set $Λ$ of $n$ non-vertical planes in $R^{3}$ in general position.

Definition 1.

Let $a, b, c \in Λ$ . The open region between the lower envelope and the upper envelope of $a, b, c$ is called the corridor of $a$ , $b$ , $c$ and is denoted by $C_{a, b, c}$ .

We will be mostly interested in corridors $C_{a, b, c}$ for which the point $p_{a, b, c} = a \cap b \cap c$ (the unique vertex of $C_{a, b, c}$ ) is at level $k$ . We will refer to such corridors as $k$ -corridors, and define $C^{k}$ as the collection of $k$ -corridors in $A (Λ)$ ; $k$ -corridors serve as a dual version of $k$ -triangles.

Definition 2.

We say that a corridor $C_{1}$ is immersed in a corridor $C_{2}$ in $A (Λ)$ if they share exactly one plane, and the intersection line of the other two planes of $C_{1}$ is fully contained in $C_{2}$ . Let $X^{k}$ denote the number of ordered pairs of immersed corridors in $C^{k}$ .

Immersion of $k$ -corridors is the dual notion of crossings of $k$ -triangles. Note that if a corridor $C_{1}$ is immersed in a corridor $C_{2}$ , it cannot be that $C_{2}$ is also immersed in $C_{1}$ .

In Section 2 we provide more details of this dual setup. We present the derivation of the upper bound and the lower bound on $X^{k}$ , the number of ordered pairs of immersed $k$ -corridors, and the derivation of the bound $O (n^{8 / 3})$ , and then the bound $O (n k^{5 / 3})$ . We remark, though, that this translation to the dual context, although routine in principle, is rather involved and nontrivial, and requires careful handling of quite a few details.

In Section 3 we consider in detail the extension to the case of pseudoplanes, which is our main topic of interest. In our context, a family $Λ$ of $n$ surfaces in $R^{3}$ is a family of pseudoplane, if, in addition to the general definition of a pseudoplane family (namely, the surfaces are graphs of total bivariate continuous functions, and each triple of them intersect exactly once), it satisfies the following conditions:

The intersection of any pair of surfaces in $Λ$ is a connected $x$ -monotone unbounded curve.
The $x y$ -projections of the set of all $(\frac{n}{2})$ intersection curves of the surfaces form a family of pseudolines in the plane.

The second condition is a nontrivial assumption on a general pseudoplane family (although it trivially holds in the case of planes).

We generalize the definition of $X^{k}$ for the number of ordered pairs of immersed $k$ -corridors, where $Λ$ is a family of pseudoplanes as above. We then generalize the analysis in Section 2, for obtaining lower and upper bounds for $X^{k}$ . Comparing those bounds, where the main technique is the same, gives us the bound $O (n^{8 / 3})$ on the complexity of the $k$ -level.

2 The case of planes

In this section we present our technique for the simpler case of planes. This might be useful for readers that would like to get familiar with the machinery of this paper, in the simpler, and easier to visualize, context of planes. On the face of it, in the case of planes this is just a translation to the dual setting of classical arguments used in the primal analysis of $k$ -sets in three dimensions [9, 15]. Still, this translation is fairly nontrivial, so getting familiar with it in this simple context might be helpful for absorbing the more general arguments in our analysis.

2.1 A dual version of the Lovász Lemma

The following lemma is a dual variant of the antipodality of the set of $k$ -triangles in the primal setup, as reviewed above.

Figure 3: A $z$ -vertical line $h$ that meets $l_{a, b} = a \cap b$ at a point $p$ , and one of the rays $ρ_{1}$ that emanates from $p$ and is contained in $l_{a, b}$ . In this case, ${d_{1}, d_{3}, d_{5}} \subseteq h_{u p}$ and ${d_{2}, d_{4}} \subseteq h_{d o w n}$ , and $D_{2, 4} = {d_{2}, d_{3}, d_{4}}$ .

Lemma 1.

Let $a, b \in Λ$ , and let $l_{a, b} = a \cap b$ denote their intersection line. Let $h$ be a $z$ -vertical line (orthogonal to the $x y$ -plane) that intersects $l_{a, b}$ at some point $p$ . Let $D = Λ ∖ {a, b}$ , and $D^{k} = {d \in D ∣ C_{a, b, d} \in C^{k}}$ . Denote $h_{u p} = {d \in D ∣ z_{d \cap h} > z_{p}}$ and $h_{d o w n} = {d \in D ∣ z_{d \cap h} < z_{p}}$ (by choosing $p$ generically, we may assume that all these inequalities are indeed sharp). We then have $∣ ∣ | h_{u p} \cap D^{k} | - | h_{d o w n} \cap D^{k} | ∣ ∣ \leq 2$ .

Proof.

