Abstract
In this paper, we consider a class of plane curves called logaesthetic curves and their generalization which is used in CAGD. We consider these curves in the context of similarity geometry and characterize them in terms of a “stationary” integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the EulerLagrange equation. As an application of the formalism developed here, we propose a discretization of both the curves and the associated variational principle which preserves the underlying integrable structure. We finally present an algorithm for the generation of discrete logaesthetic curves for given data. The computation time to generate discrete logaesthetic curves is much shorter than that for numerical discretizations of logaesthetic curves due to the avoidance of fine numerical integration to calculate their shapes. Instead, only coarse summation is required.
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LogAesthetic Curves: Similarity Geometry,
Integrable Discretization and Variational Principles
[5mm] Junichi Inoguchi
Institute of Mathematics, University of Tsukuba
Tsukuba 3058571, Japan
email: inoguchi@math.tsukuba.ac.jp
[2mm] Kenji Kajiwara
Institute of Mathematics for Industry, Kyushu University
744 Motooka, Fukuoka 8190395, Japan
email: kaji@imi.kyushuu.ac.jp
[2mm] Kenjiro T. Miura
Graduate School of Science and Technology,Shizuoka University
351 Johoku, Hamamatsu, Shizuoka, 4328561, Japan
email: miura.kenjiro@shizuoka.ac.jp
[2mm] Hyeongki Park
Graduate School of Mathematics, Kyushu University
744 Motooka, Fukuoka 8190395, Japan
email: hpaku@math.kyushuu.ac.jp
[2mm] Wolfgang K. Schief
School of Mathematics and Statistics, The University of New South Wales
Sydney, NSW 2052, Australia
email: w.schief@unsw.edu.au
1 Introduction
In this paper, we consider a class of plane curves in CAGD called logaesthetic curves (LAC) and their generalization called quasi aesthetic curves (qAC), and present a new mathematical characterization based on the theory of integrable systems and similarity geometry. We then construct the discrete analogue of LAC and qAC within the abovementioned framework, which gives a new implementation of LAC and qAC with a sound mathematical background as discrete curves.
In the previous paper [4], we have announced the similarity geometric framework of LAC and qAC, where these curves have been characterized by the stationary integrable flow of plane curves preserving the turning angle. More precisely, the evolution of the curves is governed by the curvature which is characterized by the stationary solutions of an integrable nonlinear partial differential equation arising from the geometric setting. In addition, we have introduced a fairing energy functional and formulated LAC and qAC in terms of a variational principle. Here, we first present a detailed account of those results.
Secondly, we construct the discrete analogue of LAC and qAC based on the above formalism, where these curves are characterized by the stationary discrete integrable flow of discrete plane curves preserving the turning angle. We then introduce a discrete fairing functional and formulate these discrete curves in terms of a discrete variational principle. The discrete curves obtained in this manner are not naïve approximations of the original LAC and qAC but admit their own natural geometric characterization.
Finally, we give an implementation of discrete LAC and qAC obtained above for given endpoints and associated tangent vectors. One of the advantages of discrete LAC for practical purposes is that the computation time to generate them is much faster than that for numerical discretization of LAC because we do not have to perform numerical integration to calculate their shapes and only require coarse summation.
2 Logaesthetic curves and similarity geometry
Originally, LAC has been studied in the framework of Euclidean geometry. Before proceeding to LAC, we give a brief account of the treatment of plane curves in Euclidean geometry. Let be an arc length parametrized plane curve and be arc length. We introduce the Frenet frame by
(1) 
where and are the tangent and the normal vector fields, respectively, and is the positive rotation. Since by definition of arc length, we may write , where is called the turning angle. The Frenet frame satisfies the Frenet formula
(2) 
where is the curvature. Note that the curvature is related to the signed radius of curvature and the turning angle by
(3) 
According to [6], an arc length parametrized plane curve is said to be a LAC of slope if its signed curvature radius satisfies
(4) 
The class of LAC includes some well known plane curves. For instance, the logarithmic spiral, the clothoid, the Nielsen spiral are included as LAC of slope , , and , respectively. The LAC of slope is also known as the circle involute curve. These examples are illustrated in Figure 1.
