This vignette is a long-winded mathematical exposition on zonotopes that concentrates on inversion. Discussion of software is delayed until the end. Featured functions are: zonoseg()
, zonogon()
, zonohedron()
, invert()
, and invertboundary()
.
The emphasis in this vignette are the concepts needed to understand the inversion functions in the zonohedra package. Much of this is based on [5].
Note that the 2 properties imply that the intersection \(P \cap C\) is a subset of the boundary of \(C\).
If the compact set is a convex body, i.e. is convex with interior, then 2 equivalent properties are given by this:
Proof:
not ii. \(\implies\) not 2.
Let ii. be false, so hyperplane \(P\) does intersect \(int(B)\), at a point \(p\). Then there is an open ball centered at \(p\), and the ball is inside \(B\). There are clearly points in the ball in both halfspaces, and so 2. is false.
not 2. \(\implies\) not ii.
Let 2. be false, so there are points \(b^-, b^+ \in B\) that are in different open halfspaces. Let \(b_i\) be a point in \(int(B)\). If \(b_i \in P\) then ii. is false and we are done. Otherwise, either \(b^-\) or \(b^+\) are in a different halfspace than \(b_i\). Take it to be \(b^+\), w.l.o.g. Since \(b_i\) and \(b^+\) are in opposite halfspaces, the segment \([b_i,b^+]\) intersects \(P\); let \(c_i\) be this point of intersection. There is an open ball in \(B\) centered at \(b_i\). Let \(C\) denote the convex hull of \(b^+\) and this ball - a partial open cone. By convexity of \(B\), \(C\) is in \(B\). There is a scaled down open ball centered at \(c_i\) and contained in \(C\). Thus \(c_i \in P \cap int(B)\) and ii. is false. \(\square\)
The equivalence 1 and 2 is straightforward, and the equivalence of 2 and 3 is trivial.
The entire set \(C\) is considered to be an (improper) face.
The dimension of a face is the dimension of the affine subspace spanned by the face. From now on, We always assume that the dimension of \(C\) is \(n\), which is equivalent to \(C\) having an interior.
A d-face is a face of dimension d. So a 0-face is a vertex, and a 1-face is an edge. A facet is an (\(n{-}1\))-face, and a maximal proper face.
Note that every face of the cube \([0,1]^n \subset \mathbb{R}^n\) is a cube of smaller dimension; in fact the dimension is the number of 0s in the normal vector \(w\) from part 2 of the above definition.
Let \(A : \mathbb{R}^n \twoheadrightarrow \mathbb{R}^m\) be a surjective affine map (so \(m \le n\)), and let \(C':= A(C)\).
Theorem: If \(F'\) is a face of \(C'\), then \(A^{-1}(F')\) is a face of \(C\).
Stated in words, the affine preimage of a face is a face.
Proof: Use part 3 of the above definition so \(F' = \mathop{\mathrm{argmax}}\limits_{y \in C'} \lambda(y)\), where \(\lambda\) is a non-zero linear functional on \(\mathbb{R}^m\). Let \(\mu := \mathop{\mathrm{max}}\limits_{y \in C'} \lambda(y)\). Now \(A^{-1}(F') = A^{-1}( \lambda^{-1}(\mu) ) = (\lambda \circ A)^{-1}(\mu)\). But \(\lambda \circ A\) is a non-zero linear functional on \(\mathbb{R}^n\), plus a constant. \(\square\)
See also [5], Lemma 7.10.
Definition: A zonotope \(Z\) is a set of the form \(L([0,1]^n) + z_0\) where \(L : \mathbb{R}^n \twoheadrightarrow \mathbb{R}^m\) is a surjective linear map.
Simply stated, a zonotope is an linear image of a cube plus a translation (an affine image of a cube).
Since the cube is convex, the zonotope is also convex.
The \(n\) generators of \(Z\) are the images of the \(n\) elementary vectors \(L(e_1), ... , L(e_n)\). A point \(z \in Z\) iff \(z = \alpha_1 L(e_1) ~+~ ... ~+~ \alpha_n L(e_n) + z_0\) with all \(\alpha_i \in [0,1]\).
A zonotope is centrally symmetric about the point \(L(1/2,...,1/2) + z_0\). By reflecting through the center of symmetry, each facet of \(Z\) has a corresponding antipodal facet. The facets come in antipodal pairs.
Given a face \(F\) of zonotope \(Z\), the preimage of \(F\) is a face \(F'\) of the cube. But every face of a cube is also a cube, and so \(F\) is also a zonotope. Let the normal vector of the supporting hyperplane of \(F'\) be \(w\). Then the vectors \(\{ \ L(e_i) | w_i=0 \ \}\) are all parallel to the face \(F\), and in fact they generate the linear subspace parallel to \(F\). We call \(\{ \ L(e_i) | w_i=0 \ \}\) the generators of \(F\). And important fact: the number of generators of \(F\) is the dimension of the preimage \(F'\). Note that a face of dimension \(d\) may have more than \(d\) generators, because they may be linearly dependent. For a parallelogram face, even if no generators are 0, the face may have more than 2 generators because some may be multiples of others.
