Basis of discrete calculus (DEC: discrete exterior calculus)

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"If people don't believe in the simplicity of mathematics, they just don't realize the complexity of life ." -- Johnvon norann

Dec mainly discusses extra points in discrete situations, which are used in the computer field. We know that it is impossible to use a computer to process geometric images completely smooth (the computer is a discretization world consisting of 0 and 1 ), the concept of Dec also provides us with a better tool to portray discrete ry. For example, in the commonly used "Finite Element Method" in geometric analysis, the method based on Dec can use a surface without uniform, which is more convenient and simple.

Exterior Algebra) 

First, let's talk about exterior calculus, which is also called an external algebra.

External algebra is a class of algebra about exterior product, which is also called wedge product because its operator number is wedge. Traditional linear algebra involves algebra addition and scalar multiplication, but it is not enough to meet actual requirements. Therefore, it defines extra points and inner points with special characteristics.

The outer product has a unique algorithm, for example, in matrix multiplication:

A = [A1, A2, A3, A4] ^ t

B = [B1, B2, B3]

C is the outer product of A and B, then

C = [b1a1, b2a1, b3a1]

[B1a2, b2a2, b3a2]

[B1a3, b2a3, b3a3]

[B1a4, b2a4, b3a4]

The points have other characteristics and properties. You can refer to the relevant textbooks. External points have many different purposes in different situations, so they become a system called external algebra.

Although there are external algebra, there is no inner algebra and only inner product. They are often defined as follows:

A = [A1, A2, A3,] ^ t

B = [B1, B2, B3]

C is the inner product of A and B, then

C = A1 * B1 + A2 * B2 + A3 * B3

Differential Form) 

Dec mainly discusses the external integral in discrete conditions. Its Differential Form is as follows:

0 form: indicates that the integral is in the scalar field (scalar field.

1 form: indicates that the integral is in a one-dimensional field, such as a vector field ).

2: Same as 1.

3 form: The same as 0 form.

Newton-leibnitzformula)

Also called first fundamental theorem, it is a common method for solving points. In the form of 1:

INT (A, B) (DF) = int (A, B) (f (x) dx) = F (B)-f ()

Lex and chain

First, explain the Convex Shell (convex hull, also called convex envelope ). Given many points, the Convex Shell is a set in the vector space, which contains all points and is the smallest (minimal ).

Simplex is the basic unit of DEC and has the following types:

0-simplex: Point.

1-simplex: edges, or convex shells with two different points.

2-simplex: triangle, or Convex Shell with three different points.

3-simplex: a triangle or a Convex Shell with four different points.

Here we have an interesting discovery that a n-simplex must have n + 1 points. In addition, an N-simplex contains smaller simplex. For example, a triangle is a 2-simplex with three vertices in total. If two vertices are taken to form a new 1-simplex, it is also called a 1-face (1-face) of 2-simplex ). Obviously, a triangle has three 0-face, three 1-face, and one 2-face (itself ).

So when we consider a certain N-simplex, we need to think that it is actually a lot of smaller simplex.

A n-simplex has many (n-1)-faces. All (n-1)-faces can form a (n-1)-chain. All (n-1)-faces can also be called the n-simplex (n-1)-face.

Simplicial Complex) 

Simplicialcomplex is a collection of many simplex instances. The dimension with the highest dimension in this set is the dimension of Simplicial Complex. For example, a two-dimensional simplicialcomplex must have at least one 2-simplex, that is, a triangle, but there cannot be any higher-dimensional simplex, such as a triangle.

If K is a Simplicial Complex, it must meet the following conditions:

1. All the faces of Simplex in this set must also be in K. In other words, no broken Simplex;

2. If any two simplex interfaces exist in the Set, the intersection must be a public plane rather than anything else. That is to say, if two triangles exist in a two-dimensional Simplicial Complex, the two triangles will not be connected. If they are connected, they can only be connected from the vertex to the vertex, one vertex cannot be placed directly on another edge.

Therefore, simplicial complex is a fully geometric and reasonably spliced graph. It helps us to directly analyze its relationship between points, edges, and surfaces and restore its topological structure. As long as we find an analytical method, any complex object that can be simplified to a simple complex can be used.

Chain is actually a kind of directionality Simplicial Complex.

Discrete Manifold) 

It is also called simplicial manifold. A n-dimensional simplicial manifold is an n-dimensional simplicial complex. In addition, the following conditions are met:

For all n-dimensional simplex, if they are bounded, they must form an n-dimensional "ball" or "hemisphere" (Half Ball) if it is a boundary point. In this case, each (n-1)-simplex can be associated with a maximum of two n-simplex instances. For example, a triangle is a two-dimensional simplicial manifold. If only one triangle frame is left when the middle of the triangle is dug, manifold is no longer, because each isolated point cannot be full of football or hemisphere definitions.

Such a mesh is also called the manifold mesh. Through the above introduction, we can intuitively understand that this mesh (usually three-dimensional) has no isolated points, lines, and surfaces, and each line is only one (when the boundary is used) or two sides (non-boundary) border. At the same time, it also has all the features of Simplicial Complex (no damage, public surface restriction ). Very pure and neat.

Homology) 

Homology is a basic topology concept. When talking about this concept, it must be set to simplicial complex. Homology, a tool used to discuss graphical features (such as similarity.

Let's first look at an example. Given three figures: Circle, square, and ring, we think the circle is similar to the square, while the ring is special, because the circle and square can be converted by deformation, the ring is not possible.

If two Simplicial Complex have similar structures, they have the same number of "holes" (hole) internally, which are called homology ). Homology provides an important method for studying invariants.

Boundary) 

Each N-simplex has a direction. Its direction is defined as follows: n + 1 (n-1)-simplex (or (n-1)-faces) of N-simplex) there are two sort orders, one of which is its direction.

For example, a triangle is a 2-simplex triangle with three sides (1-simplex). These three sides can form a triangle clockwise or counterclockwise.

Generally, for the convenience of observing an N-simplex, it is assumed that it has only one direction (usually the clockwise direction). All content in this series will follow this rule.

Boundary refers to any face in a n-simplex (n-1)-faces. If the boundary direction is the same as the specified direction, it is recorded as the boundary of "+" and expressed as "+ 1"; otherwise, it is the boundary, it is represented by "-1.

Boundary Operators 

Define the operator P as the boundary operator. When this operator acts on an N-simplex, the result is all its forward boundary.

For example, a triangle (a, B, c) has a boundary relationship: A-> B-> C->. Then P (A, B, C) = (a, B) + (B, c) + (C, ). If it is a-> B-> C & A-C, P (A, B, C) = (a, B) + (B, c)-(C, ). Readers can draw pictures on their own.

It is worth noting that if n-simplex is an empty set, that is, 0-simplex, P (0-simplex) = (-1)-simplex.

As mentioned above, a set of simplex is a chain, and the boundary operator actually converts a simplex into a chain. If the boundary operation is performed on the chain, every simplex operation on the chain is taken and then added up. Interestingly, if "+ 1" and "-1" are used to represent the boundary, P (chain) must be equal to zero!

Because the boundary operator maps N-simplex to (n-1)-simplex space, it can be seen as a (N-1) * n matrix. This is a sparse matrix, because the boundary is only related to the bounded part. In the + 1-1 notation, this matrix contains only 0, + 1, and-1 elements.

Basis of discrete calculus (DEC: discrete exterior calculus)