# Directional tutorial notes

## Introduction¶

Directional is a C++ geometry processing library written on the basis of libigl, with a specialization in directional-field processing. The functionality is based on the definitions and taxonomy surveyed theoretically in 18, and through it by the relevant papers in the literature. It contains tools to edit, analyze, and visualize directional fields of various degrees and symmetries.

The underlying structure extends the general philosophy of libigl: the library is header only, where each header contains a set of functions closely related (for instance, the precomputation and computation of some directional quantity over a mesh). For the most part, one header contains only one function. The data structures are, for the most part, simple matrices in Eigen, and the library avoids complicated and nested structures, relying instead on standalone functions. The visualization is done through a specialized class DirectionalViewer, on the basis of libigl viewer, with many extended options that allow the rendering of directional fields.

The header files contain documentation of the parameters to each function and their required composition; in this tutorial we will mostly tie the functionality of Directional to the theoretical concepts of directional fields and the methods to process and visualize them.

### Installing the tutorial examples¶

This tutorial comprises an exhaustive set of examples that demonstrates the capabilities of Directional, where every subchapter entails a single concept. The tutorial code can be installed by going into the tutorial folder from the main Directional folder, and typing the following instructions in a terminal:

mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../
make


This will build all tutorial chapters in the build folder. The necessary dependencies will be appended and built automatically. To build in windows, use the cmake-gui .. options instead of the last two commands, and create the project using Visual Studio, with the proper tutorial subchapter as the “startup project”.

To access a single example, say 202_Sampling, go to the build subfolder, and the executable will be there. Command-line arguments are never required; the data is read from the shared folder directly for each example.

Most examples require a modest amount of user interaction; the instructions of what to do are given in the command-line output upon execution.

### Discretization¶

Input: Directional requires the input to always be an orientable triangle mesh in a single connected-component. There are no general limitations on the genus or the boundaries. Specific algorithms may have additional requirements or consistency issues (for instance, index prescription requires that the singularity indices ad dup to the Euler characteristic). If your input comprises several connected components altogether, you can just run any algorithm on each of them, loaded individually in sequence.

There are several ways to represent tangent planes on triangle meshes, although not all of them compatible with the different information that the fields convey. Mainstream discretizations roughly categorize into face-based, edge-based, and vertex-basex discretizations (see 9 for an in-depth analysis). The only discretization currently supported by Directional is that of tangent face-based fields, where the discrete tangent plane is taken to be the supporting plane to each (naturally flat) face. We also use a local basis (provided by igl::local_basis()) to parameterize each tangent plane. Notions like connection, parallel transport, and consequently smoothness, are measured on dual edges between adjacent faces.

### Representation¶

The representation of a directional field is the way it is encoded in each discrete tangent plane, and across adjacent tangent planes. Directional uses several different representations to describe directional fields. We denote the number of faces in the mesh as $|F|$, the set of inner edges (adjacent to two triangles) as $E_I$, and the degree of the field as $N$ (must be fixed for the entire field). The supported representations are as follows, where the taxonomy is based on that of the directional field course 18:

1. Raw - A $|F|\times 3N$ double matrix, representing an $1^N$-vector field (a directional with $N$ independent vectors in each face) in the form $X_1, Y_1, Z_1, X_2, Y_2, Z_2, \cdots X_N, Y_N, Z_N$ per face. Vectors are assumed to be ordered in counterclockwise order in most Directional functions that process raw fields.
2. Representative. A $|F| \times 3$ double matrix that represents a rotationally symmetric $N$-vector field, known as an $N$-RoSy. The single vector is an arbitrary “first” vector in the face, and the rest of the vectors are deduced by rotations of $\frac{2\cdot\pi}{N}$. That means the all functions should also accept $N$ as input to decode the full field.
3. Rotation Angles. A $|E_I|$-sized double vector representing the rotation angle between two directions (without magnitude information) on two adjacent triangles. The rotation represents the deviation from the Levi-Civita parallel transport 6,12. This representation may only encode $N$-direction fields (no magnitude). Note that the effort (sum of all rotations) is then $N$ times rotation angles. Since this is a differential quantity, an extra global rotation needs to be given to uniquely decode the full face-based field.
4. Power Field - An $|F|$-sized complex vector, encoding an $N$-RoSy object as a single complex number $y=u^N$ encoded in the local basis, where the $N$-RoSy is the set of roots $u \cdot e^{\frac{2\pi i k}{N}}, k \in [0,N-1]$. The magnitude is also encoded this way, though it may be neglected in some applications. The representation depends on a local $2D$ basis, such as one that could be obtained from igl::local_basis().
5. PolyVector - A $|F| \times N$ complex matrix, representing the coefficients $a$ of a monic complex polynomial $f(z)=z^N+\sum_{i=0}^{N-1}{a_iz^i}$, which roots $u$ are the encoded $1^N$-vector field. Every row is encoded as $a_{0},\cdots, a_{N-1}$, where $a_0$ is the free coefficient. In case where the field is an $N$-RoSy, all coefficients but $a_0$ are zero. Note: A PolyVector that represents a perfect $N$-RoSy would have all $a_i=0,\ \forall i>0$, but $a_0$ would have opposite sign from the power-field representation of the same $N$-RoSy. This is since the power field represents $u^N$ directly, whereas a PolyVector represents the coefficients of $z^N-U^N$ in this case.