If the dimension of \(Z\) is \(m\), we call it an m-zonotope.
Theorem: If \(K\) is the convex hull of a finite set of points in \(\mathbb{R}^n\), with \(n \ge 3\). Then \(K\) is an \(n\)-zonotope iff all facets of \(K\) are (\(n{-}1\))-zonotopes.
For a proof of this hard result, plus much more, see [1].
A zonotope of dimensions 1, 2, and 3 is called a zonoseg , zonogon, and zonohedron, respectively. The term “zonoseg” is mine, since I could not find a term for it in the literature. Geometrically a zonoseg is just a line segment.
A zonoseg has only two faces - the endpoints of the segment.
A zonogon is a convex polygon with 0-faces (vertices) and 1-faces (edges). Since the dimension of an edge is 1 less than the dimension of the zonogon, an edge of a zonogon is also a facet. It can be shown that a convex polygon is a zonogon iff it is centrally symmetric.
A zonohedron has 0-faces (vertices), 1-faces (edges), and 2-faces (facets). All the facets are zonogons. A parallelogram facet is called trivial, and facets with more than 4 edges are non-trivial.
Let \(\mathbb{R}^n_{\ge 0} := \{ \ (x_1,x_2,...x_n) \ | \ x_i \ge 0 \ \}\) denote the non-negative orthant in \(\mathbb{R}^n\).
A convex cone \(K\) is a set of the form \(K = L(\mathbb{R}^n_{\ge 0})\), where \(L : \mathbb{R}^n \twoheadrightarrow \mathbb{R}^m\) is a surjective linear map. The \(n\) generators of \(K\) are the images of the \(n\) elementary vectors \(L(e_1), ... , L(e_n)\). \(K\) is the set of all non-negative linear combinations of the generators.
If \(K\) is a subset of a closed linear halfspace, it is called salient. If \(K\) is a subset of an open linear halfspace (except for the vertex 0), it is called pointed. These properties are equivalent to 0 being in the boundary of \(K\), and being a vertex of \(K\), respectively. Obviously, pointed implies salient, but salient does not imply pointed.
Given a zonotope \(Z = L([0,1]^n) + z_0\) the map \(L\) also defines a convex cone \(K\). \(Z\) is a subset of \(K\), after translating \(Z\) by \(-z_0\). We carry the two above properties of \(K\) over to \(Z\). It is straightforward to show that \(Z\) is salient iff \(z_0\) is in the boundary of \(Z\), and \(Z\) is pointed iff \(z_0\) is a vertex of \(Z\).
If \(Z\) is pointed then there is a “cutting plane” that has \(z_0\) on one side, and all the other vertices on the other side. The intersection of this cutting plane and \(Z\) is called the vertex figure of \(Z\) at \(z_0\). The vertex figure is actually more general, and is defined for any vertex of a convex polyhedron, see [5], p. 54. In the case that \(Z\) is a zonohedron, the vertex figure at \(z_0\) is a a convex polygon that we call the generator polygon. The polygon is only unique up to a 2D projective transformation.
This section assumes some knowledge of matroids; for background on them see the matroids vignette.
Given a zonotope \(Z = L([0,1]^n) + z_0\) as above, the generators define a matroid \(M\). Since \(L\) is surjective, \(\mathrm{rank}(M) = m\). A hyperplane of \(M\) corresponds to a pair of antipodal facets of \(Z\) (the concept of hyperplane here is the the one used in matroid theory).
Assume now that \(m=3\), so \(Z\) is a zonohedron and all its facets are zonogons. If \(M\) is simple, then the number of sides of a zonogon facet is twice the number of points in the corresponding hyperplane. So a parallelogram corresponds to a hyperplane with 2 points, which is called a trivial hyperplane.
As before, let \(L : \mathbb{R}^n \twoheadrightarrow \mathbb{R}^m\) be a surjective linear map, and let \(Z := L([0,1]^n) + z_0\) for some \(z_0 \in \mathbb{R}^m\).
From this setup, given a point \(z \in Z\), we know that the equation \[\begin{equation}\tag{$\star$} L(x) + z_0 = z ~~~~ \textrm{for} ~ x \in [0,1]^n \end{equation}\] has a solution \(x\). The rest of this section looks at the solutions of (\(\star\)) in more depth.
In this section we consider the question: When is the solution of \((\star)\) unique ?
In the interior case, the answer is straightforward.
Lemma: If \(z\) is in the interior of \(Z\), then the solution of \((\star)\) is unique iff \(n=m\).