Denote one of the rays that emanates from $p$ and is contained in $l_{a, b}$ by $ρ_{1}$ , and the other one by $ρ_{2}$ , and denote $D_{ρ_{1}} = {d \in D ∣ d$ intersects $ρ_{1}}$ , $D_{ρ_{2}} = {d \in D ∣ d$ intersects $ρ_{2}}$ . Clearly, with a generic choice of $p$ , $D_{ρ_{1}} \cup D_{ρ_{2}} = D$ . Enumerate the planes in $D_{ρ_{1}}$ as $d_{1}, \dots, d_{j}$ , according to the order in which their respective intersection points with $ρ_{1}$ , denoted $p_{1}, \dots, p_{j}$ , appear on $ρ_{1}$ in the direction from $p$ to infinity. Assume there are $1 \leq r < s \leq j$ such that $d_{r}, d_{s} \in D_{ρ_{1}} \cap (h_{d o w n} \cap D^{k})$ , and denote $D_{r, s} = {d_{i} \in D_{ρ_{1}} ∣ r \leq i \leq s}$ . Each $d_{i} \in h_{u p} \cap D_{r, s}$ with $r < i < s$ is above $p_{r}$ and below $p_{s}$ , and each $d_{i} \in h_{d o w n} \cap D_{r, s}$ with $r < i < s$ is above $p_{s}$ and below $p_{r}$ ; the same also holds for $d_{r}$ and $d_{s}$ , except that $d_{r}$ passes through $p_{r}$ and $d_{s}$ passes through $p_{s}$ (see Figure 3).

It is easy to show that, for each point $p_{i}$ , the following properties hold (see Figure 4):

$λ (p_{i}) = λ (p_{i - 1}) - 1$ if and only if both $d_{i - 1}, d_{i} \in h_{d o w n}$ .
$λ (p_{i}) = λ (p_{i - 1}) + 1$ if and only if both $d_{i - 1}, d_{i} \in h_{u p}$ .
$λ (p_{i}) = λ (p_{i - 1})$ if and only if one of $d_{i - 1}, d_{i}$ is in $h_{u p}$ and the other one is in $h_{d o w n}$ .

Figure 4: Both $d_{i}, d_{i - 1}$ intersect the same ray of $l_{a, b}$ from $p$ : (i) The case where both $d_{i}, d_{i - 1} \in h_{d o w n}$ . (ii) The case where both $d_{i}, d_{i - 1} \in h_{u p}$ . (iii) a-b The case where one of $d_{i}, d_{i - 1}$ is in $h_{d o w n}$ , and the other one is in $h_{u p}$ .

We claim that there must exist a plane $d^{'} \in D_{r, s}$ that is in $h_{u p} \cap D^{k}$ . If $d_{r + 1} \in h_{u p}$ , by (iii) above, $λ (p_{r + 1}) = λ (p_{r}) = k$ . Since $d_{r + 1} \notin h_{d o w n}$ , $r + 1 < s$ , and therefore $d_{r + 1} \in h_{u p} \cap D^{k}$ . Otherwise, $d_{r + 1} \in h_{d o w n}$ , and by (i) above, the level of $p_{r + 1}$ is $k - 1$ . Note that $r + 1 < s$ because the level of $p_{s}$ is $k$ . Because the level can change only by $0$ , $+ 1$ , or $- 1$ between two consecutive points $p_{i}$ , $p_{i + 1}$ , there must be a point $p_{i} \in D_{r, s} \subseteq D_{ρ_{1}}$ , so that the level of $p_{i}$ is $k$ and the level of the previous point on $l_{a, b}$ is $k - 1$ , which means, by (ii) above, that $d_{i} \in h_{u p}$ . That is, between each pair $p_{r}, p_{s} \in ρ_{1}$ so that $d_{r}, d_{s} \in h_{d o w n} \cap D^{k}$ , there exists $p_{i}$ so that $d_{i} \in h_{u p} \cap D^{k}$ , and our claim is established (see Figure 5).

Similarly, between each pair $p_{r}, p_{s} \in ρ_{1}$ so that $d_{r}, d_{s} \in h_{u p} \cap D^{k}$ , there exists $p_{i}$ , for some $r < i < s$ , so that $d_{i} \in h_{d o w n} \cap D^{k}$ . Both of these properties are easily seen to imply that

∣ ∣ | h_{u p} \cap D^{k} \cap D_{ρ_{1}} | - | h_{d o w n} \cap D^{k} \cap D_{ρ_{1}} | ∣ ∣ \leq 1.

The same reasoning applies to $ρ_{2}$ , and yields $∣ ∣ | h_{u p} \cap D^{k} \cap D_{ρ_{2}} | - | h_{d o w n} \cap D^{k} \cap D_{ρ_{2}} | ∣ ∣ \leq 1$ . Thus, $∣ ∣ | h_{u p} \cap D^{k} | - | h_{d o w n} \cap D^{k} | ∣ ∣ \leq 2$ . ∎

Figure 5: (a) The case where both $d_{r}, d_{s} \in h_{d o w n} \cap D^{k}$ . (b) There exists an in-between plane $d_{i}$ that belongs to $h_{u p} \cap D^{k}$ .