Directional provides a number of conversion functions to switch between different representations. Each of the functions is of the form rep1_to_rep2, where rep1 and rep2 are the representation names in the above list. e.g., rotation_to_representative() and polyvector_to_raw(). Some possible combinations are given by composing two functions in sequence. However, note that not every conversion is possible; for instance, it is not possible to convert between PolyVectors and rotation angles, as they do not possess the same power of expression (with current state-of-the-art…). For $N$-RoSy fields, for instance, you will most likely work primarily with the power field, representative, or rotation-angle representation. converting into the more explicit raw representation is often needed for I/O and visualization.

## Chapter 1: I/O and Visualization¶

Directional uses a specialized class called DirectionalViewer which inherits and extends the libigl Viewer class, augmenting it with functionality that pertains to directional fields. A mesh is then stored with its accompanying geometric quantities: the field, edge, vertex, and face-based scalar data, isolines, and more, that we detail in the tutorial per chapter in context. Like libigl viewer, Directional supports independent multiple meshes, each with its own set of quantities. Internally, the visualization schemes carete sub-meshes which serve as layers on the original meshes: arrows for glyphs, bars for edge highlights, etc. In practice this is encapsulated from the user and does not need to be controlled directly.

### 101 Glyph Rendering¶

The most basic operation on directional fields is reading them from a file and drawing them in the most explicit way. In Example 101, a mesh and a field are read from a file and visualized as follows:

 directional::read_raw_field(TUTORIAL_SHARED_PATH "/bumpy.rawfield", N, rawField);

directional::DirectionalViewer viewer;

viewer.set_mesh(V,F);
viewer.set_field(rawField);
viewer.set_singularities(singVertices, singIndices);


The field is read in raw format (see File Formats), which is detailed in the Introduction. The field is face-based, and the singularities are consequently vertex-based, where singVertices are the singular vertices, and singIndices are the corresponding integer indices, so that the actual fractional indices are $\frac{singIndices}{N}$. If the mesh number is not prescribed explicitly, the default single mesh (number 0) is assumed.

The singularities and glyphs (and most other properties) can be toggled by functions of the type DirectionalViewer::toggle_field() and DirectionalViewer::toggle_singularities().

### 102 Picking and editing¶

This example demonstrates the editing paradigm in Directional, based on libigl picking. A face and a vector within the face are chosen, and clicking on a new direction for the vector changes it. Note the different colors for glyphs and selected faces. The specificiation of selection is done via the following code in Example 102.

directionalViewer->set_selected_faces(selectedFaces);
directionalViewer->set_selected_vector(currF, currVec);


### 103 Streamline Tracing¶

Vector fields on surfaces are commonly visualized by tracing [streamlines] (https://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines). Directional supports the seeding and tracing of streamlines, for all types of directionals. The seeds for the streamlines are initialized using DirectionalViewer::init_streamlines(), and the lines are traced using streamlines_next. Each call to DirectionalViewer::advance_streamlines() extends each line by one triangle, allowing interactive rendering of the traced lines, as demonstrated in Example 103. The streamline have the same colors as the initial glyphs, where the colors fade into white as the streamline advance.

### 104 Scalar quantities on meshes¶

It is possible to set and visualize scalar quantities on meshes at different discretization locations: either face based quantities that appear as flat colors per face, vertex-based (pointwise) quantities that interpolate linearly on faces, appearing smooth, and edge-based (integrated) quantities, that appear as flat quantities on a diamond mesh associates with each edge (taking a $\frac{1}{3}$ of the area of each adjacent triangle). The is controlled by the DirectionalViewer::set_X_data() functions, that also allow the setting of the viewable range of the function (the rest is clipped).