LAC are now maturing in industrial and graphics design practices. Figure 2 shows the practical example of a car designed using LA splines. Figure 2(a) shows freeform surface isoparametric lines generated using LA splines and corresponding zebra maps. Figure 2(b) shows the geometric model with a special lighting condition and 2(c) are photos of a manufactured mockup based on the geometric model. Note that the roof of the car is designed by an LA spline curve with three segments and its zebra maps indicate that the surface is of high quality. Based on our experience, LA splines are generated with most Hermite data. Another direction of application is developed in architecture design [12]. For more details of the LAC, we refer to [7, 8].
Those studies have been carried out based on the basic characterization (4) in the framework of Euclidean geometry. However, (4) is too simple to identify the underlying geometric structure. Consequently, we do not have a good guideline as to how to generate a larger class of aesthetic geometric objects including LAC based on a sound mathematical background. As we have announced in the previous paper [4], it is natural to adopt the framework of similarity geometry, which is a Klein plane geometry associated with the group of similarity transformations, i.e., isometries and scalings:
The natural parameter of plane curves in similarity geometry is the turning angle . Let be a plane curve in similarity geometry parametrized by . We introduce the similarity Frenet frame by
(5) 
where
(6) 
are the similarity tangent and normal vector fields, respectively.
Note that which follows from and (see Figure 3). Then, the (Euclidean) Frenet formula (2) implies that the similarity Frenet frame satisfies the similarity Frenet formula
(7) 
for some function which is called the similarity curvature. Moreover, the similarity curvature is related to the signed curvature radius by the ColeHopf transformation:
(8) 
One can check that a plane curve in similarity geometry is uniquely determined by the similarity curvature up to similarity transformations.
The notion of LAC may be shown to be invariant under similarity transformations. For instance, the slope is expressed as . LAC is reformulated in terms of similarity geometry as follows [3, 10]. A plane curve in similarity geometry is said to be a LAC of slope if its similarity curvature satisfies the Bernoulli equation
(9) 
Based on this reformulation, qAC is introduced in the following manner [11]. A plane curve in similarity geometry is said to be a qAC of slope if its similarity curvature obeys the Riccati equation
(10) 
3 Burgers flows on similarity plane curves
One of the key techniques to understand LAC and qAC is to consider the integrable (time) evolution of plane curves that preserves the invariant parameter of similarity geometry, which is known to be described by the Burgers hierarchy [1]. The simplest evolution is given by
(11) 
which is rewritten in terms of the similarity Frenet frame as
(12) 
The compatibility condition of (7) and (12) yields the Burgers equation
(13) 
Therefore, the evolution (11) is referred to as the Burgers flow. The Burgers equation is linearized in terms of the signed curvature radius via the ColeHopf transformation (8) according to
Imposing the stationarity ansatz reduces the Burgers equation (13) to the Riccati equation
(14) 
In particular, putting , we recover the Riccati equation (10) with . We note that (10) is obtained formally from (14) by making the substitution . In this sense, qAC are characterized as the stationary curves of the Burgers flow. We also note that the parameter corresponds to a rotation of the curve.
4 Fairing energy in similarity geometry
In this section, we present the details of a variational formulation of LAC and qAC. To this end, we introduce the fairing energy functional [4]
(15) 
where , is an arbitrary constant and (). The above functional is invariant under similarity transformations and its name “fairing energy” is motivated by the fairing procedure in digital style design of industrial products. To compute the variation, we consider a deformation of parametrised by
(16) 
where is the variation of . We distinguish the quantities relevant to the deformed curve from their undeformed counterparts by adding an overbar . For example, the turning angle and the similarity curvature of are denoted by and , respectively. In order to obtain the variation of and from (16), we first compare the similarity Frenet formula for and :
(17) 
where we used (7). Setting
(18) 
(19) 
where we omitted the higher order terms in . We next compute in two ways by using (7) and (19):
(20) 
where we used (7), then (19). On the other hand, using (19) then (7), we have
(21) 
Comparing (20) and (21), we have
(22) 
In order to determine consistently, and must satisfy the equation obtained from (22) by eliminating :
(23) 
Substituting (19), we obtain from the term
(24) 
Thus, and are seen to be
(25) 
We compute the variation of by using , where is the Euclidean inner product, so that . Then, is computed from (18) and (20) as
Therefore, we have
(26) 
Now, we are ready to calculate the variation of :
(27) 
We have, by virtue of (26),
(28) 
where
(29) 
Then, we obtain the first variation formula from (27) after some straightforward calculations by using (25) and (28):
(30) 
where
(31) 
The first variation formula implies that if is a critical point of the fairing energy for deformations which respect the boundary condition eliminating the first term in (30), then satisfies
(32) 
which is equivalent to the Riccati equation for qAC (10) together with (8). Indeed, elimination of via differentiation and evaluation modulo (8) lead to the stationary Burgers equation obtained by differentating (10). Hence, the parameter plays the role of a constant of integration.