We now prove the promised dual version of the Lovász Lemma.

Lemma 2.

Any nonvertical line that is not parallel to any of the planes of $Λ$ and does not intersect any of the edges of $A (Λ)$ , is fully contained in at most $n (n - 1) / 2$ corridors in $C^{k}$ .

Proof.

Let $l_{0}$ be a line as in the lemma. Consider the vertical plane through $l_{0}$ , and let $l_{1}$ be a parallel line to $l_{0}$ contained in that plane, that is not contained in any of the corridors in $C^{k}$ . We can find such a line, for example, by translating $l_{0}$ sufficiently far upwards (in the positive $z$ -direction). For each $0 \leq τ \leq 1$ , define $l_{τ} = τ l_{0} + (1 - τ) l_{1}$ . As $τ$ increases from $0$ to $1$ , $l_{τ}$ translates from $l_{1}$ down to $l_{0}$ . During the translation, we maintain an upper bound on the number of corridors from $C^{k}$ that the translating line is fully contained in, until we reach $l_{0}$ at $τ = 1$ , and argue that this bound remains $O (n^{2})$ .

At the beginning of the process, the number of corridors from $C^{k}$ that $l_{1}$ is contained in is $0$ . We say that the line $l_{τ^{'}}$ is about to enter (resp., exit) the corridor $C_{a, b, d}$ , if there is $ε^{'} > 0$ so that for every $0 < ε \leq ε^{'}$ , $l_{τ^{'} + ε}$ (resp., $l_{τ^{'} - ε}$ ) is fully contained in the interior of $C_{a, b, d}$ , but this does not hold for $l_{τ^{'}}$ itself. In order to reach a position where it is fully contained in a corridor $C_{a, b, d}$ , during the translation downwards in the negative $z$ -direction, the translating line $l_{τ}$ has to reach a position where it is about to enter $C_{a, b, d}$

. If at some moment

l_{τ^{'}}

reaches a position at which it is about to enter

C_{a, b, d}

, one of the following two situations must occur:

One of the planes contains $l_{τ^{'}}$ .
There are two planes among ${a, b, c}$ , say they are $a$ and $b$ , such that $l_{τ^{'}}$ touches the line $a \cap b$ at some point $p$ , at a position that lies on $\partial C_{a, b, d}$ , $l_{τ^{'}}$ lies below $a$ on one side of $p$ and below $b$ on the other side, and $p$ lies above the third plane $d$ .

Conversely, if the properties in (2) hold, $l_{τ^{'}}$ is about to enter $C_{a, b, d}$ .

Since $l_{0}$ is not parallel to any plane of $Λ$ , and $l_{τ}$ is parallel to $l_{0}$ , $l_{τ}$ cannot be contained in any plane of $Λ$ . Hence, the first scenario cannot happen. Assume that the second case occurs. Since, immediately below $l_{τ^{'}}$ , the translating $l_{τ}$ starts being fully contained in the interior of $C_{a, b, d}$ , it follows that $l_{τ^{'}}$ must pass through an edge of $C_{a, b, d}$ , which is contained in an intersection line of two of the planes $a, b, d$ . Assume without loss of generality that $l_{τ^{'}}$ touches $a \cap b$ , at some point $p$ . Moreover, just before reaching this position, $l_{τ}$ has a portion that lies above the upper envelope of $a$ and $b$ , and this portion shrinks to the single point $p = l_{τ^{'}} \cap (a \cap b)$ , for otherwise crossing $a \cap b$ would not make $l_{τ}$ being fully contained in $C_{a, b, d}$ . It follows that $l_{τ^{'}}$ is fully contained in the “horizontal” dihedral wedge of $a, b$ (the wedge that does not contain the $z$ -vertical direction), denoted by $W_{a, b}$ . See Figure 6. Moreover, the third plane $d$ must pass below $p$ .

Figure 6: The horizontal dihedral wedge $W_{a, b}$ in which the translating line $l_{τ}$ fully lies at $τ = τ^{'}$ . For small enough $ε > 0$ , for each $d \in Λ$ that meets the $z$ -vertical line through $p$ at a point $p^{'}$ that satisfies $z_{p^{'}} > z_{p}$ (resp., $z_{p^{'}} < z_{p}$ ), we have $l_{τ^{'} - ε} \subseteq C_{a, b, d}$ (resp., $l_{τ^{'} + ε} \subseteq C_{a, b, d}$ ) (both possibilities are shown in the figure).

In a similar manner, $l_{τ^{'}}$ is about to exit $C_{a, b, d}$ if and only if it touches an intersection line of two of these planes, say $a \cap b$ , and the third plane $d$ passes above the contact point.

