can be abstracted from concrete data structures using programming techniques such as generic programming [79]. However, generic algorithms still need to make certain assumptions about the data they operate on. The question remains what these concepts are that describe “data”: what properties should be expected by some algorithm from any kind of data provided for scientific visualization? Moreover, consistency among concepts shared by independent algorithms is also required to achieve interoperability among algorithms and eventually (independently developed) applications. While any particular problem can be addressed by some particular solution, a common concept allows to build a framework instead of just a collection of tools. Tools are what an end user needs to solve a particular problem with a known solution. However, when a problem is not yet clearly defined and a solution unknown, then a framework is required that allows exploration of various approaches and eventually adaption toward a specific direction that does not exist a priori.

### Differential Geometry: Manifolds, Tangential Spaces, and Vector Spaces

_{P }(M) at each point P which has vector space properties.

#### Tangential Vectors

_{q(s)}(M) on the n-dimensional manifold M at each point q(s) ∈ M:

^{ μ }are the components of the tangential vector in the chart {x

^{μ }} and {∂

_{μ }} are the basis vectors of the tangential space in this chart. In the following text the Einstein sum convention is used, which assumes implicit summation over indices occurring on the same side of an equation. Often tangential vectors are used synonymous with the term “vectors” in computer graphics when a direction vector from point A to point B is meant. A tangential vector on an n-dimensional manifold is represented by n numbers in a chart.

#### Covectors

^{0}, ∂

_{0}> = 1 must be sustained under coordinate transformation, dx

^{0}must shrink by the same amount as ∂

_{0}grows when another coordinate scale is used to represent these vectors. In higher dimensions this is expressed by an inverse transformation matrix.

_{z }and as covector field dz. (a) Vector field ∂

_{z }. (b) Duality relationship among ∂

_{z }and dz. (c) Co-vector field dz

_{r }. (b) Azimuthal field dϕ ∂

_{ϕ }view of the equatorial plane (z-axis towards eye). (c) Altitudal field dθ ∂

_{θ }slice along the z-axis

#### Tensors

_{n }

^{m }of rank n × m is a multi-linear map of n vectors and m covectors to a scalar

^{n+m }numbers for a k-dimensional manifold. Tensors of rank 2 may be represented using matrix notation. Tensors of type T

_{1}

^{0}are equivalent to covectors and called co-variant; in matrix notation (relative to a chart) they correspond to rows. Tensors of type T

_{0}

^{1}are equivalent to a tangential vector and are called contra-variant, corresponding to columns in matrix notation. The duality relationship between vectors and covectors then corresponds to the matrix multiplication of a 1 × n row with a n × 1 column, yielding a single number

_{2}

^{0}with , as they can be used to define a metric or inner product on the tangential vectors. Its inverse, defined by operating on the covectors, is called the co-metric. A metric, same as the co-metric, is represented as a symmetric n × n matrix in a chart for an n-dimensional manifold.

#### Exterior Product

^{n }(V ).

^{k }(V ) are called k-vectors, whereby two-vectors are also called bi-vectors and three-vectors tri-vectors. The number of components of a k-vector of an n-dimensional vector space is given by the binomial coefficient {n}{k}. For n = 2 there are two one-vectors and one bi-vector, for n = 3 there are three one-vectors, three bi-vectors, and one tri-vector. These relationships are depicted by the Pascal’s triangle, with the row representing the dimensionality of the underlying base space and the column the vector type:

#### Visualizing Exterior Products

^{μ }→ −x

^{μ }, they flip sign, whereas a “true” scalar does not. An example known from Euclidean vector algebra is the allegedly scalar value constructed from the dot and cross product of three vectors which is the negative of when its arguments are flipped:

_{μ }– any volume must be a scalar multiple of this basis volume element but can flip sign if another convention on the basis vectors is used. This convention depends on the choice of a right-handed versus left-handed coordinate system and is expressed by the orientation tensor . In computer graphics, both left-handed and right-handed coordinate systems occur, which may lead to lots of confusions.

_{0}the electrostatic, A

_{k }the magnetic vector potential), take the form

#### Geometric Algebra

^{2}= g(v, v). It can be shown that the sum of the outer product and the inner product fulfill these requirements; this defines the geometric product as the sum of both:

^{2}= 4. A multivector V, constructed from tangential vectors on a two-dimensional manifold, is written as

^{μ }the four components of the multivector in a chart. For a three-dimensional manifold, a multivector on its tangential space has 2

^{3}= 8 components and is written as

^{μ }the eight components of the multivector in a chart. The components of a multivector have a direct visual interpretation, which is one of the key features of Geometric Algebra. In 3D, a multivector is the sum of a scalar value, three directions, three planes, and one volume. These basis elements span the entire space of multivectors. Geometric Algebra provides intrinsic graphical insight to the algebraic operations. Its application for computer graphics will be discussed in Sect. 4.

#### Vector and Fiber Bundles

_{1}, which is a function that maps each element of a product space to the element of the first space:

_{b }⊂ B of each point b ∈ B is homeomorphic to U

_{b }× F such that the projection pr

_{1}of U

_{b }× F is U

_{b }again:

_{p }(M) on a manifold M together with the manifold . Every differentiable manifold possesses a tangent bundle . The dimension of is twice the dimension of the underlying manifold M, its elements are points plus tangential vectors. T

_{p }(M) is the fiber of the tangent bundle over the point p.

