A boundary-partition-based Voronoi diagram of d-dimensional balls: definition, properties, and applications

Abstract

In computational geometry, different ways of space partitioning have been developed, including the Voronoi diagram of points and the power diagram of balls. In this article, a generalized Voronoi partition of overlapping d-dimensional balls, called the boundary-partition-based diagram, is proposed. The definition, properties, and applications of this diagram are presented. Compared to the power diagram, this boundary-partition-based diagram is straightforward in the computation of the volume of overlapping balls, which avoids the possibly complicated construction of power cells. Furthermore, it can be applied to characterize singularities on molecular surfaces and to compute the medial axis that can potentially be used to classify molecular structures.

Appendices

Appendix 1. Proof of Claim 2.2

First, we consider a sphere $S_{i_{t}} \ni \mathbf {x}$ for a fixed 1 ≤ t ≤ m. According to the definition (24) and Eq. 26, we can compute

$$ \begin{array}{llll} &\displaystyle|\zeta_{s}(x)-\mathbf{c}_{i_{t}}|^{2} - |\mathbf{x}-\mathbf{c}_{i_{t}}|^{2} \\ & \displaystyle{ = |x\mathbf{n}_{s}+\alpha(x)\mathbf{v}|^{2}-2\left( \mathbf{v}_{t}, x\mathbf{n}_{s}+\alpha(x)\mathbf{v}\right)}\\ & \displaystyle{ = \left( \alpha^{2}(x)|\mathbf{v}|^{2} + x^{2}\right)+ 2x\alpha(x)\left( \mathbf{n}_{s},\mathbf{v}\right)-2\alpha(x)(\mathbf{v},\mathbf{v}_{t})-2x(\mathbf{v}_{t},\mathbf{n}_{s})}\\ & \displaystyle{ =2\alpha(x)(\mathbf{v},\mathbf{v}-\mathbf{v}_{t})-2x(\mathbf{v}_{t},\mathbf{n}_{s})}\\ & \displaystyle{ = 2x\left[-(\mathbf{v}_{t},\mathbf{n}_{s}) + \frac{x}{|\mathbf{v}|^{2}(2-\alpha(x))}(\mathbf{v},\mathbf{v}-\mathbf{v}_{t})\right]}, \end{array} $$

(56)

where in the third and forth equality, we use the fact that α²(x)|v|² + x² = 2α(x)|v|² and $\left (\mathbf {n}_{s},\mathbf {v}\right ) =0$ as $\mathbf {v}_{r}\in \mathcal A_{s}$.

Since (x + λv_t, n_s) ≤ b_s, ∀λ ≥ 0, we know that (v_t, n_s) ≤ 0. In the case of (v_t, n_s) < 0, according to Eq. 56, there exists a small enough number $\varepsilon _{i_{t}}>0$ such that $\forall x\in [0,\varepsilon _{i_{t}}]$,

$$|\zeta_{s}(x)-\mathbf{c}_{i_{t}}|\geq |\mathbf{x}-\mathbf{c}_{i_{t}}|=r_{i_{t}}.$$

Besides, in the case of (v_t, n_s) = 0 implying $\mathbf {v}_{t} \in \mathcal {A}_{s}$, according to Eq. 28 and (56), we have ∀0 ≤ x ≤|v|,

$$|\zeta_{s}(x)-\mathbf{c}_{i_{t}}|\geq |\mathbf{x}-\mathbf{c}_{i_{t}}|=r_{i_{t}}.$$

In particular, when t = t₀, we have both (v_t, n_s) = 0 and (v,v −v_t) = 0. As a consequence,

$$|\zeta_{s}(x)-\mathbf{c}_{i_{t}}|= |\mathbf{x}-\mathbf{c}_{i_{t}}|=r_{i_{t}},$$

which means that $\zeta _{s}(x) \in S_{i_{t_{0}}}$. So far, we have proved that $\forall x\in [0,\varepsilon _{i_{t}}]$, ζ_s(x) does not lie in the interior of $S_{i_{t}}$ and lies on the sphere $S_{i_{t_{0}}}$.