Assume then that $l_{τ^{'}}$ touches an intersection line $a \cap b$ of two planes $a, b \in Λ$ , and fully lies in the horizontal dihedral wedge $W_{a, b}$ . Let $h$ be the vertical line through $p$ . The argument just given implies that at this contact, $l_{τ^{'}}$ is about to enter a $k$ -corridor $C_{a, b, d}$ (resp., about to exit $C_{a, b, d}$ ) if and only if $d \in h_{d o w n} \cap D^{k}$ (resp., $d \in h_{u p} \cap D^{k}$ ). Hence, the net increase in the number of containing $k$ -corridors, as we pass through $a \cap b$ , is $| h_{d o w n} \cap D^{k} | - | h_{u p} \cap D^{k} |$ , and by Lemma 1, the absolute value of this difference is at most 2. There are $(\frac{n}{2}) = \frac{n (n - 1)}{2}$ horizontal dihedral wedges formed by the planes of $Λ$ . Assume without loss of generality that for at most half of them, the intersection line of the two planes defining the horizontal dihedral wedge pass above $γ_{0}$ . Than the sweeping line can encounter at most $\frac{1}{2} \cdot \frac{n (n - 1)}{2}$ horizontal dihedral wedges. Thus, at the end of the process, $l_{0}$ is fully contained in at most $n (n - 1) / 2$ $k$ -corridors and the claim follows. ∎

Lemma 3.

The number $X^{k}$ of ordered pairs of $k$ -corridors such that the first corridor is immersed in the second one, in the arrangement $A (Λ)$ , is at most $\frac{3 n^{4}}{4}$ .

Proof.

Fix an intersection line $l_{a, b} = a \cap b$ of two planes from $Λ$ . by Lemma 2, $l_{a, b}$ is fully contained in at most $n (n - 1) / 2$ $k$ -corridors. For each containing $k$ -corridor $C_{c, d, e}$ , $l_{a, b}$ can contribute at most three ordered pairs to $X^{k}$ , namely an immersion of $C_{a, b, c}$ in $C_{c, d, e}$ , of $C_{a, b, d}$ in $C_{c, d, e}$ and of $C_{a, b, e}$ in $C_{c, d, e}$ . Since there are only $(\frac{n}{2})$ intersecting lines in $A (Λ)$ , we get that there are at most $3 \frac{n (n - 1)}{2} (\frac{n}{2}) < \frac{3 n^{4}}{4}$ ordered pairs of immersed $k$ -corridors. ∎

2.2 The dual version of the Crossing Lemma

In this subsection we derive a lower bound on $X^{k}$ , using a dual version of the Crossing Lemma. For each plane $a \in Λ$ , denote by $z_{a}$ the intersection point of $a$ with the $z$ -axis. By our general position assumption, we may assume that all the values $z_{a}$ are distinct.

Definition 3.

Let $a \in Λ$ . Denote by $Γ_{a}$ the collection of intersection lines of $a$ and the other planes $b \in Λ$ with $z_{b} > z_{a}$ . That is, $Γ_{a} = {l_{b} = a \cap b ∣ b \in Λ, z_{b} > z_{a}}$ .

Definition 4.

Let $a \in Λ$ , and let $Γ_{a}$ be as above. For each pair of distinct planes $b, c \in Λ ∖ {a}$ such that $l_{b}, l_{c} \in Γ_{a}$ , define the horizontal wedge $W_{b, c}^{a}$ as the region in the plane $a$ that is contained in exactly one of the halfplanes that are induced by $l_{b}, l_{c}$ and contain $z_{a}$ (see Figure 7).

Figure 7: The horizontal wedge $W_{b, c}^{a}$ on the plane $a$ .

Note that without loss of generality, we may assume that the $z$ -axis meets $a$ at a point that lies above all the lines $l_{b}$ , in a suitably defined coordinate frame within $a$ . In this case, the wedges $W_{b, c}^{a}$ are indeed ‘horizontal’ within this frame.

When extending our analysis to the case of pseudoplanes, we will use an extension of Euler’s formula, derived in Tamaki and Tokuyama [17], for an arrangement of pseudolines in $R^{2}$ . For the case of lines that we are currently considering, this is simply a dual variant of Euler’s formula for planar graphs. Specifically, consider a plane $a \in Λ$ , let $Γ_{a}$ be as defined above, let $E_{a}$ be a subset of vertices of $A (Γ_{a})$ , and let $G_{a} = (Γ_{a}, E_{a})$ denote the graph whose vertices are the lines in $Γ_{a}$ and whose edges are the pairs that form the vertices of $E_{a}$ . Define a diamond in $G_{a}$ as two pairs ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ on $a$ , both belonging to $E_{a}$ , with all four planes $b, c, d, e$ distinct, such that $l_{b} \cap l_{c} \in W_{d, e}^{a}$ and $l_{d} \cap l_{e} \in W_{b, c}^{a}$ (see Figure 8).

The following lemma is an easy consequence of planar duality (within $a$ ).

Figure 8: The blue area belongs to $W_{b, c}^{a}$ , the red area belongs to $W_{d, e}^{a}$ , and the gray area is common to both $W_{b, c}^{a}, W_{d, e}^{a}$ . The two pairs ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ on $a$ induce a diamond in $G_{a}$ , as $l_{b} \cap l_{c} \in W_{d, e}^{a}$ and $l_{d} \cap l_{e} \in W_{b, c}^{a}$ .

