**3.6 Ruler-dimension method**

A standard method used to estimate the length of a curve in a plane is counting how many lengths of a ruler are used to move from the start to the end of the curve. Since the ruler has a finite length, the details of the curve are smaller than the ruler with which we obtain the jumps, and therefore, the length we measure is generally lower than the real length. We intuitively believe our rulers are shorter, but our estimation of the actual length is more exact and increases.

The estimation of the length on the right is closer to reality since successive rulers are closer. The estimation improves each time the length of the ruler decreases.

The ruler or compass dimension of a curve is calculated between the ratio of a ruler with length *L* and the number of rulers required to measure the length of a curve, given as *N*(*L*)=c *L*D, where *c* is a constant. Taking the logarithms of both sides, we have: log(*N*(*L*)) = log(*c*) *D* log(*L*). Therefore, if we graph log(*N*(*L*)) *vs* log(*L*) within a range valued as *L*, the slope will be an estimate of the fractal dimension *D*, see [14, 15]. It must also be said that after making the corresponding scaling of *Status 0* of the fracture chosen up to *Status 3*, the necessary calculations were made through the methods mentioned above. The fractal dimension, thus obtained, is 1.01606, **Table 1**.

**Table 1** shows that there is a great similarity among three of the methods regarding the results: box dimension, information dimension, and mass dimension; because the fracture is proportional as it expands along the slab. We may observe that the fractal dimension is between 1 and 2. In addition to this, from the values in **Figure 6**, it was


#### **Table 2.**

*Results of calculation of the slope of fracture radius dot increase.*

**Figure 5.** *Increase in the slope of straight line and equation of the slope.*

**Figure 6.** *(a) Fracture chosen and (b) its vectorization.*

possible to obtain the equation of the pending line of the increments that the fracture had, **Table 2**. The values obtained are represented in **Figure 5**.

## **4. Intersection of fractal curves**

Let us now focus on a phenomenon commonly present in this type of tests or cases: the intersection of fractal curves, as shown in **Figure 2**. Since the intersection of fractals *F* and *G* is often a fractal, it is natural to relate the dimension of such an intersection with one of the original sets, but, in general, it may not be said that A is true. This is because if *F* is limited, there may exist a copy *F*<sup>1</sup> of *F* such that dim (*F* ∩ *F*1) – dim-*F* exists, but in the copy, if the sets are disjoint, dim (*F* ∩ *F*1) = Φ. In addition, if *F* and *F*<sup>1</sup> are collinear, then *F* ∩ *F*<sup>1</sup> is a line segment. However, if they intersect at a given angle (as in **Figure 2**), such an intersection is a singular or sole point (also shown in **Figure 2**). The latter case is the most frequent. If the transformation σ of the plane is congruent; that is, a rigid movement, it transforms any set *F* into a congruent σ(*F*) without reflection. Rigid movements may be parameterized through coordinates (*x*, *y*, θ), with θ being the rotation angle.

The objective of the study of fractals is to know the dimension of fractal curves intercepted. Since the ways to obtain the fractal dimension are varied, we will use the Hausdorff-Besicovitch dimension (see below), which is based on considering an open cover of *n*-spheres of the fractal set, that is, for a fractal contained in the Euclidean plane open circles are considered (in mathematics we talk about balls, but everything depends on the dimension in which such exist). Of the possible coatings, the minimum formed by balls with a smaller diameter equal to a given size ε, Eq. (9) is considered. Once such a minimum is calculated, its limit is considered when ε ! 0, therefore formally defining Hausdorff content as:

$$\mathcal{H}\_{\delta}^{\epsilon}(F) = \min \left\{ \sum\_{i=1}^{\infty} |U\_i|^{\epsilon} \right\} \exists \mathcal{H}^{\epsilon}(F) = \lim\_{\delta \to 0} \mathcal{H}\_{\delta}^{\epsilon}(F) \tag{7}$$

where *Ui* j j = diam (*Ui*Þ<*δ*, complying with the fact that Hausdorff content defines a function of the set power R*<sup>n</sup>* in non-negative real numbers, Eq. (10).

$$\mathcal{H}': P(\mathfrak{R}'') \to \{0\} \cup \mathfrak{R}^+ \cap \{\infty\} \tag{8}$$

For any set, the function defined in Eq. (10) has the property to be null for any s > s0 and infinite for s < s0, and for s = s0 the Hausdorff-Besicovitch dimension is obtained, shown below *dimHB*.

Given *F* and *G* are two fractal curves to which the dimension mentioned above has been applied, then, in general, it is true that Eq. (11):

$$\dim\left(G\cap\sigma(F)\right)\le\max\left\{0,\dim G+\dim F-n\right\}\tag{9}$$

And it is also true that Eq. (12):

$$
\dim(G \cap \sigma(F)) \ge \dim G + \dim F - n \tag{10}
$$

Therefore, it is possible to obtain higher limits for dimð Þ *G*∩*σ*ð Þ *F* with a given translation or similarity group. Such limits are kept for large groups of translations and similarities.

Based on **Figure 7**, and as was mentioned in the first paragraph of this section, parameterization allows a natural measure in the space where the rigid movement is made with the measure of a rigid movement set given in space, for instance, tridimensional, and if we assign the origin to a certain point in the rectangle [1, 2]\*[0, 2] we obtain measure 1\*2\*2π.

Algebraically, in R*<sup>n</sup>* where two smooth variables *G* and *F* are completely intersected, in general, they intersect in a sub-variety of dimension *max* f g 0, *dim* � *G* þ *dim* � *F* � *n* . If dim*G* þ *dim F* � *n*>0, therefore Eq. (13):

$$\dim(G \cap \sigma(F)) = \dim - G + \dim - F - n \tag{11}$$

For a rigid movement set with positive measure, but it is 0 for everything else.

The study of fractal intersections moving between them is known as integral fractal geometry. In the optics of classical geometry, see [16], they may be

**Figure 7.** *Intersection of fractals.*

consulted, and the ones who have contributed the most to the fractal intersection are seen [17–20].

Although the vectorization of the fractal image was chosen here, there is an article that proposes a method of rapid fractal coding based on the fractal dimension. The fractal dimension in fractal theory can be used to describe the texture of the image. The larger the fractal dimension, the rougher the surface of the corresponding graph, and vice versa. The method is based on the differential box count that is chosen specifically for texture analysis. Since the search space is reduced and the sorting operation is simple and computationally efficient, improving the encoding speed and preserving the quality of the decoded image; see [21].
