Aesthetic curve generating apparatus and aesthetic curve generating method

Abstract

To provide a new type of aesthetic curve with more flexible expression than conventional logarithmic aesthetic curves so as to make it possible to open up applications in a field of design.SOLUTION: An aesthetic curve generating apparatus comprises a curve generating unit 12 for generating a target curve by performing a curve generating process based on an initial curve acquired by an initial curve acquiring unit 11 so as to satisfy conditions that an LCG is represented by a piecewise smooth function f and the differentiation of the function f with respect to the logarithm of the curvature or the logarithm of the radius of curvature changes monotonically, and enables the generation of a variety of aesthetic curves in which the LCG is allowed to be a parabola or other function, rather than being limited to the conventional logarithm aesthetic curves in which the LCG is a straight line, thus allowing to obtain a new type of aesthetic curve with more flexible expression and relaxed restrictions on curvature change compared to the logarithm aesthetic curves which have strong restrictions on curvature change.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to an aesthetic curve generating device and an aesthetic curve generating method, and is particularly suitable for use in a device and method for generating an aesthetic curve from a given initial curve. [Background technology]

[0002] In general product design, product shape data is created using CAD (Computer Aided Design). One method that has attracted attention is the use of reverse engineering technology to reconstruct the shape of curves / surfaces from measurement data of the model object shape. For example, automobile manufacturers embody the designer's ideas in clay models, and then create CAD data from the measurement data.

[0003] The CAD curve data to be created must satisfy two requirements: (i) it must be able to adequately approximate the measured data within a specified tolerance, and (ii) the curvature must change smoothly. In (ii), the smoothness of the curvature change is generally characterized by good curvature monotonicity.

[0004] One type of monotonic curvature curve is a log-aesthetic curve, which is expected to be used to express aesthetic curves in nature and man-made objects. A log-aesthetic curve is a curve that satisfies the following equation (0a).

[0005]

number

[0006] Here, ρ is the radius of curvature, and s is the arc length parameter. The vertical and horizontal axes of a logarithmic curvature graph (LCG) are defined as log(ρds / dρ) and logρ, respectively, and the slope α of the LCG is called the shape parameter. The shape parameter α of a log-aesthetic curve is the slope α of the straight line equation (0a), and this value characterizes the shape of the curve. As examples of log-aesthetic curves, α=-1 is called a clothoid curve, and α=1 is called a logarithmic spiral. Patent documents 1 and 2 disclose methods for generating log-aesthetic curves.

[0007] The CAD system described in Patent Document 1 divides input curve data into a plurality of segments, and generates a log-aesthetic curve (an approximation curve) for each divided segment so that the radius of curvature or the power of the curvature of each segment is expressed by a linear expression of the arc-length parameter, that is, so as to minimize an objective function based on a κ graph defined by an expression including the second derivative of the curve.

[0008] Patent Document 2 discloses that in a system for generating a space curve expressed by a plurality of parameters, a plurality of parameters including values ​​indicating the slope of the space curve in an LCG, the slope of the space curve in a logarithmic torsion graph, and an offset component in the logarithmic torsion graph (the ratio of the radius of curvature to the radius of torsion at a reference point) are determined, undetermined parameters are further searched for from the determined parameters, and a log-aesthetic curve is generated using the searched undetermined parameters.

[0009] Furthermore, Non-Patent Document 1 discloses a case where an LCG is expressed as a parabola as an extension of a conventional log-aesthetic curve, in which the LCG is expressed as a straight line with a slope of α by equation (0a). The extended curve is defined by a quadratic term of coefficient γ, a linear term of coefficient α, and a constant c, as shown in the following equation (0b).

[0010]

number

[0011] [Patent Document 1] Patent No. 5177771 [Patent Document 2] Patent No. 5487416 [Non-patent literature]

[0012] [Non-Patent Document 1] Abstract of the IEEJ Research Meeting 14.02 (0), p104-109, 2015, Norimasa Yoshida and Takafumi Saito, "Curves whose logarithmic curvature graphs are quadratic curves" Summary of the Invention [Problem to be solved by the invention]

[0013] However, log-aesthetic curves, including the extended type, have not been fully utilized in industrial fields. The following points are considered as the reasons for this. The log-aesthetic curve has a strong restriction on the change in curvature due to the "LCG slope α" being a constant, which results in the curve giving an impression of being relatively hard, making it difficult to reflect the designer's intention. In addition, although the basic properties of the curve are shown for the extended type described in Non-Patent Document 1, no method suitable for application to reverse engineering or curve representation in a CAD system is shown.

