2.1.1. Original Definition

One curve is typically represented by a function. A definition must be provided to study the problem of the curve similarity. In geometry, there is a strict definition for shape similarity, which is an accurate similarity. In engineering applications, due to the large number of error factors, the definition of fuzzy curve similarity is adopted. Taking a 2D plane curve as an example, a kind of curve similarity is defined as follows:

**Definition 1** ([32])**.** *Given functions f* <sup>1</sup>*(x) and f* <sup>2</sup>*(x),d*(*f*1, *<sup>f</sup>*2) = *<sup>C</sup>*<sup>2</sup> *C*1 *<sup>f</sup>*1(*x*) <sup>−</sup> *<sup>f</sup>*2(*x*) *dx is the distance between two functions, and also known as the function similarity distance or the curve similarity distance, where [C1, C2] is the function definition domain or the definition domain*.

**Definition 2** ([32])**.** *For a given threshold* ε*, if d(f1,f2)* < ε*, then f1(x) and f2(x) are similar, otherwise they are not.*

As mentioned above in the definitions of curve similarity distance (CSD) and curve similarity, the two functions have the same definition domains, that is to say, two curves must be aligned first, which is very limited in practical application. In most cases, the definition domains of two functions are different, and it is necessary to perform a truncation, translation, stretching, or even rotation transformation on a function to calculate the similarity distance.

As shown in Figure 1, given a curve *L*, the curves *L*<sup>1</sup> to *L*<sup>3</sup> are new curves obtained after applying different transformations such as translation or stretching. If the above calculation method is adopted, the distances between *L*<sup>1</sup> to *L*<sup>3</sup> and *L* are different. Similar problems are available in the calculation of DTW and Fréchet distances of the curves.

**Figure 1.** Translation and stretching transformation of curves.

From the perspective of transformation, the above several curves are similar, and the distance is 0. Therefore, when to measure the similarity between curves, the transformation of the curve should be taken into consideration. Firstly, a curve similarity distance definition based on curve transformation may be suggested.