### Sparse Glyph View¶

On big meshes, it might appear cumbersome to view all glyphs on every face. It is possible to only view the glyphs on a subsample of faces, by using the sparsity parameter in DirectionalViewer::set_field().

## Chapter 2: Discretization and Representation¶

In the following sections, we show some effects of working with different representations and converting between them.

### 201 Principal Matching¶

One of the fundamental operations in directional-field processing is matching. That is, defining which vectors in face $f_i$ correspond to those in adjacent face $f_j$. In Directional, we only work with order-preserving matchings: if vector $k$ in face $f_i$ is matched to vector $m$ in face $f_j$, then for any $l \in \mathbb{Z}$, vector $k+l$ is matched to vector $m+l$ (modulu $N$) in the respective faces. Suppose that the orientation of the dual edge is $f_i \rightarrow f_j$, then the matching is encoded as $m-k$. Some representations, like rotation angles, already encode the matching explicitly, but others do not. Therefore, it needs to be devised from the field.

Given a raw field (in assumed CCW order in every face), it is possible to devise the rotation angles $\delta_{ij}$ by the process of principal matching 7. Principal matching is defined as the matching with minimal effort, always putting it within the range of $[-\pi, \pi)$ (and therefore denoted as “principal”). It corresponds to the “smallest angle” matching for $N$-RoSy fields.

principal matching is done through the function principal_matching() (from in Example 201) as follow:

directional::principal_matching(V, F,EV, EF, FE, rawField, matching, effort, singVertices, singIndices);

The function also computes the index of each vertex from the effort around it. The index of a vertex is the amount of rotations a directional object undergoes along a cycle around the vertex. A directional must return to itself after a cycle, and therefore the index is an integer $I$ when a vector $m$ in the face ended up in vector $m+I$. Note that this can also include multiple full rotations (i.e., this is not taken modulu $N$), where the index is unbounded. The fractional part of the index is encoded by the matching; however, matching alone cannot encode integral indices (for instance, a single vector field has trivial (Zero) matching anywhere, but can have singularities). singVertices and singIndices only enumerate the singular vertices.

### 202 Sampling¶

This is an educational example that demonstrates the loss of information when moving between a polar (in this case, rotation angle) representation to a Cartesian representation, where the matching between vectors in adjacent faces is done with principal matching. In that case, low valence cycles and undersampling cause aliasing in the perceived field. There are three modes seen in the example:

1. In the polar mode, the user can prescribe the index of a singularity directly. With this, the rotation angles between adjacent faces become arbitrarily large, and appear as noise in the low valence cycles.

2. In the principal matching mode, the rotations are reconstructed from the field, without prior knowledge of the polar-prescribed rotations from the previous mode. The large rotation between adjacent faces is lost, which gives rise to a “singularity party”: many perceived singularities or a lower index.

3. In the Cartesian mode, the field is interpolated on the free faces (white) from the constrained faces (red), keeping the red band fixed from the polar mode. We see a field that is smooth in the Cartesian sense, with more uniformly-placed singularities.

### 203 Combing¶

Given a matching (in this case, principal matching), it is possible to “comb” the field. That is, re-index each face (keeping the CCW order), so that the vector indexing aligns perfectly with the matching to the neighbors—then, the new matching on the dual edges becomes trivially zero. This operation is important in order to prepare a directional field for integration, for instance. In the presence of singularities, the field can only be combed up to a forest of paths that connect between singularities, also known as seams. Note that such paths do not necessarily cut the mesh into a simply-connected patch, but may only connects subgroups of singularities with indices adding up to an integer; as a trivial example, a 1-vector field is always trivially combed, even in the presence of integral singularities, and the set of seams is zero. The combing is done through the function directional::combing() as follows, taken from Example 203

irectional::combing(V,F, EV, EF, FE, rawField, matching, combedField);


where combedField is the re-indexed rawField, done according to the input matching. We can recompute the matching on the combed field to retrieve the seams:

  directional::principal_matching(V, F,EV, EF, FE, combedField, combedMatching, combedEffort,singVertices, singIndices);


## Chapter 3: Cartesian Methods¶

### Cartesian Fields¶

The Cartesian representation is a meta-category for representation of vectors in explicit coordinates, either $\left(x,y\right)$ in some local 2D basis on a tangent plane, or $\left(x,y,z\right)$ in the ambient coordinates of the 3D space. The raw, representative (of an $N$-RoSy), power field, and PolyVector representations are all such examples. Cartesian fields often do not automatically contain information about the matching, or rotation, of a field between one face and the next, and it needs to be computed using principal matching. This chapter focuses on computing fields with this representation.