Lemma 4.

Let $Γ_{a}$ and $G_{a}$ be as defined above, $| Γ_{a} | > 3$ . If $Γ_{a}$ is diamond-free, then $G_{a}$ is planar, and hence $| E_{a} | \leq 3 | Γ_{a} | - 6$ .

As a corollary of the lemma, we obtain the following dual version of the Crossing Lemma (see [4]). For completeness, we include the proof, with suitable adjustments to accommodate the duality.

Lemma 5.

Let $Γ_{a}$ and $G_{a}$ be as defined above, so that $| Γ_{a} | = m$ and $| E_{a} | > 4 m$ . The number of diamonds in $G_{a}$ is $Ω (\frac{| E_{a} |^{3}}{| Γ_{a} |^{2}})$ .

Proof.

Let $Γ$ be a collection of nonvertical lines in the (standard) plane, and $E$ a subset of the vertices of $A (Γ)$ . Denote by $Δ$ the number of diamonds in $G = (Γ, E)$ (where the wedges used to define the diamonds are the standard horizontal wedges, relative to the $y$ -direction). We now repeat the following process, until there are no diamonds left in $G$ : For a surviving diamond, remove from $E$ one of the two points that form the diamond. This eliminates the diamond and maybe some other diamonds too. Continue the process with the new $G$ . The dual version of Euler’s formula implies that if $| E | > 3 | Γ | - 6$ then there is a diamond in $G$ . We stop the process after removing at most $| E | - 3 | Γ | + 6$ vertices, each time removing at least one diamond from $G$ . Hence, for the original graph $G$ , we have $Δ \geq | E | - 3 | Γ |$ .

Denote by $δ_{a}$ the number of diamonds in $G_{a}$ . Consider a random subgraph of $G_{a}$ , in which each vertex (which is a line in $Γ_{a}$

) is chosen independently with the same probability

p

. The expected number of vertices, edges and diamonds in the induced subgraph of

G_{a}

p m

p^{2} | E_{a} |

, and

p^{4} δ_{a}

, respectively. Using linearity of expectation, we have

p^{4} δ_{a} > p^{2} | E_{a} | - 3 p m

, which implies that

δ_{a} > \frac{| E_{a} |}{p^{2}} - \frac{3 m}{p^{3}}

. For

p = \frac{4 m}{| E_{a} |}

(note that, by assumption,

| E_{a} | > 4 m

, so

p < 1

), we obtain that

δ_{a} > \frac{| E_{a} |^{3}}{64 m^{2}}

. ∎

For each $a \in Λ$ consider $E_{a}^{'} = {p_{a, b, c} = l_{b} \cap l_{c} ∣ l_{b}, l_{c} \in Γ_{a}, C_{a, b, c} \in C^{k}}$ , and the graph $G_{a}^{'} = (Γ_{a}, E_{a}^{'})$ defined as above. By Lemma 5 we conclude the following:

Lemma 6.

The number $X^{k}$ of ordered pairs of immersed $k$ -corridors in the arrangement $A (Λ)$ is at least $\sum a \in Λ (\frac{| E_{a}^{'} |^{3}}{64 | Γ_{a} |^{2}} - | Γ_{a} |)$ .

Proof.

Let $a \in Λ$ , $G_{a}^{'} = (Γ_{a}, E_{a}^{'})$ be as above, and define $Δ_{a}$ as the number of diamonds in $G_{a}^{'}$ . Let ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ be a pair that form a diamond as in Figure 9; that is, $p_{a, b, c}$ is contained in the halfplane of $l_{d}$ which contains $z_{a}$ and is contained in the halfplane of $l_{e}$ which does not contain $z_{a}$ , and $p_{a, d, e} \in W_{b, c}^{a}$ .

Figure 9: The two pairs ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ on $a$ form a diamond in $G_{a}^{'}$ .

Notice the projection of the intersection line $l_{b, c}$ on $a$ is fully contained in the region on $a$ that consists of the following three regions: the region on $a$ where both $b$ and $c$ are above $a$ , the region on $a$ where both $b$ and $c$ are below $a$ and the intersection point $p_{a, b, c}$ (see Figure 10(i)). The same implies for projection of the intersection line $l_{d, e}$ on $a$ : it is fully contained in the region on $a$ that consists of the region on $a$ where both $d$ and $e$ are above $a$ , the region on $a$ where both $d$ and $e$ are below $a$ and the intersection point $p_{a, d, e}$ (see Figure 10(ii)).

