1. Introduction
The curve is the basic design element used to obtain shapes and silhouettes of industrial products. This is why designers try to make it aesthetic in order to improve the quality of the shape design. The aesthetic aspect of a curve is one of the most studied topics [
1,
2]. Farin considers that one of the main aspects that must be used in the evaluation of an aesthetic curve is its curvature distribution [
3,
4]. If the changes of the curvature are constant, i.e., the second derivative of the curve is monotonic, then the curve is beautiful [
4,
5,
6]. Among them, a family of curves called the log-aesthetic curves has been deeply investigated with respect to the relation between designers′ visual languages and mathematical properties; see for example [
7]. The Logarithmic Distribution Diagram of Curvature (LDDC) was used to analyze the characteristics of planar curves with monotonic curvature by Harada et al. in [
8]. LDDC describes the relation between the length frequencies of a segmented curve with respect to its radius of curvature plotted in log-log coordinate system. The aesthetic value of a curve for automobile design can be identified based on these graphs [
6,
8]. In 2003, Kanaya et al. [
9] proposed the generation of Logarithmic Curvature Histogram (LCH) to substitute LDDC; see also [
10,
11]. Gobithaasan et al. [
8] proposed the formula of the Logarithmic Curvature Graph’s (LCG) gradient to define an aesthetic curve: a curve with a constant or a linear LCG gradient is called aesthetic curve. Moreover, he obtained a classification of aesthetic curves taking into consideration the sign of the gradient. In [
12], the authors obtain a relation for the curvature if the gradient of the curve is linear. New aesthetic standards based on the Golden Ratio and its generalization have been proposed in [
1,
2].
Several research studies have focused on determining the aesthetics of an object and on identifying how certain object features evoke feelings during their perception. Despite the large number of definitions and approaches, there is not a generally valid model for determining the aesthetic value of an object because “the visual perceptions are accompanied by a certain intuitive feeling of value” called aesthetic feeling [
13,
14]. Since people’s perceptions of aesthetics are not necessarily objective, we propose two new notions in this paper, namely
approximate aesthetic curve and
approximate neutral curves, and we study their relation to an aesthetic curve. For each class of curves, the paper includes a definition, a characterization theorem as well as illustrative applications. The notion of approximate aesthetic curve, corresponding to an approximate solution of an equation, is connected to the notion of Ulam Stability of that equation. An equation is called Ulam stable if for every approximate solution of it there exists an exact solution of the equation near it.
2. Aesthetic Curves
Let
be an arc-length
parameterized curve defined by the equations:
where
is the total length of the curve and
is a point of coordinates
,
on
.
The Logarithmic Curvature Graph (LCG) gradient is given by the relation:
where
is the curvature of
and
is a nonconstant function (see [
7]). The notion of aesthetic curve is defined in [
8] as follows:
Definition 1. [
8].
A curve is said to be an aesthetic curve if its LCG gradient is either constant or the gradient is represented by a straight line with a nonzero slope. A classification of aesthetic curves is given below.
Definition 2. [
8].
An aesthetic curve is called:Convergent: if the LCG gradient is positive;
Divergent: if the LCG gradient is negative;
Neutral: if the LCG gradient is zero.
In the next theorem, we give a characterization of convergent, divergent, and neutral curves and obtain their representation.
Theorem 1. Letbe a curve given by (1). Then:
is convergent if and only ifis strictly decreasing;
is divergent if and only ifis strictly increasing;
is neutral if and only if, and c1, c2 are real constants.
Proof. 1.
is convergent
, which is equivalent to:
Let be defined by . Then ,. Then the relation (3) becomes or , which is equivalent to . Therefore is a strictly decreasing function. We conclude that is a strictly decreasing function since log is strictly increasing.
2. Similarly, we get, for divergent curves, . So is an increasing function.
3. If
is a neutral curve, then
or
and integrating with respect to
it follows:
where
are arbitrary constants.
The representations of the curves can be obtained from the following relation (see [
8]):
where
is the measure in radians of the angle from the positive axis
to the unit tangent vector to the curve. □
3. Approximate Neutral Curves
In [
8] is defined the notion of an aesthetic curve as a curve with the LCG gradient constant or a straight line. The authors also mention that the aesthetic value of the curve increases when the LCG gradient approximates to a straight line, and, in particular, for neutral curves to zero. In what follows, we propose a definition for measuring the aesthetic value for a neutral curve in the sense mentioned above.
Definition 3. Let.
We say that a curve:
isneutral curve if its LCG gradientsatisfies the relation:for all.
A curve satisfying (5) for some positive is called approximate neutral curve.
For an neutral curve we will give an estimation of its curvature. Since in the definition of the gradient we suppose that for all , we deduce that or on the interval . So, according to Theorem 2, the curve is either convergent or divergent.