### 301 Power Fields¶

This representation is offered in 11, but they did not give it a specific name (the method in general is called “globally optimal”). We use the name “power fields” coined in 1.

A power field representation uses a complex basis in each tangent plane (face in our implementation), and represents an $N$-RoSy using a power vector—a single complex number $y$ per face so that its root set $y=u^N$ comprises the vectors of the $N$-RoSy.

By prescribing constraints $y_B$ on a set of faces $B$, the algorithm interpolates the field to the rest of the faces $y_I$ by minimizing the face-based quadratic Dirichlet energy:

Y_I=\text{argmin}\sum_{e=(f,g)\in F \times F}{\omega_e\left|Y_fe_f^N - Y_ge_g^N\right|^2},

where $e_f$ is the representation of the vector of edge $e$ in the basis of $f$, and $e_g$ is for $g$ respectively. The weights $\omega_e$ are the harmonic weights as given by 5. The field is computed through the function directional::power_field(). It is possible to alternatively only softly prescribe the constraints $\left\{Y^*_C\right\}$ with alignment weights $\omega_c$, solving the following minimization problem:

y_I=\text{argmin} \left[\lambda_S\sum_{e=(f,g)}{\omega_e\left|Y_fe_f^N - Y_ge_g^N\right|^2}+\lambda_B\sum_{c \in C}{\omega_c \left|Y_c - Y^*_c\right|^2}\right],

where $\lambda_S,\lambda_C, \omega_{\forall c\in C}$ are user-controlled.

If the set $B$ is empty, then the computed field is the eigenvector of the power-field Laplcian associated with the smallest non-zero eigenvalue.

### 302 PolyVectors¶

A Polyvector field 7 is a generalization of power fields that allows to represent independent vectors in each tangent plane. The representation is as the coefficient set $a_{0 \cdots N-1}$ of a monic complex polynomial in the local compex basis:

P(z) = X_0 + X_1z + \ldots + X_{N-1} z^{N-1} + z^N,

where the roots $P(z)=0$ are the vectors of the face-based directional object, represented as complex numbers in the local basis. The Dirichlet energy is as for power fields, except with a term for each $a_i$, with the appropriate power $i$. Note that an $N$-RoSy is represented as a polynomial where all $a$ are zero except $a_0$. Principal matching, combing, and effort are well-defined on PolyVectors as well.

With the function Directional::polyvector_field() one can solve the linear Polyvector problem in its full capacity; the input is a set of prescribed constraints $v_b$ per face set $B \subset F$, where you can prescribe any amount of vectors smaller of equal than $N$ (unless there is symmetry) per face; that means we consider set $B$ to have repeating faces, and the enumerator in this set is the vectors. Further consider the soft-alignment vectors $v_c$ for face set $c \subset C$, accompanied by an alignment weight $\omega_c$ (again with repeating faces). We then solve the following quadratic optimization problem:

y_I = \text{argmin} \left(\lambda_S \sum_{n=0}^{N-1}{\sum_{e=(f,g)}{\omega_e\left|X_{n,f}e_f^{N-n} - X_{n,g}e_g^{N-n}\right|^2}}\right)+\left(\lambda_R\sum_{n=1}^{N-1}{\sum_{f}{A_f\left|X_{n,f}\right|^2}}\right)+\left(\lambda_C\sum_{C,n}{\omega_c\left|X_{n,f}-Q_{c}Q_{c}^{\dagger}(X_{n,f}-q_{c})+q_{c}\right|^2}\right).

So that the set $B$ is perfectly interpolated. The matrices $Q_{c}$ and vectors $q_{c}$ are designed to create a linear subspace of polynomials for which the respective vectors $v_c$ are roots; for details see Appendix A in 14. The last term then measures the soft alignment, also weighted by $\omega_c$ per vector. The same technique is used as a hard reduction for the set $B$. The middle term subdues all non-free powers of $P(z)$, thus optimizing $P(z)$ to be as much a power vector as possible. The different energies are also controlled globally by user parameters $\lambda_S, \lambda_R$ and $\lambda_C$. It is also possible to constrain the field to be a perfect power field; in fact Directional::power_field() is implemented by calling Directional::polyvector_field(). See an example that also allows playing with the weights in Example 302.