Let us examine the boundary condition required in the above computation. Noticing that is a first integral of (32), we put . Then, the boundary term in (30) gives
(33) 
where , . If we require preservation of the total turning angle, that is, , which is the analogue of the preservation of arc length in Euclidean geometry, then it follows from (25) that so that, by virtue of (26), the boundary term vanishes if
(34) 
Hence, we conclude that , namely, the ratio of length of tangent vectors at the endpoints is preserved by the variation. Note that this condition is invariant with respect to similarity transformations. Summarizing the discussion above, we obtain the following theorem:
Theorem 4.1.
If a plane curve is a critical point of the fairing energy (15) under the assumption of preservation of the total turning angle and the boundary condition that the ratio of length of tangent vectors at the endpoints is preserved, then the similarity curvature satisfies , where is a constant. Therefore, quasi aesthetic curves of slope are critical points of the fairing functional.
5 Discrete LAC and qAC
One of the benefits of the formalism developed in the preceding sections is that one is led to the construction of a natural discrete analogue of LAC and qAC which preserves the underlying integrable nature of these curves. It is expected that these discrete curves obtained on the principle of structure preservation have better quality as discrete curves compared to other existing discretizations regarded as approximations (cf. Section 7). In this section, we construct the discrete analogue of LAC and qAC by using the framework of integrable evolutionss of discrete plane curves in similarity geometry as discussed in [5].
Let , be a discrete plane curve. As shown in Figure 4, we introduce the similarity Frenet frame according to
(35) 
where and are discrete tangent and normal vectors, respectively, and we write
(36) 
Then, satisfies the discrete similarity Frenet formula
(37) 
where plays the role of a discrete counterpart of the similarity curvature of smooth plane curves. Hereafter, we assume that the discrete turning angle , and the associated discrete curves may be regarded as the similarity geometric analogues of arc length parameterized discrete curves in Euclidean geometry.
Remark 5.1.
It is interesting to remark on the radii of osculating circles for both arc length parameterized discrete curves in Euclidean geometry and discrete curves of constant turning angle in similarity geometry (see Figure 5). In Euclidean geometry, a discrete plane curve is said to be an arc length parameterized discrete curve if the segment length is constant, i.e., . Then, there exists a circle touching the two segments and at their midpoints, and its radius is given by , where . On the other hand, in similarity geometry, there exists a circle touching simultaneously the three consecutive segments , , with the second segment being touched at its midpoint. The radius of the circle is given by , which is the same expression as in the Euclidean case.
We consider a discrete (time) evolution of a discrete curve preserving the constant turning angle . We denote the original discrete curve by and the curve obtained after discrete time steps is labelled by . The quantities relevant to these discrete curves are written in a similar manner. For example, and . Then, the simplest evolution is known to be given by [5]
(38) 
where the frame satisfies
(39) 
and is a constant. Note that the first equation is nothing but the discrete similarity Frenet formula. The compatibility condition of (39), , yields the discrete Burgers equation [2, 9]
(40) 
which is linearized in terms of according to
(41) 
Note that the continuum limit (13) of (40) with is obtained by setting
(42) 
Imposing the stationarity ansatz on the discrete Burgers equation and neglecting the superscript , we obtain the discrete stationary Burgers equation
(43) 
whose continuum limit gives the stationary Burgers equation
(44) 
Equation (43) can be integrated to yield the discrete Riccati equation
(45) 
The existence of the continuum limit of (45) requires the parametrisation , leading to
(46) 
In order to construct the discrete analogue of (9) and (10), we replace by , where , to obtain
(47) 
and
(48) 
This dependence is consistent with the parametrization (42) which comes from a geometric restriction on the continuum limit. Actually, noticing that , we see that (47) and (48) reduce to (9) and (10), respectively, if we set . Let us consider the solution of (48), which may be linearized according to
(49) 
by putting
(50) 
In the case , the solution of (49) is given by with being arbitrary constants to yield
(51) 
where . It is evident that (51) yields the original expression for the similarity curvature of LAC,
(52) 
by applying the continuum limit (42). The above discussion motivates the following natural definition.
Definition 5.2.