Proof: If \(n=m\) then \(L\) is invertible so we are done. If \(n>m\) the nullspace of \(L\) has positive dimension \(n-m\). By Theorem 4.2 of [2], we can pick an \(x \in \operatorname{int}([0,1]^n)\) that satisfies \((\star)\). Let \(U \subset [0,1]^n\) be an open ball around \(x\); the intersection of the nullspace with \(U\) is an infinite set. \(\square\)
For \(z \in Z\), let \(F_z\) be the smallest face that contains \(z\), i.e. the intersection of all faces that contain \(z\). It is clear that \(z\) is in the relative interior of \(F_z\), i.e. \(z \in \operatorname{relint}(F_z)\). At the extremes, if \(z\) is a vertex then \(F_z\) is \(\{ z \}\), and if \(z\) is in the interior of \(Z\), then \(F_z\) is \(Z\) (here we allow \(Z\) itself as an improper face). The relative interiors of the faces form a partition of \(Z\), see [5], p. 61.
Theorem: Let \(z\) and \(Z\) and \(F_z\) be as above. Then the solution of \((\star)\) is unique iff the number of generators of \(F_z\) is equal to the dimension of \(F_z\).
Proof: The condition says that the dimension of the preimage of \(F_z\) is the dimension of \(F_z\). Now apply the Lemma, with \(Z\) replaced by the preimage of \(F_z\). \(\square\)
It is useful to reformulate the uniqueness theorem for the specific case when \(Z\) is a zonohedron (\(m=3\)).
Theorem: Let \(z\) be in a zonohedron \(Z\). If \(n>3\), then the solution of \((\star)\) is unique iff none of the generators of \(Z\) are 0 and:
\(z\) is a vertex of \(Z\)
or \(z\) is in an edge of \(Z\), and the edge has one generator
or \(z\) is in a parallelogram facet of \(Z\), and the parallelogram has two generators
If the matroid of \(Z\) is simple, i.e. no generators of \(Z\) are 0 or multiples of each other, then this simplifies to:
Theorem: Let \(z\) be in a zonohedron \(Z\) whose matroid is simple. If \(n>3\), then the solution of \((\star)\) is unique iff \(z\) is in an edge or a parallelogram facet.
And for a zonogon we have:
Theorem: Let \(z\) be in a zonogon \(Z\) whose matroid is simple. If \(n>2\), then the solution of \((\star)\) is unique iff \(z\) is in the boundary of \(Z\) (denoted by \(\partial Z\)).
library(zonohedra)
Let \(Z\) be a zonogon whose matroid is simple. By the previous theorem there is a unique function \(\sigma : \partial Z \to [0,1]^n\) that is a right inverse for \(x \mapsto L(x)+z_0\). We have \(L( \sigma(z) ) + z_0 = z\) for all \(z \in \partial Z\).
Question: Is \(\sigma()\) continuous ?
Well, on each edge it is linear, and so certainly continuous. Moreover, on each vertex is is uniquely defined, and so the separate linear maps on each edge must match up on the vertices. So yes, \(\sigma()\) is continuous; in fact it is piecewise-linear.
It is instructive to consider a very specific case. Consider the figure:
= polarzonogon( 14, 4 )
zono = par( omi=c(0,0,0,0), mai=c(0.8,0.7,0.7,0.2) )
oldpar plot( zono, elabels=T )
par( oldpar )
Denote the 4 generators by \(z_1, z_2, z_3, z_4\); these are labeled in the figure by just the indexes. Along the bottom edge, \(x_1\) increases from 0 to 1, while the other \(x\)’s are 0. When the first vertex \(z_1=(1,0)\) is reached, \(x_1\) remains at 1, and on the 2nd edge \(x_2\) increases from 0 to 1, until the next vertex \(z_1+z_2\), etc. At any point on the lower boundary \(x = (1,...,1,\alpha,0, ... ,0)\); i.e. a run of 1s, then an arbitrary \(\alpha \in [0,1]\), and then a run of 0s. Both runs are allowed to be empty. Similarly, along the upper boundary \(x = (0,...,0,\alpha,1, ... ,1)\). These two “low-pass” and “high-pass” filters are analogous to Goethe’s edge colors (Kantenfarben), see [4] p. 17.
In the previous section, the right inverse \(\sigma()\) was only defined on \(\partial Z\).
Question: Is there a way to extend \(\sigma()\) across the interior ?
The answer lies in parallelogram tilings. Consider the figure:
= par( omi=c(0,0,0,0), mai=c(0.8,0.7,0.7,0.2) )
oldpar plot( zono, tiling=T, elabels=T, tlabels=T )
par( oldpar )
The labels inside the parallelogram tiles are the generators of the tiles. For points inside the 3 tiles that meet 0, the values of \(x_i\) are obvious. For a point inside tile 1,3, \(x_2=1\) and \(x_1\) and \(x_3\) vary in [0,1].
