Drawable Region of the Generalized Log Aesthetic Curves

Abstract

The main characteristic of visually pleasing curves used for product design is a monotonic curvature profile. Recently, a planar curve called Generalized Log Aesthetic Curve (GLAC) has been extended from the Log Aesthetic Curve (LAC), and it has an additional shape parameter, ?. This curve preserves the monotonicity of curvature and is said to produce visually pleasing curves. This paper delves on the drawable region of the GLAC segment which indicates the probable solutions of shape parameters from given interpolating points and the direction of travel at those points. The first section reviews the formulation of GLAC and its related bounds. The section describes the algorithm for identifying the drawable region. It is followed by the section describing how small changes of ? widen the drawable boundaries. The final section discusses the superiority of GLAC compared to LAC for use in industrial product design.

1. Introduction

The importance of aesthetic shapes to design industrial products leads to the studies of planar aesthetic curves [1]. High quality curves are also known as fair curves, minimal energy curves, aesthetic curves, and so forth. The main characteristic of these curves is that it has a monotonic curvature profile. The mathematical term used to describe planar curves with monotonic (either increase or decrease) curvature is called a spiral [2]. A curve is categorized as fair if its curvature plot consists of relatively minimum number of monotone pieces.

In 1999, Harada et al. presented a quantitative method of investigating curves used in automobile design which is called Logarithmic Distribution Diagram of Curvature (LDDC). They defined aesthetic curves as curves with a constant LDDC gradient (denoted as𝛼). Consequently in 2003, Kanaya et al. simplified the formulation of LDDC to Logarithmic Curvature Graph (LCG). In 2005, Miura analyzed the features of aesthetic curves and derived a general formula of aesthetic curves called Log Aesthetic Curve (LAC). Yoshida and Saito [3] further investigated LAC to identify the overall shapes of LAC. They proposed a method to draw LAC curve segment interactively by using two endpoints and their respective tangent vectors. In 2009, Levien and Sequin stated that the LA curve is the most promising curve for aesthetic design. In 2012, Yoshida and Saito derived a method to render the drawable boundary for LAC segments.

The number of research papers produced since the introduction of LAC has been increasing exponentially such as the reformulation of 2D and 3D LAC in the form of variational principal [4], analytic representation of LA curve using Incomplete Gamma function for efficient computation [5], and the formulation of G2 LA spline for automobile design application [6].

Recently, a planar curve called Generalized Log Aesthetic Curve (GLAC) has been developed by extending the formulation of Generalized Cornu Spiral (GCS) [7] in a similar manner to the LAC [8]. The family of GLAC comprises planar curves of high quality such as GCS, LAC, clothoid, Nielsen's spiral, logarithmic spiral, circle involute, and so forth. The GLAC segment has an additional parameter (denoted as 𝜈) to determine its shape as compared to GCS and LAC segment. Hence, an extra constraint can be satisfied when shaping the GLAC segment. Recent advancements of GLAC include the extension of spatial GLAC [9] and the shape analysis of GLAC [10].

Interactive control of GLAC segment has been developed similar to interactive LAC segment [11]. However, it is necessary to analyze the drawable region of interactive GLAC segment so that the users are aware of the flexibility of GLAC for shape design. The identification of drawable region of GLAC is similar to the method employed for LAC [12]. However, a slight modification has been carried out to determine the shape parameter Λ. We modify the arc length instead of modifying the 𝜃_𝑑 directly. We will modify arc length in the range of 0<𝑠<∞, which will give 𝜃_𝑑 such that 0<𝜃_𝑑<𝜋.

1.1. The Formulation of GLAC

GLAC is derived via the curve synthesis process where a formulation of a curve is derived from defined curvature function. The advantage of GLAC is that it has an extra shape parameter (𝜈) which can be used to dictate the curvature radius (𝜌) at the starting point of the GLAC. GLAC becomes LAC when 𝜈 becomes 0. The LCG and its gradient function, 𝜆(𝑠) are derived for GLAC as follows [8]: 𝐋𝐂𝐆_{𝐆𝐋𝐀𝐂}(𝐬)=𝟏𝐥𝐨𝐠(Λ𝜶𝐬+𝟏)^−𝟏/𝜶,𝟏+𝝂𝐥𝐨𝐠Λ(Λ𝜶𝐬+𝟏)𝟏+𝝂(Λ𝜶𝐬+𝟏)^𝟏/𝜶,𝜆(𝑠)_GLAC=𝛼+𝜈(Λ𝛼𝑠+1)^1/𝛼(1+𝛼). (1)

Arc length and tangential angle function of GLAC are shown below, respectively: 𝑠_GLAC𝜌=1Λ1log𝜌⁻¹1−𝜈if𝛼=0,𝜌Λ𝛼⁻¹−𝜈^−𝛼𝜃−1otherwise,_GLAC(𝑠)=1Λ1−𝑒^−Λ𝑠1+𝜈𝑠if𝛼=0,Λ[]1logΛ𝑠+1+𝜈𝑠if𝛼=1,Λ()()𝛼−1Λ𝛼𝑠+1^{(𝛼−1)/𝛼}−1+𝜈𝑠otherwise. (2)

The parametric form of GLAC is 𝐶_GLAC(𝑠)=^𝑠₀cos𝜃_GLAC(𝑢)𝑑𝑢,^𝑠₀sin𝜃_GLAC(𝑢)𝑑𝑢, (3) where {Λ,𝛼,𝜈}∈ℝ are the variables that can be used to shape the GLAC segment.