### 303 PolyCurl Reduction¶

Vector-field guided surface parameterization is based on the idea of designing the candidate gradients of the parameterization functions (which are tangent vector fields on the surface) instead of the functions themselves. Thus, vector-set fields ($N$-Rosy, frame fields, and polyvector fields) that are to be used for parameterization (and subsequent remeshing) should be as integrable as possible: it should be possible to locally comb them into individual vector fields that are approximately gradients of scalar functions. Fields obtained by “as-smooth-as-possible” design methods (eg. 12, 11, 7, 2, 17) do not have this property in general. In 8, a method for creating integrable polyvector fields was introduced by the process of curl reduction. This method takes as input a given field and improves its integrability by iteratively reducing the PolyCurl of the field; that is, the coefficients of a dual-edge-based polynomial, whose roots are the curl of the matched vectors in the two adjacent faces. By working with PolyCurl instead of matching, the optimization can be done on the PolyVector itself, allowing for singularities to naturally emerge. However, the optimization is nonlinear—it reduces the PolyCurl iteratively, while preserving the CCW order of the vectors (for a bijective parameterization), and opting for as smooth and orthogonal as possible result.

A field that has zero PolyCurl everywhere is locally (away from singularities) integrable into $N$ different scalar functions; globally, it is integrable into a rotationally-seamless multi-branched function, which we further demonstrate in Chapter 5. In Example 303 we demonstrate the PolyCurl-reduction optimization.

### 304 Conjugate Fields¶

Two tangent vectors $u$ and $v$ are conjugate if

$$k_1 (u^T d_1)(v^T d_1) + k_2(u^T d_2)(v^T d_2) = 0,$$ where $k_1$ and $k_2$ are the principal curvatures and $d_1$ and $d_2$ are the respective principal directions.

Conjugate vector fields are very important in architectural geometry: their integral lines form, informally speaking, an infinitesimal planar quad mesh. As such, the finite quad mesh that results from discretizing conjugate networks is a good candidate for consequent planarity parameterization 13.

Finding a conjugate vector field that satisfies given directional constraints is a standard problem in architectural geometry, which can be tackled by deforming a $2^2$ PolyVector field to the closest conjugate field. Such an algorithm was presented in 7, which alternates between a global smoothness and orthogonality step, and a local step that projects the field on every face to the closest conjugate field (Example 304).

## Chapter 4: Polar Methods¶

### Polar Fields¶

Polar fields are represented using angles. These angles may encode the rotation from some given basis on a tangent plane (and so it is a “logarithmic” representation, when compared to Cartesian methods), or an angle difference between two neighboring tangent planes (in the sense of deviation from parallel transport). The former usually requires integer variables for directional field design. The latter does not, but state-of-the-art methods require the prescription of indices around independent dual cycles in the mesh. Currently, Directional supports the latter.

### 401 Index Prescription¶

The notion of encoding rotation angles on dual edges, as means to encode deviation from parallel transport between adjacent tangent planes, appeared in several formats in the literature 6,12. The formulation and notation we use in Directional is that of Trivial Connections 6. Trivial connection solves for a single rotation angle $\delta_{ij}$ per (dual) edge $e_{ij}$ between two faces $f_i$ and $f_j$, encoding the deviation from parallel transport between them. The algorithm first computes a spanning set of basis cycles, around all of which the sum of $\delta_{ij}$ has to be prescribed. The summation is defined as a matrix $H$. Every such cycle (row in the matrix) has an original curvature $K_0$, defined as a discrete angle defect, and the prescribed index defines an alternative curvature. The algorithm solves for the smoothest field, in the 2-norm least squares sense, as follows:

\delta = \text{argmin}\ |\delta_{ij}|^2\ s.t.\ H\delta = -K_0 + K.

$H$ is the matrix that defines the basis-cycles sum, $K_0$ is a vector of the original curvatures of every basis cycle, and $K$ is the prescribed curvatures, which result from prescribed singularity indices: for regular cycles, we prescribe $K=0$, and for a singular cycle with prescribed singularity index $\frac{i}{N}$, we set $K=\frac{2\pi i}{N}$. the sum of $K$ has to conform to the Poincaré index theorem. However, generator (handle) cycles can admit unbounded indices. See 6 for exact details. If the input obeys the sum, the result obeys the prescribed indices around the cycles everywhere. The representation is differential, where the single global degree of freedom is resolved by setting a single direction in a single arbitrary face.