Second, we consider a sphere $S_{j}\not \in \{S_{i_{1}}, S_{i_{2}},\ldots ,S_{i_{m}}\}$ that does not contain x. In this case, we have |x −c_j|− r_j > 0. Therefore, there exists a small number ε_j > 0 such that ∀x ∈ [0,ε_j],

$$|\zeta_{s}(x)-\mathbf{x}|\leq\frac{1}{2}(|\mathbf{x}-\mathbf{c}_{j}|-r_{j}).$$

This yields that ∀x ∈ [0,ε_j],

$$|\zeta_{s}(x)-\mathbf{c}_{j}|\geq|\mathbf{x}-\mathbf{c}_{j}|-|\zeta_{s}(x)-\mathbf{x}|\geq\frac{1}{2}(|\mathbf{x}-\mathbf{c}_{j}|+r_{j})>r_{j}, $$

that is to say, ζ_s(x) lies outside the sphere S_j.

In summary, there exists a possibly small number ε > 0 such that ∀x ∈ [0,ε], ζ_s(x) does not cross any sphere S_i and lies on the sphere $S_{i_{t_{0}}}$, which implies that ζ_s(x) ∈Γ.

Appendix 2. Proof of Claim 3.1

Recall the definition of the face

$$ F_{ij} := \text{conv}\left( {\gamma}_{i}^{(0)},\mathbf{c}_{\mathcal{I}(\gamma_{ij}^{(1)})}\right),\qquad 1\leq i\leq n_{0},~1\leq j\leq K_{i}. $$

(57)

Given a nondegenerate intersection point $\gamma _{i}^{(0)}$ and the associated 1-patch $\gamma _{ij}^{(1)}$, the (d − 1)-face F_ij is actually a subset of $\widetilde R_{0}$, since, according to Lemma 3, F_ij is a subset of $\mathcal {R}({\gamma }_{i}^{(0)})$. Further, taking any point $\mathbf {y}\in \gamma _{ij}^{(1)}$, we have the following result:

$$ \mathcal{R}(\mathbf{y}) = \text{conv}\left( \mathbf{y}, \mathbf{c}_{\mathcal{I}(\gamma_{ij}^{(1)})}\right) \subseteq \mathcal{R}\left( \gamma_{ij}^{(1)}\right). $$

(58)

As y tends to $\gamma _{i}^{(0)}$, $\mathcal {R}(\mathbf {y})$ tends to F_ij. Any interior point of $\mathcal {R}(\mathbf {y})$ has y as a unique closest point, which implies that the interior of $\mathcal {R}(\mathbf {y})$ lies completely outside $\widetilde {R}_{0}$. For any point x ∈ F_ij, we can then find a sequence of points {x_n} outside $\widetilde {R}_{0}$ converging to x. Therefore, we have $F_{ij} \subseteq \partial \widetilde {R}_{0}$.

It is sufficient to prove that any point $\mathbf {x}\in \partial \widetilde {R}_{0}$ belongs to either some face F_ij or some set F₀ with $\dim (F_{0})\leq d-2$. $\partial \widetilde {R}_{0}$ can be divided into two sets

$$ U_{1} := \{\mathbf{x} \in \partial \widetilde{R}_{0}\mid \text{ all closest points of } \mathbf{x} \text{ belong to } \widetilde{P}_{0}\}, $$

(59)

and

$$ U_{2} := \{\mathbf{x} \in \partial \widetilde{R}_{0}\mid \text{ there exists a closest point of } \mathbf{x} \text{ not contained in } \widetilde{P}_{0}\}. $$

(60)

In the following content, we prove that if x ∈ U₁, then x belongs to a certain face F_ij, while if x ∈ U₂, then x belongs to F₀ which will be defined later.

