Isoptics of log-aesthetic curves Ferenc Nagy1 1University of Debrecen, Faculty of Informatics, 26 Kassai St., 4028 Debrecen, Hungary 1University of Debrecen, Doctoral School of Informatics, 26 Kassai St., 4028 Debrecen, Hungary January 2021 Abstract The log-aesthetic curve has a significant factor in the field of aesthetic design to meet the high industrial requirements. It has much exceptional property and a large number of research papers are published, since its introduction. It can be parametrized by tangential angle, and it is au- toevolute, meaning that its evolute is also a log-aesthetic curve. In this paper, we will examine the isoptics of the log-aesthetic curve and deter- mine if it is autoisoptic as well. We will show that the log-aesthetic curve only coincides its isoptic in some specific cases. We will also provide a new formula to calculate the isoptic curve of any tangential angle parametrized curves. 1 Introduction To design highly aesthetic objects, the use of high-quality curves is very important. The main property of such curves is the curvature, which should be monotonically varying. Among them, the log-aesthetic stated as the most promising for aesthetic design [10] because its curvature and torsion graphs are arXiv:2104.11327v1 [math.GM] 20 Apr 2021 both linear [19]. The log-aesthetic curve is originated from Harada et al. [7, 6]. They revealed that natural aesthetic curves have such a property that their logarithmic dis- tribution diagram of curvature (LDDC) can be approximated by straight lines. Based on their work Miura et al. [12, 11] have defined the logarithmic curvature graph (LCG), an analytical version of the LDDC as the following: when a curve (given with arc length s and radius of curvature ρ) is subdivided into infinites- imal segments such that ∆ρ/ρ is constant, the LCG represents the relationship between ρ and ∆s in a double logarithmic graph. The authors formulated the curve whose LCG is strictly expressed by a straight line (with slope α). Such 1 curves satisfy the following equation: ds log ρ = α log ρ + c, (1) dρ where c is a constant. Eq. (1) is the fundamental equation of log-aesthetic curves. The log-aesthetic curve can also be considered as the generalization of the clothoid, logarithmic spiral, Nielsen’s spiral, and the circle involute. The LCG of these curves is a straight line with different slopes that can be drawn as log- aesthetic curves with different shape parameter α (see Figure 1). The general equations of the log-aesthetic curve will be presented in Section 2.1. 10 5 -10 -5 5 10 -5 -10 Figure 1: The logarithmic curvature graph of several log-aesthetic curves with different α slopes: clothoid (α = −1), Nielsen’s spiral (α = 0), logarithmic spiral (α = 1), circle involute (α = 2), circle (α = ±∞). The curves are generated using parametric equations (on the same [−3π, 3π] interval), which parameter is limited (red dots), except in the cases of α = 1 and α = ±∞. Besides the linear curvature, the log-aesthetic curve has another interesting 2 property. It is autoevolute, that means the evolute of a log-aesthetic curve with shape parameter α is also a log-aesthetic curve with shape parameter −1/(α−2) [21]. This characteristic led us to study the isoptic curve of the log-aesthetic curve. The isoptic curve of a curve is constructed by the involving lines with a given angle intersect each other at a certain point of the isoptic curve. Some results of calculating the isoptics of various plane curves will be presented in Section 2.2. The similar attribute to autoevolute is called the autoisoptic, meaning that the given curve coincide its isoptic curve. In [13], there are some autoisoptic curves, among the logarithmic spiral, which is the log-aesthetic curve with α = 1. It is also evident that the isoptic of a circle is also a circle (log-aesthetic curve with α = ±∞). In Section 3, we will present a new formula for calculating the isoptic curve of the log-aesthetic curve (and for all the tangential angle parametrized curves). We will also prove that for arbitrary α ∈ R the log-aesthetic curve is not au- toisoptic. 