Theorem 2. Suppose thatis anneutral convergent curve given by (1). Then its curvature satisfies the following relation Proof. The curve
satisfies the relation
, which is equivalent to
Let
. Then (7) becomes
, which leads to
It follows
or
□
Integrating the previous relation on
and taking account that
we get
or
Remark 1. .
An analogous result holds for convergent curves.
Theorem 3. Suppose thatis anneutral divergent curve given by (1) with. Then its curvature satisfies the following estimation: Proof. The gradient of
satisfies the relation:
and the function
defined in the proof of the Theorem 3 satisfies the double inequality:
The curvature
fulfills the relation
which leads to
or
The theorem is proved. □
Application 1. Letbe anneutral convergent curve with.
Then, from Theorem 3 we obtain: So, every curve with satisfies the previous inequality and the corresponding neutral curve has the curvature
Taking for the representation of the approximate neutral curves
we obtain a family of logarithmic spirals (
Figure 1).
Application 2. Forandwe have
Taking account of the well-known inequality , it can be easily shown that
If we consider
then
is easily computed using Legendre–Gauss quadrature for each case and the graphical representations of the curves are given in
Figure 2 together with the plot of the curve obtained for
The mentioned numerical integration method has been used for the computation of all definite integrals within this paper.
If we consider
then
is easily computed for each case and the graphical representations of the curves are given in
Figure 3.
Application 3. Letbe anneutral divergent curve with. Then from Theorem 3 we obtain
So, every curve with
satisfies the inequality above. Taking for the representation of the approximate neutral curves
we obtain a family of Euler spirals (
Figure 4). The corresponding neutral curve has the curvature
4. Approximate Aesthetic Curves
Similarly to neutral curves we define the notion of aesthetic curve in spirit of Definition 3.
Definition 4. Let.
A curveof LCG gradientis calledaesthetic curve if there exist real numbers a, b such that: Clearly if
then
for all
Therefore,
is an aesthetic curve. If
then we get the notion of
neutral curve studied in
Section 3 of this paper. A curve that is
aesthetic for some
is called approximate aesthetic curve.
Our goal is to obtain an estimation of the curvature for an approximate aesthetic curve. For this we need some auxiliary results.
Theorem 4. Letbe a curve of gradientwith. Then its curvatureis given by the relation:where C1, C2 are real constants. Moreover, Proof. The curvature
is the solution of the differential equation
The function
satisfies the equation
So
Then
and integrating the previous relation on
we get
Therefore
where
The theorem is proved. □
We also recall the following result (see [
15]):
for every real numbers
a,
b,
c,
a ≠ 0,
δ =
b2 − 4
ac.
The main result is contained in the next theorem.
Theorem 5. Letbe a curve of gradientand. Suppose that there exist the real numbers a, b such that (9) is satisfied, i.e.,is anaesthetic curve and Let Then
- (i)
If
we have
- (ii)
- (iii)
Proof. Suppose that there exists
a, b such that (9) is satisfied and let
Then
for every
and
The relation
leads by integration on
to
So, we get
for all
where
□
The relation (13) implies that the left and the right hand in (14) are different of zero and have the same sign on
thus (14) is equivalent to
Now the estimations of in (i), (ii), (iii) follow immediately from (10), (12), and (15), respectively.
Remark 2. Analogous estimations can be made for the other cases:oretc.
Application 4. Letbe anaesthetic curvewith.
Then we obtainand The two new functions that are the bounds of
are represented in
Figure 5.
5. Conclusions
Since the perception of beauty by people is not necessarily an objective process, in this paper we proposed two new notions that approximate the aesthetic in some sense. Therefore, we defined the notions of neutral curve as well as of aesthetic curve based on the approximation of their LCG gradient. Aesthetic perceptions by people can essentially depend on aspects such as geographical location, level of education, people’s traditions, etc. By generalizing the definition of aesthetics through the proposed notions, it is possible that more people can perceive an object as beautiful.
We gave estimations between the curvature of an approximate aesthetic curve and the curvature of an aesthetic curve. Moreover, for each category of aesthetic curves we provided illustrative examples. The proposed notions can be used as aesthetic standards to ensure a wider target group of consumers in industrial products markets. They can be easily used in CAD systems for designing products that meet high aesthetics requirements in order to ensure their market success. When assembling the basic elements used to obtain an aesthetic product, symmetry has an essential role.
Author Contributions
Conceptualization, I.C., D.P., F.S. and L.T.; Data curation, I.C. and F.S.; Formal analysis, I.C. and F.S.; Software, F.S.; Visualization, F.S.; Writing—original draft, I.C., D.P., F.S. and L.T.; Writing—review & editing, I.C., D.P., F.S. and L.T. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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