[0014] The present invention has been made to solve such problems, and aims to open up new applications of aesthetic curves in the field of industrial design by providing a new type of aesthetic curve with more flexible expressive power. [Means for solving the problem]

[0015] In order to solve the above-mentioned problems, in the present invention, a target curve is generated by performing a curve generation process based on an initial curve to be processed so as to satisfy the conditions that an LCG is represented by a piecewise smooth function f and the derivative of the function f with respect to the logarithm of the curvature or the logarithm of the radius of curvature changes monotonically. Effect of the Invention

[0016] According to the present invention configured as described above, a new type of aesthetic curve can be generated as a curve in which the differential of the logarithm of the curvature or the logarithm of the radius of curvature of a piecewise smooth function f exhibits a monotonous change. The new type of aesthetic curve obtained by the present invention is composed of various curves such as log-aesthetic curves in which the LCG is a straight line (the slope α of the LCG is constant), n-th degree (n≧2) polynomials including parabolas, and even exponential or logarithmic functions (both of which have variable slope α). As a result, compared to log-aesthetic curves in which the curvature change is strongly restricted, the new aesthetic curves have relaxed restrictions on the curvature change, and the flexibility of expression is increased. [Brief description of the drawings]

[0017] [Figure 1] 1 is a block diagram showing an example of a functional configuration of an aesthetic curve generating device according to an embodiment of the present invention; [Diagram 2] FIG. 1 is a diagram for explaining a log-aesthetic curve. [Diagram 3] FIG. 1 illustrates an example of a log-curvature graph and an alpha graph for a log-aesthetic curve. [Figure 4A] FIG. 13 is a diagram showing an example of an initial curve and each graph. [Figure 4B] FIG. 4B shows an α curve and each graph generated from the initial curve of FIG. 4A. [Figure 5A] 13A to 13C are diagrams showing other examples of the initial curve and each graph. [Figure 5B] FIG. 5B shows an α curve and each graph generated from the initial curve of FIG. 5A. [Figure 6] FIG. 13 is a diagram for explaining a gap of curvature vectors relating to Application Example 1. [Figure 7]FIG. 2 is a block diagram showing an example of a functional configuration of an aesthetic curve generating device according to Application Example 1 of the present embodiment. [Figure 8] 11 is a diagram for explaining the process of the α gap detection unit according to Application Example 1 of the present embodiment. FIG. [Figure 9] FIG. 13 is a diagram showing an example of an initial curve and each graph having an extreme value of curvature. [Figure 10] FIG. 11 is a block diagram showing an example of a functional configuration of an aesthetic curve generating device according to an application example 2 of this embodiment. [Figure 11] 10 is a diagram showing a curve and each graph generated from the initial curve of FIG. 9. [Figure 12] 12A and 12B are diagrams showing curves and graphs generated from the curves in FIG. 11. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0018] An embodiment of the present invention will be described below with reference to the drawings. Fig. 1 is a block diagram showing an example of the functional configuration of an aesthetic curve generating device 1 according to this embodiment. As shown in Fig. 1, the aesthetic curve generating device 1 of this embodiment includes an initial curve acquiring unit 11 and a curve generating unit 12 as functional components.

[0019] These functional blocks 11 and 12 can be configured by any of hardware, DSP (Digital Signal Processor), and software. For example, the functional blocks 11 and 12 are realized by the operation of a program stored in a storage medium such as a RAM, ROM, hard disk, or semiconductor memory under the control of a microcomputer configured with a CPU, RAM, ROM, etc.

[0020] The initial curve acquisition unit 11 acquires data of an initial curve to be processed. For example, the initial curve acquisition unit 11 acquires data of the initial curve by a user operation or by executing a batch processing type program. The initial curve is represented by a typical parametric curve such as a point sequence, a polyline, or a NURBS curve. A polyline is a broken line obtained by linearly interpolating a point sequence, and is synonymous with providing a point sequence as the initial curve.

[0021] The curve generating unit 12 generates a target curve by performing a curve generating process based on the initial curve acquired by the initial curve acquiring unit 11 so that an α-curvature logarithm graph (hereinafter referred to as α-LCG), which indicates the relationship between the slope α of the LCG and the logarithm of the curvature or the logarithm of the radius of curvature, exhibits a monotonous change. The target curve generated in this embodiment is an aesthetic curve in which the slope α of the LCG is a variable value relative to the arc length. In this embodiment, this aesthetic curve is called an "α curve." The curve generating process performed by the curve generating unit 12 is also called "α curve approximation."