Note that the correct definition for “cycle curvature” corresponds to the so-called “cycle holonomy”, only up to integer multiples of $2\pi$. However, in the discrete setting, the curvature should theoretically be computed as the exact discrete angle defect, in which for inner vertices we use $2\pi-\sum{\alpha}$, and for boundary vertices we use $\pi - \sum{\alpha}$ ($\alpha$ are the angles at the corners of a vertex). For a cycle aggregating many vertices, such as a boundary-loop cycle, we add up all the defects. That is required for exact discrete Poincaré index consistency. Note that the boundary indices define how many rotations of the vector field the boundary loop “sees”. As an example, a constant field on a simple disc in the plane has indices $0$ for all inner vertices, but the boundary index is in fact $1$—This obeys the total index sum $\chi = 2-2g-b = 2-0-1=1$ ($g$ stands for genus and $b$ for number of boundary loops)

#### Basis Cycles¶

The basis cycles form the cycles around which curvatures (and singularities) have to be prescribed on the mesh. The sum on basis cycles is described in a sparse matrix $H$ of size $|cycles|\times |E_I|$, where $E_I$ is the number of inner edges in the mesh. Each row in the matrix describes the sum over one cycle, and contains $1$ or $-1$ values depending on the (arbitrary) orientation of the dual edge participating in the cycle to the respective face. There are three types of cycles, so ordered in the rows of $H$:

1. $1$-ring dual cycles around each inner vertex, on which vertex-based singularities can be encoded (the relevant part of $H$ is basically $\left(d_0\right)^T$ in discrete exterior calculus, restricted to inner edges).
2. Cycles around mesh boundary loops.
3. Cycles around the $2g$ topological generators (independent handles).

The method directional::dual_cycles() computes the proper basis cycles and matrix $H$. To be able to intuitively prescribe singularities to inner vertices, the method also returns a conversion vector vertex2cycle, and the list of indices of inner edges from the list of edges.

The singularity indices that are prescribed contain the singularity index corresponding to each basis cycle. A value of $k \in \mathbb{Z}$ represents an $\frac{2\pi k}{N}$ rotation around the respective cycle. If the prescribed indices do not conform to the Poincaré index theorem, a result will still be computed by least squares, but it will be unpredictable. The algorithm is performed through the function directional::index_prescription(), which can also accept a solver for precomputation, for the purpose of prefactoring $H$ only once.

## Chapter 5: Seamless Integration and Meshing¶

The full details of the method implemented in this chapter can be found in a technical report 20. Many of the therotical ideas for general $N$-functions are explored in21.

$N$-Directional fields are commonly used as candidate gradients to seamless $N$-Functions, which are in turn used to generate meshes that are aligned to the original fields 2,10,15,21. Recall that combing trivializes the matching everywhere except a sparse set of seams. We augment these seams so that the mesh is cut into a topological disc. Then, we treat a combed $N$-directional $\left\{u_0,\cdots,u_{N-1}\right\}$ as a set of $N$ candidate gradients for $N$ vertex-based scalar functions $\left\{F_0,\cdots,F_{N-1}\right\}$ on the cut mesh. We then solve the Poisson problem:

F = argmin{\sum_{i=0}^{N-1}{\left|\nabla F_i - u_i\right|^2}}

Consider a seam edge $e_{ij}$ between original vertices $v_i$ and $v_j$, and between adjacent faces $f_k$ and $f_l$. The two vertices are then cut into four corners $v_{i,k},v_{j,k},v_{i,l},v_{j,l}$. Note that some corners might be identical, if the seam edge is at a singularity. Across the seam edge, we enforce the (linear) seamless conditions:

F_{i,k}= \pi_e \cdot F_{i,l} + T_e,

where $\pi_e:N \times N$ is a permutation matrix attached to the (dual) edge $e$, matching values in the integrated function $F$ as it did for the directional field $v$. and $T_e:N \times 1$ is a translational jump (also: period jump), that encodes the discontinuity in $F$ across the seam. For quick intuition, this encodes the integration of the function over a loop around the mesh beginning and ending with the seam edge: in a $4$-function, leading to a quad mesh, it is the number of quads in such a loop. If $T_e \in \mathbb{Z}^N$, then the $N$-function is fully seamless: the integer isolines of a function connect perfectly along a seam. Otherwise, it is only permutationally seamless: the gradients match, which means they are only co-oriented.

Seamless $N$-functions are denoted as such for that obeying the seamless constraints; it can be easily shown 10 that the translational jumps $T_e$ is in fact uniform across seam curves between singularities. Thus, the number of such translational jump variables is the number of seam curves in the mesh ($\times N$).