Let be a discrete plane curve of constant turning angle . is said to be a discrete LAC (dLAC) of slope if satisfies
(53) 
is said to be a discrete qAC (dqAC) of slope if satisfies
(54) 
In both cases, .
Figure 6 illustrates some qAC and dqAC with the same parameters and .
6 Variational formulation of dLAC and dqAC
It turns out that, as in the continuous case, the dLAC and dqAC proposed in Section 5 may be obtained via a variational principle. Indeed, in the following, we demonstrate that dLAC and dqAC may be characterized as the stationary curves of constant turning angle of the discrete fairing energy functional given by
(55) 
with respect to an arbitrary variation of the discrete curve which we write as
(56) 
To this end, we first compute the variation of the frame by using the discrete similarity Frenet formula (37) as
(57) 
or
(58) 
where
(59) 
The variation of the frame (58) must be compatible with the similarity Frenet formula (37). Accordingly, the associated compatibility condition results in the pair
(60) 
from which we obtain the variation of and as
(61) 
Taking the variation of , we have . Then, from (58), we obtain the variation of as
(62) 
Note that can also be calculated by using , which is consistent with (61).
On use of the variations (61) and (62), the variation of the discrete fairing energy functional is seen to be
(63) 
where
(64) 
The first variation formula (63) implies that if is a critical point of the discrete fairing energy for deformations which respect the boundary condition, then satisfies
(65) 
which is equivalent to (47) or (45) together with in the same manner as in the continuous case. The boundary term vanishes iff , which implies that from (62). This means that the ratio of length of segments at the endpoints is preserved by the variation, which is the discrete analogue of the boundary condition in the smooth curve case.
Theorem 6.1.
If a discrete plane curve is a critical point of the discrete fairing energy (55) under the boundary condition that the ratio of length of segments at the endpoints is preserved, then satisfies (43). Therefore, discrete quasi aesthetic curves of slope are those discrete curves of constant turning angle which constitute critical points of the discrete fairing functional.
Remark 6.2.
Since does not enter the variation (63) of the discrete fairing functional, whether the variation of the curve preserves the constancy of the turning angle or not does not affect the discrete EulerLagrange equation. However, if is to be preserved by the variation then, by virtue of (61), is no longer arbitrary but constrained by . It is also observed that the structure of the variation (63) may be interpreted in a simple geometric manner. Since, up to Euclidean motions, a discrete curve is uniquely determined by the angles and the lengths of the segments, we may regard as independent quantities in the variation of the energy functional. More precisely, in order to respect invariance under similarity transformations, appropriate independent variations are given by , where . Hence, since the energy functional depends on only with , its variation may be expressed entirely in terms of . In this manner, one retrieves the variation (63) if one takes into account that, for instance, .
7 Generation of dLAC and dqAC
In this section, we consider the problem of Hermite interpolation by using dLAC, namely, we generate the dLAC with specified endpoints and the direction of segments (tangent vectors) at the endpoints. This problem was formulated and solved for LAC in [13].
For simplicity, we first construct dLAC consisting of four points for given endpoints, and , and the direction of the segments at those points with the specified parameter . Consider the triangle on the plane shown in Figure 7. The problem is equivalent to determining on AB and on BC such that , where . In other words, the length of the segments is subject to the constraints
(66)  
(67) 
where we have chosen the coordinates such that and without loss of generality. Moreover, satisfies
(68) 
for specified . Therefore, the three unknown variables , and are determined from equations (66), (67) and (68), in principle, and , are given by
(69) 
It is straightforward to generalize the above procedure to generate dLAC with points, , for given and being on the respective edges of the specified triangle depicted in the second picture of Figure 7. Then, () satisfy the following equations:
(70)  
(71)  
(72) 
where . It is possible to determine in principle, since we have equations for unknown variables (). Then, we have
(73) 
Now, equations (70)–(72) may be solved numerically as follows:

Substituting the above expressions into (72), we have a nonlinear equation in . We then solve the equation by a suitable solver, e.g., the bisection method.

Compute () by using (74).

Compute the lefthand side of (71) and put it as .

Apply the scaling so that (71) is satisfied.
In this method, we are able to reduce the nonlinear equations for () to a single nonlinear equation for only one variable , by using the linearity of (70) in and the scaling property of (71) and (72).
Figure 8 illustrates the examples of dLAC generated by using the above method. Despite the different values, the shape of the curves in the top and bottom rows of the middle picture are similar. When (total number of vertices is ), the triangle cut by the vertices polyline of is a little bit