Since the planes in $Λ$ are in general position, the two intersection lines $l_{b, c}$ and $l_{d, e}$ are neither parallel nor coincident. Assume without loss of generality that $l_{b, c}$ passes below $l_{d, e}$ , and let $l$ be the unique $z$ -vertical line that meets both $l_{b, c}, l_{d, e}$ , so that the points $p_{1} = l \cap l_{b, c}$ , $p_{2} = l \cap l_{d, e}$ satisfy $z_{p_{1}} < z_{p_{2}}$ . ${l_{b}, l_{c}}$ and ${l_{d}, l_{e}}$ form a diamond, and therefore the intersection of the region on $a$ where $b, c,$ are above $a$ and $d, e$ are below $a$ is empty. The same implies for the intersection of the region on $a$ where $d, e,$ are above $a$ and $b, c$ are below $a$ . So $l$ intersects $a$ either in the region on $a$ where all the planes $b, c, d, e$ are above $a$ , or in the region on $a$ where all the planes $b, c, d, e$ are below $a$ .

Figure 10: The diamond formed by ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ on $a$ . (i) The gray area is the region where the projection of the intersection line $l_{b, c}$ on $a$ must be contained in, including $p_{a, b, c}$ . (ii) The gray area is the region where the projection of the intersection line $l_{d, e}$ on $a$ must be contained in, including $p_{a, d, e}$ .

Assume without loss of generality that $l$ intersects $a$ as in Figure 11; that is, $l$ intersects $a$ in the region on $a$ induced by the intersection of the halfplanes containing $z_{a}$ and induced by the lines $l_{b}$ , $l_{c}$ , $l_{d}$ and $l_{e}$ . The case where $l$ intersects $a$ in the region on $a$ induced by the intersection of the halfplanes that are not containing $z_{a}$ and induced by the lines $l_{b}$ , $l_{c}$ , $l_{d}$ and $l_{e}$ , is symmetric. Notice that since $l_{d, e}$ is above $p_{1}$ , it follows that both $d$ and $e$ themselves are above $p_{1}$ . Moreover, because $z_{e} > z_{a}$ and $p_{a, b, c}$ lies in the halfplane of $a$ which is bounded by $l_{e}$ and does not contain $z_{a}$ , as in Figure 11, $e$ must lie below $p_{a, b, c}$ . Hence $e$ must intersect $l_{b, c}$ at some point $q$ between $p_{a, b, c}$ and $p_{1}$ . Moreover, since $d$ satisfies $z_{d} > z_{a}$ and $p_{a, b, c}$ lies in the halfplane of $a$ which is bounded by $l_{d}$ and contains $z_{a}$ , as in Figure 11, $d$ is above $p_{a, b, c}$ . Since $d$ is also above $p_{1}$ , it must intersect $l_{b, c}$ at one of the two rays that form the complement of the segment $p_{a, b, c} p_{1}$ .

Figure 11: The two pairs ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ on $a$ form a diamond in $G_{a}^{'}$ . The intersection line $l_{b, c}$ is below the intersection line $l_{d, e}$ , and $l$ is the unique $z$ -vertical line that intersects $l_{b, c}, l_{d, e}$ at $p_{1}, p_{2}$ respectively. The plane $e$ (resp. $d$ ) meets $l_{b, c}$ at a point $q$ between $p_{a, b, c}$ and $p_{1}$ (resp., outside this segment).

We claim that $l_{b, c} \subseteq C_{a, d, e}$ . Indeed, the intersection line $l_{b, c}$ is fully above the lower envelope of ${a, d, e}$ : the points on $l_{b, c}$ that are higher than $p_{a, b, c}$ , are above the plane $a$ , since $l_{b, c}$ intersects $a$ at $p_{a, b, c}$ . In addition, the points on $l_{b, c}$ that are lower than $p_{a, b, c}$ , including $p_{a, b, c}$ itself, are above $e$ . That is because the intersection point $q$ of $l_{b, c}$ and $e$ is higher than $p_{a, b, c}$ , and thus $e$ is below the points on $l_{b, c}$ that are lower than $q$ . The intersection line $l_{b, c}$ is also fully below the upper envelope of ${a, d, e}$ . That is because the points on $l_{b, c}$ that are higher than the intersection point $q$ of $l_{b, c}$ and $e$ , are below the plane $e$ , since $e$ passes above $p_{1}$ and $z_{p_{1}} > z_{q}$ . Moreover, the points on $l_{b, c}$ that are lower than $p_{a, b, c}$ , are below the plane $a$ , since $l_{b, c}$ intersects $a$ at this point. Finally, the points on $l_{b, c}$ that are higher than $p_{a, b, c}$ and lower than $q$ , including $p_{a, b, c}, q$ , are below $d$ , since $d$ is above both $p_{a, b, c}, p_{1}$ and therefore must be above the complete segment $p_{a, b, c} p_{1}$ , and in particular $d$ is above the smaller segment $p_{a, b, c} q$ .