2 Previous results In this section, we are going to summarize the work of Yoshida and Saito [20]. They analyzed the properties of the log-aesthetic curve and derived a general formula to draw it. Their related results will be presented in the first subsection. Besides, in the second part, we will introduce the isoptic curves and show some results in the topic. 2.1 Generating log-aesthetic curves The general formula is obtained from the relationship between the arc length and the radius of curvature of the log-aesthetic curve, using a reference point (at the origin) and placing the following constraints on it: the tangent vector is T 1 0 , and the radius of curvature is 1. By transforming the curve such that the above constraints are satisfied, a point P (θ) of the log-aesthetic curve whose tangential angle is θ can be expressed on the complex plane as ([20]): θ (1+i)Λψ 0 e dψ if α =1 P (θ)= 1 (2) θ α 1 iψ (α − 1)Λψ +1 − e dψ otherwise, R0 R RR+ where α ∈ and Λ ∈ are parameters. α is the slope of the LCG. When α 6= 1, all the aesthetic curves are congruent under similarity transformations without depending on the value of Λ(6= 0). Since similarity transformations preserve angles, in our examples we focus on the cases of Λ = 1. The radius of curvature as the function of θ can be expressed as ([20]): eΛθ if α =1 ρ(θ)= 1 (3) α 1 (α − 1)Λθ +1 − otherwise. 3 When θ = 0, ρ = 1. The function increases monotonically when Λ 6= 0. In case of Λ = 0, ρ is constant 1 and the log-aesthetic curve is a circle. Similarly, when α = ±∞ because taking the limit of Eq. (3) when α approaches ±∞ we get ρ = 1. The tangential angle θ and the arc length s are related by ([20]): Λs 1 e− −Λ if α =0 log(Λs+1) θ(s)= Λ if α =1 (4) (1 1 ) (Λαs+1) − α 1 Λ(α 1) − otherwise. − Therefore, a point C(s) on the aesthetic curve whose arc length is s, can be defined on the complex plane as ([20]): Λu s 1 e− 0 exp(i − Λ ) du if α =0 s exp(i log(Λu+1) ) du if α =1 C(s)= R0 Λ (5) (1 1 ) s (Λαu+1) − α 1 R0 exp(i Λ(α 1) − ) du otherwise. − Eq. (5) and Eq. (2) representR the same curve. Using Eq. (4), the radius of curvature can also be expressed from the arc length s ([20]): eΛs if α =0 ρ(s)= 1 (6) Λαs +1 α otherwise. Since ρ can change from −∞ to +∞, the tangential angle θ and arc length s may have upper or lower bound depending on the value of α because of the negative bases of the fractional exponents (see Table 1). Besides the general formula, the authors of [20] also analyzed the properties of the curve. They called the case of Λ = 1 as the standard form I of the curve. Let us highlight some important characteristics in this case regarding the isoptic of the curve, based on the value of α: α> 1 As θ approaches ∞, the curve spirally diverges towards the point at ρ = ∞ (the arc length to that point is infinite). However, θ has a lower 1 bound of Λ(1 α) (see Figure 2(a)–(d)). − α< 1 As θ approaches −∞, the curve spirally converges to the point at ρ =0 (the arc length to that point is infinite). However, θ has an upper bound 1 of Λ(1 α) (see Figure 2(f)–(l)). − α =1 As θ approaches −∞, the curve spirally converges to the point at ρ =0 (the arc length to that point is finite), and as θ approaches ∞, the curve spirally diverges towards the point at ρ = ∞ (the arc length to that point is infinite, see Figure 2(e)). T α = ±∞ The log-aesthetic curve is a circle centered at 0 1 . 4 Tangential angle (θ) Arc length (s) α< 1 α =1 α> 1 α< 0 α =0 α> 0 1 1 Upper bound: Λ(1 α) - - − Λα - - − 1 1 Lower bound: - - Λ(1 α) - - − Λα − Table 1: Upper and lower bound of θ and s. 5 5 5 5 -5 5 -5 5 -5 5 -5 5 -5 -5 -5 -5 (a) α = 3 (b) α = 2.5 (c) α = 2, (d) α = 1.5 Circle involute 3 3 3 5 2 2 2 1 1 1 -5 5 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -1 -1 -1 -5 -2 -2 -2 -3 -3 -3 (e) α = 1, (f) α = 0.5 (g) α = 0, (h) α = −0.5 Logarithmic spiral Nielsen’s spiral 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.5 1.0 1.5 -0.5 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -1.5 -1.5 -1.5 -1.5 (i) α = −1, (j) α = −1.5 (k) α = −2 (l) α = −2.5 Clothoid Figure 2: Log-aesthetic curves with different shape parameter α and Λ = 1.