The rule for a point \(z\) is to locate the tile containing \(z\) and then the origin of the tile. Next, locate a path of tile edges from 0 to the origin, and that determines the \(x\) coordinate alues that are 1. The \(x\) coordinate values for the tile generators are the 2 coordinates of \(z\) in the tile relative to the origin, and all other \(x\) values are 0. For tile 1,4 there are 2 different paths to the origin of the tile, but it doesn’t matter since the \(x\) indexes on the different paths are the same: 2 and 3. In general, two different paths are connected by a homotopy that “crosses” one parallelogram at a time, and each time, the 2 associated \(x\) indexes are the same.
It is straightforward to verify that the right inverse \(\sigma()\) defined by this rule is continuous.
The above tiling is just one of many, and each different tiling generates a different right inverse. For the 4-generator zonogon above, the number of different tilings is 8. This sequence increases very rapidly with \(n\); for \(n{=}8\) generators (16 sides) the number of tilings is already more than \(10^6\), see [6].
The above tiling is an example of the standard tiling in the zonohedra package, which is generated by the following recipe. Each generator is “lifted” to \(\mathbb{R}^3\) by the mapping \((x,y) \mapsto (x,y,\sqrt{x^2+y^2})\). The mapping “lifts” each generator to the cone \(x^2 + y^2 = z^2\). The lifted generators generate a zonohedron, which has an upper half and a lower half. The faces in the lower half are the ones that can be seen from a a viewpoint far below the \(xy\)-plane, see [5], p. 130. The parallelogram facets in the lower half are projected down to \(\mathbb{R}^2\) and these form the standard tiling.
The standard tiling has the following nice property: if the zonogon is pointed, and the generators are in order by angle (clockwise or counterclockwise), then every point \(\sigma(z)\) has 2 transitions, and the run of 1s does not wrap around. For the definition of a 2-transition point of the cube, see the The 2-Transition Subcomplex and the 2-Transition Surface vignette. The 2-transition concept is important for zonohedra coming from colorimetry. The standard tiling is denoted by \(T_{min}\) in [3] p. 13, where the 2-transition property is also noted in equation (12).
In this section, let \(Z\) be a zonohedron whose matroid is simple, with \(n{>}3\).
Assume for simplicity that all facets are parallelograms. Then by a previous theorem there is a unique function \(\sigma : \partial Z \to [0,1]^n\) that is a right inverse for \(x \mapsto L(x)+z_0\). This right inverse is unique on the edges, which implies that \(\sigma()\) is continuous.
Each parallelogram is the image of a square in \([0,1]^n\), and the squares are “glued” together on the edges to form a “surface” (in fact a topological sphere) embedded in \([0,1]^n\). We see another example of this in the The 2-Transition Subcomplex and the 2-Transition Surface vignette.
Now suppose that a facet of \(Z\) is an arbitrary zonogon, with a high number of generators. Then by rotating this non-trivial facet to the plane, and choosing the standard tiling, we can use the construction in the previous section to extend the right inverse across this facet. Once again, the right inverse is unique on the edges, which implies that the extended \(\sigma()\) is continuous.
To summarize this section, a right inverse \(\sigma()\) defined on \(\partial Z\) always exists and is continuous, but is only unique up to the selected tiling of the non-trivial facets of \(Z\).
The above sections are mathematical in nature. This sections is about the implementation of the above in the software package zonohedra.
The package only supports zonotopes of dimensions 1, 2, and 3, which are called zonosegs, zonogons, and zonohedra, respectively.
The extra generality of the translation \(z_0\) turned out to be an unnecessary complication. Thus, in the package, the constructors zonoseg()
, zonogon()
, and zonohedron()
only take a matrix argument, and not \(z_0\).
Many of the above theorems require that the matroid associated with \(Z\) is simple. In the non-simple matroid case, the package ignores all generators that are 0. And for a multiple group, when all the generators are positive multiples of each other, it replaces these generators by their sum. When some generators are negative multiples of each other, the situation is more complicated and not yet documented. The zonohedron is then computed from these “simplified” generators, which has a simple matroid.
In many calculations, the central symmetry is used to reduce storage. For example, only one facet in a pair of antipodal facets needs to be stored, and the other can easily be derived by reflection. In some cases, the symmetry is also used to reduce computation time.
The section extending the right inverse, using parallelogram tilings is implemented in the function invert()
, which takes the zonogon as argument. The section a right inverse on the boundary of a zonohedron is implemented in the function invertboundary()
, which takes the zonohedron as argument.
For a pointed zonohedron \(Z\), the function plotpolygon()
plots the generator polygon for \(Z\) at 0.
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