Thus for each pair ${l_{b}, l_{c}}, {l_{d}, l_{e}}$ that form a diamond, either $l_{b, c} \subseteq C_{a, d, e}$ , or $l_{d, e} \subseteq C_{a, b, c}$ . Either way, one of the corridors $C_{a, b, c}, C_{a, d, e}$ is immersed in the other one. Notice that every diamond in ${G_{a}^{'}}_{a \in Λ}$ yields a distinct ordered pair of immersed $k$ -corridors, because for each $k$ -corridor $C_{a, b, c}$ , the intersection point $p_{a, b, c} = a \cap b \cap c$ belongs only to the graph of the plane with the lowest intersection point with the $z$ -axis. Hence, by the dual version of the Crossing Lemma, namely Lemma 5, we have $X^{k} \geq \sum a \frac{| E_{a}^{'} |^{3}}{64 | Γ_{a} |^{2}}$ , where the sum is over all those $a$ for which $| E_{a}^{'} | \geq 4 | Γ_{a} |$ . Any other plane $a$ satisfies $\frac{| E_{a}^{'} |^{3}}{64 | Γ_{a} |^{2}} \leq | Γ_{a} |$ , which implies the following somewhat weaker lower bound

X^{k} \geq \sum a \in Λ (\frac{| E_{a}^{'} |^{3}}{64 | Γ_{a} |^{2}} - | Γ_{a} |) .

∎

2.3 The complexity of the $k$ -level of $A (Λ)$

We are now ready to obtain the upper bound on the complexity of the $k$ -level of $A (Λ)$ .

Lemma 7.

The complexity of the $k$ -level of $A (Λ)$ is $O (n^{8 / 3})$ .

Proof.

Denote the number of $k$ -corridors in $A (Λ)$ by $ξ = | C^{k} |$ . By the notations in Section 2.2 and by the definition of $G_{a}^{'}$ , each $k$ -corridor $C_{a, b, c}$ in $A (Λ)$ appears in exactly one of $E_{a}^{'}, E_{b}^{'}, E_{c}^{'}$ (in the graph of the plane that intersects the $z$ -axis at the lowest point). Thus, $\sum a \in Λ | E_{a}^{'} | = ξ$ . Notice that the number of lines in $Γ_{a}$ is the number of planes in $Λ$ that intersect the $z$ -axis higher than $z_{a}$ , and this number is at most $n - 1$ .

We compare the upper bound in Lemma 3 and the lower bound in Lemma 6 for the number $X^{k}$ of ordered pairs of immersed $k$ -corridors in $A (Λ)$ , and get: $\frac{3 n^{4}}{4} \geq X^{k} \geq \sum a \in Λ (\frac{| E_{a}^{'} |^{3}}{64 | Γ_{a} |^{2}} - | Γ_{a} |) \geq \frac{1}{64 n^{2}} \sum a \in Λ | E_{a}^{'} |^{3} - n^{2} \geq \frac{1}{64 n^{2}} \cdot \frac{}{{(\sum a \in Λ | E_{a}^{'} |)}^{3}} n^{2} - n^{2} = \frac{ξ^{3}}{64 n^{4}} - n^{2}$ ,

where the last inequality follows from Hölder’s inequality. Hence we get that $ξ^{3} \leq 48 n^{8} + 64 n^{6}$ , which implies that $ξ = O (n^{8 / 3})$ . The number of $k$ -corridors is the number of vertices of $A (Λ)$ at the $k$ -level, which implies that the complexity of the $k$ -level of $A (Λ)$ is $O (n^{8 / 3})$ . ∎

Agarwal et al. [1] present a general technique, based on random sampling and the analysis of Clarkson and Shor [7], for transforming an upper bound on the number of $k$ -sets that is independent of $k$ to a bound that does depend on $k$ . Hence, by combining the upper bound in Lemma 7 with the technique of [1], we get the following result:

Theorem 1.

The complexity of the $k$ -level of $A (Λ)$ is $O (n k^{5 / 3})$ .

We remark that this bound is weaker than the bound $O (n k^{3 / 2})$ established in [15], which was obtained using a more refined argument than the one based on the Crossing Lemma. We use the weaker analysis presented above because of the generalization of Euler’s formula to arrangements of pseudolines, due to Tamaki and Tokuyama [17], which allows us to extend our analysis to the case of pseudoplanes, as described in the following section.

3 The case of pseudoplanes

We say that a family $Λ$ of $n$ surfaces in $R^{3}$ is a family of pseudoplanes in general position if

The surfaces of $Λ$ are graphs of total bivariate continuous functions.
The intersection of any pair of surfaces in $Λ$ is a connected $x$ -monotone unbounded curve.
Any triple of surfaces in $Λ$ intersect in exactly one point.
The $x y$ -projections of the set of all $(\frac{n}{2})$ intersection curves of the surfaces form a family of pseudolines in the plane. That is, this is a collection of $(\frac{n}{2})$ $x$ -monotone unbounded curves, each pair of which intersect exactly once; see [3] for more details.

The assumption that the pseudoplanes of $Λ$ are in general position means that no point is incident to more than three pseudoplanes, no intersection curve of two pseudoplanes is tangent to a third pseudoplane, and no two pseudoplanes are tangent to each other. We note that conditions (1)–(3) are natural, but condition (4) might appear somewhat restrictive, although it obviously holds for planes. For any $a, b, c \in Λ$ , we denote the intersection curve $a \cap b$ by $γ_{a, b}$ , and the intersection point $a \cap b \cap c$ by $p_{a, b, c}$ .

