Context-Free Grammar及形状规则集

1. Context-Free Grammar

Definition − A context-free grammar (CFG) consisting of a finite set of grammar rules is a quadruple (N, T, P, S) where

N is a set of non-terminal symbols.
T is a set of terminals where N ∩ T = NULL.
P is a set of rules, P: N → (N ∪ T)*, i.e., the left-hand side of the production rule P does have any right context or left context.
S is the start symbol.

Example

The grammar ({A}, {a, b, c}, P, A), P : A → aA, A → abc.
The grammar ({S, a, b}, {a, b}, P, S), P: S → aSa, S → bSb, S → ε
The grammar ({S, F}, {0, 1}, P, S), P: S → 00S | 11F, F → 00F | ε

由上述语法可以构建对应的语法树，即Derivation Tree

2. Generation of Derivation Tree

A derivation tree or parse tree is an ordered rooted tree that graphically represents the semantic information a string derived from a context-free grammar.

Example

Let a CFG {N,T,P,S} be

N = {S}, T = {a, b}, Starting symbol = S, P = S → SS | aSb | ε

One derivation from the above CFG is “abaabb”

S → SS → aSbS → abS → abaSb → abaaSbb → abaabb

3. CGA Shape Grammar

Notation: Production rules are defined in the following form:

id: predecessor : cond -> successor : prob

id is a unique identifier for the rule;
predecessor ∈ V is a symbol identifying a shape that is to be replaced with successor;
cond is a guard (logical expression) that has to evaluate to true in order for the rule to be applied.
The rule is selected with probability prob.

3.1 Scope rules:

平移向量T: T(tx, ty, tz) is a translation vector that is added to the scope position P;

旋转向量R: Rx(angle), Ry(angle), and Rz(angle) rotate the respective axis of the coordinate system;

尺寸向量S: S(sx,sy,sz) sets the size of the scope;

类别I: I(ob jId). Typical objects include a cube, a quad, and a cylinder, but any threedimensional model can be used.

3.2 Basic split rule:

规则1：对fac立面在Y轴上进行分割，分割尺寸为3.5m, 0.3m,3m,3m,3m，形状类别为floor,ledge, floor, floor，floor。也可以是multiple axis (”XY”, ”XZ”,”YZ”, or ”XYZ”), nested splits, or nested combinations of splits and L-system rules。

3.3 Scaling of rules:

上图figure4 右图给出了两种尺寸设计（with x-length 12 and 10）：

S（12，3）和 S（10，3）

再使用3.2的分割规则，将其分为BAAB shape, 分割方向为X轴，长度为，2， lr, lr， 2.

3.4 Repeat: tile a specified element.

对floor进行平铺填充，填充物为B形状，填充方向为X轴方向，填充个数为： repetitions = [Scope.sx/2]

所以2是填充间隔？

原文：“The floor will be tiled into as many elements of type B along the x-axis of the scope as there is space. The number of repetitions is computed as repetitions = ⌈Scope.sx/2⌉ and we adjust the actual size of the element accordingly.”

3.5 Component split:

在type物体上构建诸如A,B...Z等component。

type identifies the type of the component split with associated parameters param (if any).

For example:

Comp(”faces”){A} means creating a shape with symbol A for each face of the original three-dimensional shape.
Comp(”edge”,3){A} means creating a shape A aligned with the third edge of the model.

4. Mass Modeling

如何使用上述规则生产模型？

4.1 初级生产水平：以一个box为起始或基础，使用平移规则、尺寸规则、分割规则等生成一些简单的模型，比如basic building blocks L, H, U and T。

4.2 对简单物体加入一些旋转操作。

The next level of difficulty is to use arbitrary rotations of shapes and to include a cylinder in the shape vocabulary.

4.3 一些屋顶类型

Further, it becomes necessary to include basic roof shapes, such as the examples depicted in figure 7 and more general shapes, such as general L-shapes and extruded general quadrilaterals.

4.4 Problem of complex surfaces:

4.5 Modeling strategy:

1） use three dimensional scopes to place three-dimensional shapes (volumes) to form a mass model.

2）generate two-dimensional scopes aligned with facade surfaces and roof surfaces by extracting the faces of the three-dimensional shapes with a component split.

3) The grammar can then proceed to refine the resulting quads and triangles.

参考：

https://www.tutorialspoint.com/automata_theory/context_free_grammar_introduction.htm

Müller P, Wonka P, Haegler S, et al. Procedural modeling of buildings[M]//ACM SIGGRAPH 2006 Papers. 2006: 614-623.