1.2. The Bounds of GLAC

The bounds are derived for both arc length and tangential angle function of GLACwhich are depicted in Table 1 and Table 2 respectively. For the case of 𝜈=0, the bounds of GLAC are similar to the bounds of LAC as stated in [10].

Table 1 

The lower bound and upper bound for 𝑠.

Table 2 

The lower bound and upper bound for 𝜃.

2. Drawable Region of GLAC

Interactive control of GLAC segment can be developed either by stating the value of 𝜈 or allowing the optimization techique to find a suitable value for 𝜈. It is possible to construct the drawable region for interactive GLAC if the designer chooses to set the value for 𝜈. The analysis of the drawable region can then be used to identify all possible solutions that exist for a given input.

2.1. The Method

The method used is similar to LAC where three control points will be placed in a rectangle of corners (±1,±1). Values of 𝛼<0 are used to construct the drawable region for the case of 𝜈>0. For the case of −1<𝜈<0, there is no restriction for 𝛼. Figure 1 illustrates the setup of control points in order to obtain the drawable region of GLAC.

(a) GLAC segment with new endpoints

(b) GLAC segment and a drawable boundary

Figure 1 

A drawable region of interactive GLAC segment.

Let points 𝑃₀, 𝑃₁, and 𝑃₂ be the control points for drawing a GLAC segment. The points 𝑃₀ and 𝑃₂ are placed at (−1,0) and (1,0). The point 𝑃₁ is the point on the drawable boundary in the rectangle. To note, 𝑃₁ is unknown and will be determined by the tangential angle 𝜃_𝑑 on the curve. GLAC segment with tangential angle from 0 to 𝜃_𝑑 is used for all 𝛼. Point 𝑃₁ can be found on the rectangle by performing transformation. By changing the arc length from 0 to a large number, the points of 𝑃₁ will form the drawable boundary. Note that we are modifying arc length to obtain 𝜃_𝑑 in the range of 0<𝜃_𝑑<𝜋. Hence, by applying reflection on the 𝑥-axis and 𝑦-axis, a complete drawable boundary can be found.

2.2. The Algorithm

The objective of this algorithm (Algorithm 1) is to determine the second control point through the formulation of GLAC curve segment and by plotting it on the rectangle.

Remark: Control points 𝑃0,𝑃2 are given with a preferred 𝛼. Via GLAC function, second control point is determined. New points are transformed to the originals and plotted. 𝑠 is modified from 0 to a large number such that 𝜃𝑑 is in range of 0<𝜃𝑑<𝜋 to get the complete drawable boundary. Input: 𝑃0(−1,0),𝑃2(1,0),𝛼,𝜈,𝑠. Output: Complete drawable boundary of interactive GLAC. Begin Step 1. If −1<𝜈<0 then        if 𝛼=0 then             []Λ←−log−𝜈𝑠.        else             ()Λ←−𝜈−𝛼−1𝑠𝛼.     else if 𝜈>0 and 𝛼<0 then             Λ←−1𝑠𝛼. Step 2. Determine 𝑆𝑃𝑥0,𝑦0 and 𝐸𝑃𝑥2,𝑦2 using (3). Step 3. Determine scaling factor 𝑃𝑟←2𝑃0𝑆𝑃𝐸𝑃 and angle 𝜃𝑒←cos−1𝑥2𝑥22+𝑦22. Step 4. Determine 𝑃1 using the tangent line of both end points. Step 5. Scale points to 𝑟, reflect through 𝑥-axis, rotate to −𝜃𝑒 and translate to 𝑃0,𝑃2. Step 6. Plot point 𝑃1. Step 7. Modify 𝑠 from 0 to a large number and repeat Step 1. to 6. Step 8. Reflect points via 𝑥-axis and𝑦-axis. Step 9. Output. End.