Definition 5.

Let $γ$ be a curve in $R^{3}$ . The vertical curtain through $γ$ , denoted by $Υ_{γ}$ , is the collection of all $z$ -vertical lines that intersect $γ$ . The portion of $Υ_{γ}$ above (resp., below) $γ$ is called the upper (resp., lower) curtain of $γ$ , and is denoted by $Υ_{γ}^{u}$ (resp., $Υ_{γ}^{d}$ ).

Let $γ$ be an $x$ -monotone unbounded connected curve in $R^{3}$ , and let $p \in γ$ . We call each of the two connected components of $γ ∖ {p}$ a half-curve of $γ$ emanating from $p$ .

The following lemma is derived from the general position of the pseudoplanes in $Λ$ :

Lemma 8.

Let $a, b, c \in Λ$ , and let $γ_{a, b} = a \cap b$ , $p_{a, b, c} = a \cap b \cap c$ .

One of the two half-curves of $γ_{a, b}$ that emanates from $p_{a, b, c}$ lies fully below $c$ , and the other half-curve lies fully above $c$ .
The collection of intersections between the surfaces of $Λ$ and $Υ_{γ_{a, b}}$ forms an arrangement of unbounded $x$ -monotone curves on $Υ_{γ_{a, b}}$ , each pair of which intersect at most once.²²2In a sense, this is a collection of pseudolines, except that they are, in general, not drawn in a plane.

Proof.

The proof of (a) is straightforward and is omitted here. For (b), property (4) implies that, for any $c, d \in Λ ∖ {a, b}$ , the projection on the $x y$ -plane of $γ_{a, b}$ and $γ_{c, d} = c \cap d$ intersect at most once. Thus, the intersection curves $c \cap Υ_{γ_{a, b}}$ , $d \cap Υ_{γ_{a, b}}$ intersect at most once. ∎

Another property of $A (Λ)$ , shown in Agarwal and Sharir [2], is:

Lemma 9.

The complexity of the lower envelope of $Λ$ is $O (n)$ .

The notion of corridors can easily be extended to the case of pseudoplanes. That is, for any $a, b, c \in Λ$ , denote by $C_{a, b, c}$ the open region between the lower envelope and the upper envelope of $a, b, c$ , and call it the corridor of $a, b, c$ . Refer to corridors $C_{a, b, c}$ for which the intersection point $p_{a, b, c}$ lies at level $k$ as $k$ -corridors, and define $C^{k}$ as the collection of $k$ -corridors in $A (Λ)$ . The following is an extension of Definition 2:

Definition 6.

A corridor $C_{1}$ is immersed in a corridor $C_{2}$ if they share exactly one pseudoplane, and the intersection curve of the other two pseudoplanes of $C_{1}$ is fully contained in $C_{2}$ . Let $X^{k}$ denote the number of ordered pairs of immersed corridors in $C^{k}$ .

Organization of this section.

In Section 3.1 we derive an upper bound for $X^{k}$ , using an extended dual version of the Lovász Lemma. In Section 3.2 we obtain a lower bound for $X^{k}$ , using a dual version of the Crossing Lemma. In Section 3.3 we combine those two bounds to obtain an upper bound on the complexity of the $k$ -level of the arrangement.

3.1 An extension of the dual version of the Lovász Lemma

The following lemma is an extension to the case of pseudoplanes of a dual version of the antipodality property in the primal setup.

Lemma 10.

Let $Λ$ be as above, let $a, b \in Λ$ , and let $γ_{a, b} = a \cap b$ denote their intersection curve. Let $h$ be a $z$ -vertical line (i.e., parallel to the $z$ -axis) that intersects $γ_{a, b}$ at some point $p$ . Let $D = Λ ∖ {a, b}$ , and $D^{k} = {d \in D ∣ C_{a, b, d} \in C^{k}}$ . Denote $h_{u p} = {d \in D ∣ z_{d \cap h} > z_{p}}$ and

On the Complexity of the k-Level in Arrangements of Pseudoplanes

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1 Introduction

Definition 1.

Definition 2.

2 The case of planes

2.1 A dual version of the Lovász Lemma

Lemma 1.

Proof.

Lemma 2.

Proof.

Lemma 3.

Proof.

2.2 The dual version of the Crossing Lemma

Definition 3.

Definition 4.

Lemma 4.

Lemma 5.

Proof.

Lemma 6.

Proof.

2.3 The complexity of the k-level of A(Λ)

Lemma 7.

Proof.

Theorem 1.

3 The case of pseudoplanes

Definition 5.

Lemma 8.

Proof.

Lemma 9.

Definition 6.

Organization of this section.

3.1 An extension of the dual version of the Lovász Lemma

Lemma 10.

2.3 The complexity of the $k$ -level of $A (Λ)$