[0022] Since the slope of LCG is α, the relationship between LCG and α-LCG, "LCG ⇔ α-LCG", corresponds to the "integral ⇔ differential" relationship. Therefore, if α-LCG is a straight line with a slope of zero, LCG is a straight line, which corresponds to a log-aesthetic curve with α as a constant. Also, if α-LCG is a straight line with a slope of non-zero, LCG is a parabola.

[0023] The α curve generated in this embodiment will be described in detail below, along with a comparison with a conventional log-aesthetic curve. First, the log-aesthetic curve and the shape parameter α will be described.

[0024] The LCG of a log-aesthetic curve is a straight line with a slope of α. The vertical and horizontal axes of the LCG are log(ρds / dρ) and logρ, respectively, so the linear equation for a log-aesthetic curve is given by the above formula (0a). A graph in which the radius of curvature ρ is replaced by the curvature κ (=1 / ρ) is also called an LCG, and the linear equation is given by formula (1). When ρ appears in the formula, it is assumed that the radius of curvature of the curve increases monotonically with the arc length s. Similarly, when κ appears in the formula, it is assumed that the curvature of the curve increases monotonically with the arc length s. There is no loss of generality in this assumption, since a curve with a monotonic change in curvature can be either monotonically increasing or monotonically decreasing by appropriately parameterizing it.

[0025]

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[0026] Furthermore, because the shape parameter α is defined as the slope of the LCG, the following equation (2) holds between it and κ. A graph with α in equation (2) on the vertical axis and arc length s on the horizontal axis is called an α graph. Note that a similar relationship to equation (2) holds between α and ρ. Below, the α curve will be explained using the curvature κ, but the same can be said when the radius of curvature ρ is used.

[0027]

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[0028] Figure 2 shows the log-aesthetic curve C LA FIG. 2 is a diagram for explaining the log-aesthetic curve C LA Starting point P s is the origin of the xy coordinate system, and the end point is P e , P s The tangent at is the x-axis, and P e The angle between the tangent at and the x-axis is θ. LA is an example where θ=80° and α=2, and the magnitude of the curvature κ is also illustrated in the form of a curvature comb. LA3(a) and α graph (FIG. 3(b)) for . Both graphs are straight lines, and as can be seen from the α graph, the slope α of the LCG is a constant.

[0029] <Example of α curve (when α-LCG is a straight line)> From here, the α curve generated in this embodiment will be described. The α curve is a curve in which the slope α (shape parameter α) of the LCG changes monotonically, not limited to a constant. As the simplest example of monotonous change, consider the case where the α-LCG is a straight line. Here, the vertical and horizontal axes of the α-LCG are α and -logκ in equation (3), respectively, so the linear equation that defines the monotonous change of α is expressed as the following equation (3). In this equation (3), α1 is the slope of the straight line in α-LCG, and α0 is the intercept of the straight line with respect to the vertical axis.

[0030]

number

[0031] Therefore, by substituting (Equation 3) into (1b), we obtain the following (Equation 4). This Equation (4) means that the LCG is a parabola. Note that when α1=0, it means that the LCG is a straight line, which is a log-aesthetic curve. In other words, Equation (4) shows the properties of a new aesthetic curve that is expanded to include the log-aesthetic curve.

[0032]

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[0033] As mentioned above, the curve where the LCG is expressed as a parabola is also disclosed in Non-Patent Document 1. In this document, the conventional log-aesthetic curve C LA The authors clarify the basic properties of the curve, such as the sign of γ, for equation (0b), which is obtained by simply adding a quadratic term of the coefficient γ to equation (0a) of the LCG of . However, they do not discuss the slope of the LCG when γ ≠ 0, that is, the shape parameter α (when γ = 0, the conventional log-aesthetic curve C LA(The slope of the graph coincides with the shape parameter α.) In addition, there is no discussion of α-LCG. In other words, Non-Patent Document 1 does not exhaustively discuss how to grasp parabolic LCGs by naturally extending the mathematical framework of log-aesthetic curves.