Step 1: :

In the case of x ∈ U₁, we can find a sequence of points {x_n} in Ω∖R₀ such that x_n tends to x. Correspondingly, there exists a sequence of points {a_n} on Γ, where a_n is one closest point of x_n. Since x has finitely many closest points in $\widetilde {P}_{0}$ and the total number of k-patches is finite, we can extract a subsequence of a_n such that this subsequence lies on some k-patch γ^(k) with k ≥ 1 and converges to some nondegenerate intersection point $\gamma _{i}^{(0)}$. Without loss of generality, we can therefore suppose that a_n tends to $\gamma _{i}^{(0)}$ and a_n ∈ γ^(k), ∀n. As a consequence, $\gamma _{i}^{(0)}$ is on the boundary of γ^(k) and further, there exists a 1-patch $\gamma _{ij}^{(1)}$ on $\overline \gamma ^{(k)}$, satisfying

$$ \mathbf{c}_{\mathcal{I}(\gamma^{(k)})} \subseteq \mathbf{c}_{\mathcal{I}(\gamma_{ij}^{(1)})}. $$

(61)

Due to the fact that

$$ \mathbf{x}_{n} \in \text{conv}\left( \mathbf{a}_{n},\mathbf{c}_{\mathcal{I}(\gamma^{(k)})}\right), $$

(62)

we then have

$$ \mathbf{x} \in \text{conv}\left( \gamma_{i}^{(0)},\mathbf{c}_{\mathcal{I}(\gamma^{(k)})}\right) \subseteq \text{conv}\left( \gamma_{i}^{(0)},\mathbf{c}_{\mathcal{I}(\gamma_{ij}^{(1)})}\right) = F_{ij}, $$

(63)

by taking $n\rightarrow \infty$.

Step 2: :

In the case of x ∈ U₂, we want to prove that x belongs to some F₀. According to the definition of U₂, x has at least one closest point a that is not a nondegenerate intersection point. Here, we mention the fact that for any point y belonging to the open line segment $\overline {\mathbf {a}\mathbf {x}}$ with endpoints a and x, a is the unique closest point of y on Γ, which can be easily proven by contradiction.

On the one hand, if a is not an intersection point, then a lies on some k-patch γ^(k) with k ≥ 1. According to Theorem 1, we know that

$$ \mathcal{R}(\mathbf{a}) = \text{conv}\left( \mathbf{a}, \mathbf{c}_{\mathcal{I}({\gamma^{(k)}})}\right). $$

(64)

Considering that the latter convex hull, we obtain that $\mathbf {x}\in \text {conv}(\mathbf {c}_{\mathcal {I}({\gamma ^{(k)}})})$ of dimension $\dim (\text {conv}(\mathbf {c}_{\mathcal {I}({\gamma ^{(k)}})}))\leq d-2$, since otherwise, x will has a unique closest point on Γ. On the other hand, if a is a degenerate intersection point, then we have $\mathbf {a} \in {\varLambda }_{\mathcal {I}(\mathbf {a})}$ with $\dim \left ({\varLambda }_{\mathcal {I}(\mathbf {a})}\right ) \leq d-1$. Since $\mathcal {R}(\mathbf {a})\subseteq {\varLambda }_{\mathcal {I}(\mathbf {a})}$ according to Theorem 1, it holds that $\dim \left (\mathcal {R}(\mathbf {a})\right ) \leq d-1$. Here, we actually have $\mathbf {a}\in \mathcal {R}(\mathbf {a})$ and $\mathcal {R}(\mathbf {a})$ is a convex set from Lemma 3. Due to the fact mentioned above, we obtain that $\mathbf {x} \in \mathcal {R}(\mathbf {a})$ only lies on $\partial \mathcal {R}(\mathbf {a})$ of dimension $\dim \left (\partial \mathcal {R}(\mathbf {a})\right ) \leq d-2$.

As the number of k-patches and degenerate intersection points are finite, we can conclude that

$$ \mathbf{x}\in \bigcup_{k\geq 1} \text{conv}(\mathbf{c}_{\mathcal{I}({\gamma^{(k)}})}) \bigcup_{\gamma_{i}^{(0)}\not\in \widetilde{P}_{0}} \partial\mathcal{R}(\gamma_{i}^{(0)}), \qquad \forall \mathbf{x}\in U_{2}, $$

(65)

where $\gamma _{i}^{(0)}$ in the second union is taken as all degenerate intersection points. Note that the union on the right-hand side of Eq. 65 is of dimension less than or equal to d − 2. This implies that $\dim (U_{2}) \leq d-2$. Therefore, we can define F₀ = U₂, which satisfies $\dim (F_{0}) \leq d-2$.