### 501 Seamless Integration¶

In Example 501 we demonstrate the computation of such a integration, both permutationally, and fully seamless. The computed function is a $4$-function with sign-symmetry, computing seamless $(U,V,-U,-V)$ functions that we demonstrate as a quad texture. The core functionality is in these lines:

directional::IntegrationData intData(N);
directional::setup_integration(VMeshWhole, FMeshWhole,  EV, EF, FE, rawField, matching, singVertices, intData, VMeshCut, FMeshCut, combedField, combedMatching);

intData.verbose=true;
intData.integralSeamless=false;

directional::integrate(VMeshWhole, FMeshWhole, FE, combedField, intData, VMeshCut, FMeshCut, cutUVRot ,cornerWholeUV);

//Extracting the UV from [U,V,-U, -V];
cutUVRot=cutUVRot.block(0,0,cutUVRot.rows(),2);

intData.integralSeamless = true;
directional::integrate(VMeshWhole, FMeshWhole, FE, combedField,  intData, VMeshCut, FMeshCut, cutUVFull,cornerWholeUV);

cutUVFull=cutUVFull.block(0,0,cutUVFull.rows(),2);


The data structure containing all relevant information about the integration is IntegrationData. It contains some parameters that can be tuned to control the integration. Several relevant ones are:

double lengthRatio;     //global scaling of functions

//Flags
bool integralSeamless;  //Whether to do full translational seamless.
bool roundSeams;        //Whether to round seams or round singularities
bool verbose;           //output the integration log.
bool localInjectivity;  //Enforce local injectivity; might result in failure!


lengthRatio encodes a global scale for the Poisson problem (scaling the fields uniformly), where the ratio is measured against the bounding box diagonal. Some of the other parameters are demonstrated in the other examples in this chapter. The integrator takes the original (whole) mesh, and generates a cut-mesh (in VMeshCut,FMeshCut) of disc-topology. The singularities are on the boundary of this mesh, and the function can consequently be defined without branching ambiguity on its vertices, with the appropriate permutation and translation across the cut seams.

### 502 Integration in various orders¶

Directional can handle integration for every $N$, including less common ones like the non-periodic $N \neq 2,3,4,6$. The properties of fields and integration in such unconventional $N$ are explored in 21.

In this example we demonstrate the results for $N=2,4,7,11$, for the same lengthRatio, and all fully seamless. Note that the density of the isolines increases with $N$, and that we round the singularity function values, leading to junctions of multiple isolines meeting. This is demonstrated in Example 502.

### 503 Rounding either seams or singularities¶

It is possible to choose whether to round the seam jumps $T_e \in \mathbb{Z}^N$ directly, or the function values around singularities (and of topological handles, in case of non-simply-connected topology). In both cases the seams will have integer values, but the latter case is more restrictive and will result in multiple isolines meeting at every singularity. For quad meshes with $N=4$, for instance, this is the difference between pure-quad results (round singularities), or just quad-dominant (round seams).

### 504 Linear Reductions¶

It is possible to constrain the functions to have linear relations between them, which reduce the degrees of freedom. This is done by inputting a matrix $U: N \times n$ so that $n \leq N$, and where $F = U\cdot f$ for the independent degrees of freedom encoded in $f$. This relationship should be mirrored in the integrated directional field. Warning: not every linear reduction is suitable for integration on surfaces! it needs to commute with the permutation around singularities. Feeding an incompatible linear reduction might then result in a failure of the algorithm. One popular example is triangular symmetry where $U_{3 \times 2} = [1, 0; 0, 1; -1, -1]$. Another is sign symmetry where $U = [Id_{n \times n}; -Id_{n \times n}]$, and $n = \frac{N}{2}$. The latter is always assumed when $N$ is even, and both are always valid in any singularity configuration. Symmetries can also be combined. $U$ is fed into intData through the field linRed, and there are pre-made funtcions to set it, such as set_triangular_symmetry(int N).

### 505 Meshing¶

This subchapter demonstrates how we can create an actual polygonal mesh from the arrangement of isolines on the surface. This creates an arrangement of lines (in exact numbers) on every triangle, and stitches them together across triangles to get a complete conforming mesh. The new mesh is given in libhedra format23 of $(V,D,F)$, where $D$ is a vector $|F| \times 1$ of face degrees, and $F$ is a $|F| \times max(D)$ matrix of indices into $V$.

The meshing unit is independent from the integration unit, and can be potentially used with external functions; one should fill the MeshFunctionIsolinesData structure with the input, and call mesh_function_isolines().

The input is given as functions on the vertices of the whole (original) mesh in vertexNFunction, where the user must supply two sparse matrices: orig2CutMat to produce values per corner of the cut mesh, and the equivalent exactOrig2CutMatInteger to do the same in exact numbers. This ensures that the values across seams are perfectly matching in rationals, and the robustness of the meshing.

There is a natural transition between the integrator and the mesher, which is done by calling setup_mesh_function_isolines(), which translates the IntegratorData to the meshing data.

The meshing function requires CGAL as a dependency, which is operated through the libigl CGAL dependency mechanism.