[0034] Furthermore, the display and generation of curves disclosed in Non-Patent Document 1 involves gradually extending a curve in a tangential direction from the starting point of the curve using the Frenet-Serret formula, making it difficult to generate a curve that passes through a pre-determined point such as an end point. In other words, Non-Patent Document 1 does not disclose at all a method for generating a high-quality curve, such as a parabola in LCG, using a typical parametric curve such as a NURBS curve, based on a given initial curve.

[0035] The curve generator 12 of this embodiment generates a curve called an α curve so that α-LCG and LCG are a straight line (equation (3)) and a parabola (equation (4)), respectively. Specifically, the curve generator 12 performs optimization processing on the curve C so as to minimize two objective functions Ψ(C) and Ω(C) shown in the following equations (5a) and (5b).

[0036]

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[0037] In equations (5a) and (5b), N is the number of evaluation points on curve C. n , ds / dκ| n , dκ / ds| n and d 2 κ / ds 2 | n are κ, ds / dκ, dκ / ds and d at the nth sample point. 2 κ / ds 2where α0 and α1 are the coefficients of the linear equation y = α1x + α0 obtained by linearly approximating the α-LCG of curve C using the least squares method (x and y are the horizontal and vertical axes of α-LCG, respectively). Equation (5a) is the first objective function for evaluating whether or not equation (4), which indicates that the LCG is a parabola, is satisfied. Equation (5b) is the second objective function for evaluating whether or not equation (3), which indicates that the α-LCG is a straight line, is satisfied, and α in equation (3) uses equation (2).

[0038] The α curve approximation by the curve generation process is to reconstruct a NURBS curve so that the initial curve (for example, a NURBS curve such as a Bezier curve or a B-spline curve) acquired by the initial curve acquisition unit 11 satisfies the conditions of the α curve. This α curve approximation can be said to be a new curve fairing in that a high-quality monotonic curve of curvature is obtained. In this embodiment, the above-mentioned objective functions Ψ(C) and Ω(C) are set as an example of the conditions of the α curve, and an optimal solution is searched for while gradually changing the control points and their weights so as to minimize them. As a result, a curve C in which the LCG becomes a parabola and the linearity of the α-LCG is sufficiently satisfied is obtained. Note that, if the accuracy of the α curve approximation is poor due to reasons such as a low degree of the NURBS curve, the NURBS curve may be reconstructed by converting it into a high-order formula or a rational form.

[0039] <General form of α curve (when α-LCG changes monotonically)> Next, the general form of the α curve is explained. The function ψ is defined as shown in the following equation (6). The function f used here is a piecewise smooth function, and its derivative with respect to -log κ is a monotonic function. The function f includes, for example, an n-th degree (n≧2) polynomial, an exponential function, a logarithmic function, etc. In this case, the α curve is defined as a curve that satisfies the conditions of equations (7a) and (7b). The fact that the derivative with respect to -log κ of the function f is a monotonic function means that the change in α is monotonic according to equation (7b), and this property stipulates the conditions regarding α of the α curve.

[0040]

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[0041] Here are some examples when f(logκ) is a polynomial. When it is a linear equation, i.e., a straight line f(logκ)=-a*logκ+c, Equation (7a) is the traditional log-aesthetic curve C LA For the quadratic parabola f(logκ)=a(logκ), the monotonic function of equation (7b) is the simplest constant a. 2 In the case of -b*logκ+c, equation (7a) becomes the parabolic LCG of equation (4), and equation (7b) becomes the equation of the straight line of equation (3). 3 +b(logκ) 2 In the case of +c*logκ+d, equation (7b) becomes an equation of a parabola, but because it must change monotonically, it is expressed in an interval that does not include the apex of the parabola.

[0042] When the above f(logκ) is a third-order polynomial, specifically, the curve generator 12 performs optimization processing on the curve C so as to minimize two objective functions Ψ(C) and Ω(C) shown in the following equations (8a) and (8b). The equation (8a) is a first objective function for evaluating whether or not the equation (9a) indicating that the LCG is a third-order polynomial is satisfied. The equation (8b) is a second objective function for evaluating whether or not the equation (9b) indicating that the α-LCG is a parabola is satisfied.

[0043]

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[0044] Here, α0, α1, and α2 are the equation y=α2x obtained by parabolic approximation of the α-LCG of curve C using the least squares method. 2 +α1x+α0 (x and y are the horizontal and vertical axes of α-LCG, respectively).