Algorithm 1 

3. Analysis Result

Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 show the obtained result for the drawable region of the interactive GLAC segment depending on 𝛼 and𝜈. Figures 2, 3, 4, 5, 6, and 7 show the pattern of how the decrease and increase of𝜈 affects the drawable region. Figures 8, 9, 10, and 11 show the drawable region for 𝛼=−1,0,1,2 as 𝜈 increases. The white region implies the drawable region of GLAC segment, while the gray region indicates no solution. Hence, more probable GLAC shapes can be drawn when the white region widens.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=−0.5

(e) 𝛼=−0.1

(f) 𝛼=0

(g) 𝛼=0.5

(h) 𝛼=0.8

(i) 𝛼=1

(j) 𝛼=1.1

(k) 𝛼=2

(l) 𝛼=3

Figure 2 

The drawable region of GLAC when 𝜈=−0.3.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=−0.5

(e) 𝛼=−0.1

(f) 𝛼=0

(g) 𝛼=0.5

(h) 𝛼=0.8

(i) 𝛼=1

(j) 𝛼=1.1

(k) 𝛼=2

(l) 𝛼=3

Figure 3 

The drawable region of GLAC when 𝜈=−0.1.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=−0.5

(e) 𝛼=−0.1

(f) 𝛼=0

(g) 𝛼=0.5

(h) 𝛼=0.8

(i) 𝛼=1

(j) 𝛼=1.1

(k) 𝛼=2

(l) 𝛼=3

Figure 4 

The drawable region of GLAC when 𝜈=−0.01.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=−0.5

(e) 𝛼=−0.1

Figure 5 

The drawable region of GLAC when 𝜈=0.01.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=−0.5

(e) 𝛼=−0.1

Figure 6 

The drawable region of GLAC when 𝜈=0.1.

(a) 𝛼=−3

(b) 𝛼=−2

(c) 𝛼=−1

(d) 𝛼=0.5

(e) 𝛼=−0.1

Figure 7 

The drawable region of GLAC when 𝜈=0.3.

(a) 𝜈=−0.3

(b) 𝜈=−0.1

(c) 𝜈=−0.01

(d) 𝜈=0.01

(e) 𝜈=0.1

(f) 𝜈=0.3

Figure 8 

The drawable region for 𝛼=−1 as 𝜈 increases.

(a) 𝜈=−0.3

(b) 𝜈=−0.1

(c) 𝜈=−0.01

Figure 9 

The drawable region for 𝛼=0 as 𝜈 increases.

(a) 𝜈=−0.3

(b) 𝜈=−0.1

(c) 𝜈=−0.01

Figure 10 

The drawable region for 𝛼=1 as 𝜈 increases.

(a) 𝜈=−0.3

(b) 𝜈=−0.1

(c) 𝜈=−0.01

Figure 11 

The drawable region for 𝛼=2 as 𝜈 increases.

The drawable region of GLAC shows a better result when −1<𝜈<0 compared to when 𝜈>0. This can be observed in Figure 4 where the drawable boundaries are larger compared to Figure 7. When 𝜈 is positive (small value, e.g., when 𝜈=0.01) and −1<𝛼<0, the drawable region is quite larger as compared to when −1<𝜈<0. However, when𝜈 is positive, the drawable region gets smaller for all other 𝛼. This implies that the shape variable 𝜈 plays a major role in dictating the drawable region effectively. The drawable regions can be expended remarkably with a small change of𝜈. Generally, −1<𝜈<0 gives a wider drawable region when 𝛼>1 and 𝛼<−1. We can note that when 𝛼=2, the drawable region is very large compared to LAC. LAC has regions that have no solutions, whereas GLAC provides the solution.

According to Yoshida and Saito [12], there are cases where the LAC segment is not drawable even when the second control point is within the drawable boundary. GLAC faces a similar situation where the second control point is within the drawable region yet the curve segment is not drawable.

Figure 12 is the comparison on the drawable region of LAC and GLAC when 𝛼=3. It portrays well the role of 𝜈 in GLAC. Note that GLAC becomes LAC when 𝜈 is 0. Since LAC is one of the cases in GLAC, the numerical example above clearly shows that for 𝛼>1, GLAC has a wider solution compared to LAC. Hence, GLAC provides greater flexibility where users would be able to create various shapes with a fixed start and end interpolating points.

(a) GLAC (𝜈=0)

(b) GLAC (𝜈=−0.01)

Figure 12 

Comparison on the drawable region of LAC and GLAC for 𝛼=3.

4. Conclusion and Future Works

An algorithm has been proposed to elucidate possible solutions that exist for interactive control of GLAC segment and the result is promising. The drawable region of GLAC proved to be far greater than LAC. This gives more freedom for designers to choose control points, tangent angle, and suitable shape parameters in order to design visually pleasing industrial products. The extra shape parameter 𝜈 facilitates the ability to set the curvature radius at the origin which makes the GLAC more flexible.

Future works include the development of a 𝐺² algorithm for GLAC by manipulating the shapes variables similar to the work proposed in [13, 14]. A different kind of approach will be redefining the curvature function in such a way that a curvature continuous Log Aesthetic Spline (LAS) can be created.

Acknowledgment

The authors acknowledge Faculty of Science and Technology, University Malaysia Terengganu (GGP grant) and Ministry of Higher Education Malaysia (FRGS: 59265) for providing financial aid to carry out this research.