## Chapter 6: Subdivision Fields¶

### 601 Subdivision Directional Fields¶

Directional fields can be used with subdivision surfaces in a manner which is structure preserving. That is, the subdivision of a coarse directional field into a fine directional field subdivides a coarse gradient into a fine gradient, and its coarse curl into fine curl. The challenge of doing it for piecewise-constant fields is worked by 19, which we demonstrate in Example 601. We optimize for a curl free field rawFieldCoarse, subdivide it into rawFieldFine using subdivide_field(), and compute a seamless parameterization for both. The coarse field is optimized for being curl-free fast (using the PolyCurl optimization; see Example 303), and then the fine field is curl free by design, with the same singularities, which makes for an efficient process.

## Outlook for continuing development¶

Directional is a budding project, and there are many algorithms in the state-of-the-art that we look forward to implement, with the help of volunteer researchers and practitioners from the field. Prominent examples of desired implementations are:

1. Support for 3D fields, particularly Octahedral fields 16, both in tet meshes and with the boundary-element method.

2. A discrete exterior calculus framework.

3. Differential operators and Hodge decomposition.

4. Support for tensor fields.

5. Vertex-based representations.

6. Advanced and better visualization techniques.

## References¶

1. Omri Azencot, Etienne Corman, Mirela Ben-Chen, Maks Ovsjanikov, Consistent Functional Cross Field Design for Mesh Quadrangulation, 2017.

2. David Bommes, Henrik Zimmer, Leif Kobbelt, Mixed-integer quadrangulation, 2009.

3. David Bommes, Henrik Zimmer, Leif Kobbelt, Practical Mixed-Integer Optimization for Geometry Processing, 2012.

4. Sofien Bouaziz, Mario Deuss, Yuliy Schwartzburg, Thibaut Weise, Mark Pauly, Shape-Up: Shaping Discrete Geometry with Projections, 2012.

5. Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, and Klaus Hildebrandt, Modeling n-Symmetry Vector Fields using Higher-Order Energies, 2018.

6. Keenan Crane, Mathieu Desbrun, Peter Schröder, Trivial Connections on Discrete Surfaces, 2010.

7. Olga Diamanti, Amir Vaxman, Daniele Panozzo, Olga Sorkine-Hornung, Designing N-PolyVector Fields with Complex Polynomials, 2014.

8. Olga Diamanti, Amir Vaxman, Daniele Panozzo, Olga Sorkine-Hornung, Integrable PolyVector Fields, 2015.

9. Fernando de Goes, Mathieu Desbrun, Yiying Tong, Vector Field Processing on Triangle Meshes, 2016.

10. Felix Kälberer, Matthias Nieser, Konrad Polthier, QuadCover - Surface Parameterization using Branched Coverings, 2007

11. Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder, Globally Optimal Direction Fields, 2013.

12. Nicolas Ray, Bruno Vallet, Wan Chiu Li, Bruno Lévy, N-Symmetry Direction Field Design, 2008.

13. Yang Liu, Weiwei Xu, Jun Wang, Lifeng Zhu, Baining Guo, Falai Chen, Guoping Wang, General Planar Quadrilateral Mesh Design Using Conjugate Direction Field, 2008.

14. Merel Meekes, Amir Vaxman, Unconventional Patterns on Surfaces, 2021.

15. Ashish Myles, Nico Pietroni, Denis Zorin, Robust Field-aligned Global Parametrization, 2014.

16. Justin Solomon, Amir Vaxman, David Bommes, Boundary Element Octahedral Fields in Volumes, 2017.

17. Daniele Panozzo, Enrico Puppo, Marco Tarini, Olga Sorkine-Hornung, Frame Fields: Anisotropic and Non-Orthogonal Cross Fields, 2014.

18. Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, Mirela Ben-Chen, Directional Field Synthesis, Design, and Processing, 2016.

19. Bram Custers, Amir Vaxman, Subdivision Directional Fields, 2020.

20. Amir Vaxman Directional Technical Reports: Seamless Integration, 2021.

21. Merel Meekes, Amir Vaxman, [Unconventional Patterns on Surfaces] (https://webspace.science.uu.nl/~vaxma001/Unconventional_Patterns_on_Surfaces.pdf), 2021.

22. Paolo Cignoni, Marco Callieri, Massimiliano Corsini, Matteo Dellepiane, Fabio Ganovelli, Guido Ranzuglia, , MeshLab: an Open-Source Mesh Processing Tool

23. Amir Vaxman and others, libhedra: geometric processing and optimization of polygonal meshes