_{1}). In scientific visualization, usually only trivial bundles occur. A well-known example for a nontrivial fiber bundle is the Möbius strip.

### Topology: Discretized Manifolds

_{1}, c

_{2}are incident if c

_{1}⊆ ∂ c

_{2}, where ∂ c

_{2}denotes the border of the cell c

_{2}. Two cells of the same dimension can never be incident because dim(c

_{1}) ≠ dim(c

_{2}) for two incident cells c

_{1}, c

_{2}. c

_{1}is a side of c

_{2}if dim(c

_{1}) < dim(c

_{2}), which may be written as c

_{1}< c

_{2}. The special case may be denoted by c

_{1}≺ c

_{2}. Two k -cells c

_{1}, c

_{2}with k > 0 are called adjacent if they have a common side, i.e.,

_{1}, v

_{2}are said to be adjacent if there exists a one-cell (edge) e which contains both, i.e., v

_{1}< e and v

_{2}< e. Incidence relationships form an incidence graph. A path within an incidence graph is a cell tuple: a cell-tuple within an n-dimensional Hausdorff space is an ordered sequence of k-cells of decreasing dimensions such that . These relationships allow to determine topological neighborhoods: adjacent cells are called neighbors. The set of all k + 1 cells which are incident to a k-cell forms a neighborhood of the k-cell. The cells of a Hausdorff space X constitute a topological base, leading to the following definition: a (“closure-finite, weak-topology”) CW-complex , also called a decomposition of a Hausdorff space X, is a hierarchical system of spaces , constructed by pairwise disjoint open cells c ⊂ X with the Hausdorff topology , such that X

^{(n)}is obtained from X

^{(n−1)}by attaching adjacent n-cells to each (n − 1)-cell and . The respective subspaces X

^{(n)}are called the n-skeletons of X. A CW complex can be understood as a set of cells which are glued together at their subcells. It generalizes the concept of a graph by adding cells of dimension greater than 1.

### Ontological Scheme and Seven-Level Hierarchy

^{ n }neighbors.

Hierarchy object |
Identifier type |
Identifier semantic |
---|---|---|

Bundle |
Floating point number |
Time value |

Slice |
String |
Grid name |

Grid |
Integer set |
Topological properties |

Skeleton |
Reference |
Relationship map |

Representation |
String |
Field name |

Field |
Multidimensional index |
Array index |

#### Field Properties

#### Topological Skeletons

#### Non-topological Representations

## 3 Differential Forms and Topology

### Differential Forms

_{P }(M), i.e., . They are commonly called co-variant vectors, covectors (see section “Tangential Vectors”), or Pfaff-forms. The set of one-forms generates the dual vector space or cotangential space T

_{p }

^{∗}(M). It is important to highlight that the tangent vectors v ∈ T

_{P }(M) are not contained in the manifold itself, so the differential forms also generate an additional space over P ∈ M. In the following, these one-forms are generalized to (alternating) differential forms.

^{n }and T

^{∗m }represent the n and m powered Cartesian product of the tangential space or the dual vector space (cotangential space). A tensor γ is called an (n, m)-tensor which assigns a scalar value to a set of m covectors and n vectors. All tensors of a fixed type (n, m) generate a tensor space attached at the point P ∈ M. The union of all tensor spaces at the points P ∈ M is called a tensor bundle. The tangential and cotangential bundles are specialized cases for (1, 0) and (0, 1) tensor bundles, respectively. Fully antisymmetric tensors of type (0, m) may be identified with differential forms of degree m. For m > dim(M), where dim(M) represents the dimension of the manifold, differential forms vanish.

_{p }representing an integration domain, e.g., an interval x

_{1}and x

_{2}, results in the same value f(x

_{2}) − f(x

_{1}). In the general case, a p-form is not always the exterior derivative of a p-one-form; therefore, the integration of p-forms is not independent of the integration domain. An example is given by the exterior derivative of a p-form β resulting in a p + 1-form γ = d β. The structure of such a generated differential form can be depicted by a tube-like structure such as in Fig. 7. While the wedge product of an r-form and an s-form results in an r + s-form, this resulting form is not necessarily representable as a derivative. Figure 7 depicts a two-form which is not constructed by the exterior derivative but instead by , where α and β are one-forms. In the general case, a p-form attached on an n-dimensional manifold M is represented by using (n − p)-dimensional surfaces.

_{p }(M).

#### Chains

0: if the cell is not in the complex

1: if the unchanged cell is in the complex

− 1: if the orientation is changed

_{p }on a cell complex K. Let us denote the ith p-cell as , whereby τ

_{p }

^{i }∈ K. The boundary operator ∂

_{p }defines a (p − 1)-chain computed from a p-chain: . The boundary of a cell τ

_{p }

^{j }can be written as alternating sum over elements of dimension p − 1:

_{i }is deleted from the sequence. This map is compatible with the additive and the external multiplicative structure of chains and builds a linear transformation:

_{∗}= {C

_{p }, ∂

_{p }} such that the complex property