[0045] <Two examples of α curve approximation> Figure 4 (Figure 4A and Figure 4B) shows an example of an initial curve for an automobile windshield and its α-curve approximation, as well as three related graphs. Figure 4A shows the B-spline curve created by an on-site CAD expert, while the curve in Figure 4B shows the α-curve approximation. For ease of explanation, ● marks are placed at the boundaries of each segment of the curve, but they do not exist on the actual curve (this also applies to the other figures shown below).

[0046] The curve shown in FIG. 4 is represented by a B-spline curve of three segments S1 to S3 (FIG. 4A(a) and FIG. 4B(a)), but is a typical example in which the part is represented by a single base curve. The curve generation process may be performed for each of the segments S1 to S3 that make up one base curve, but is usually performed for each base curve. FIGS. 4A(b) to (d) respectively show the LCG, α-LCG, and α graph for the initial curve. FIGS. 4B(b) to (d) respectively show the LCG, α-LCG, and α graph for the α curve.

[0047] As shown in Figure 4A(b)-(d), the LCG of the curve created by the CAD expert is not a straight line, and the α graph does not show a constant value. In other words, the curve does not have the properties of a log-aesthetic curve. Rather, the LCG suggests a parabola. On the other hand, as shown in Figure 4B(b)-(d), the LCG of the curve generated by α-curve approximation shows a parabola, and the α-LCG also shows a straight line, satisfying the conditions of an α-curve.

[0048] Figure 5 (Figures 5A and 5B) shows another example of an initial curve for a roof part of a car and its α-curve approximation and three related graphs, where Figure 5A is for an initial curve created by an on-site CAD expert and Figure 5B is for an α-curve approximation curve.

[0049] The curve shown in Figure 5 is composed of three base curves, each of which is represented by a multi-segment B-spline curve (Figures 5A(a) and 5B(a)). The curve generation process is performed for each base curve. Figures 5A(b) to (d) show the LCG, α-LCG, and α graph for the initial curve, respectively. Figures 5B(b) to (d) show the LCG, α-LCG, and α graph for the α curve, respectively.

[0050] As shown in Figure 5A(b)-(d), wavy parts can be seen in all the graphs of the curves created by the CAD expert. These wavy parts correspond to the parts where the quality of the curve is adversely affected. On the other hand, as shown in Figure 5B(b)-(d), the LCG of the curve generated by α-curve approximation is a parabola, and the α-LCG is also a straight line, satisfying the conditions of the α-curve. In other words, the α-curve approximation generates a curve of higher quality.

[0051] As described above in detail, in this embodiment, a curve called an α curve is generated based on the initial curve acquired by the initial curve acquisition unit 11 such that the α-LCG exhibits a monotonic change. As the simplest example of a monotonic change, when the α-LCG is a straight line, if the slope α is a constant, the LCG will be a straight line, which matches a conventional log-aesthetic curve. If the slope α is not a constant, the LCG can be expressed as an n-th degree (n≧2) polynomial, an exponential function, or a logarithmic function. In particular, the LCG of the curve generated to minimize the two objective functions Ψ(C) and Ω(C) shown in equations (5a) and (5b) is a parabola.

[0052] According to the present embodiment configured as described above, it is possible to obtain not only conventional log-aesthetic curves in which the LCG is expressed by a straight line (=first-order polynomial), but also new types of aesthetic curves in which the LCG is expressed by n-th-order polynomials including parabolas (=second-order polynomials), exponential functions, logarithmic functions, etc. The new types of aesthetic curves obtained by the present embodiment have monotonically changing LCG slope α, and as a result, compared to log-aesthetic curves in which the curvature change is strongly restricted, the restrictions on the curvature change are relaxed, and the flexibility of the curve expression can be increased. In addition, as described in FIG. 5, it is possible to generate curves in which the curvature change is sufficiently smooth (without waviness) and which satisfy high-level requirements for aesthetics.

[0053] As described above, the technology described in Non-Patent Document 1 is limited to aesthetic curves whose LCG is a parabola, and does not disclose a method for generating high-quality curves using representative parametric curves such as NURBS curves. In contrast, according to this embodiment, it is possible to generate high-quality curves using representative parametric curves not only for the parabolic LCG described in Non-Patent Document 1, but also for aesthetic curves expressed by more general LCGs.

[0054] <Application example 1 (division of curvature discontinuities)> 4 and 5, the initial curve acquired by the initial curve acquisition unit 11 has a continuous curvature (G2 continuous) as a whole. In contrast, as shown in FIG. 6(a) or 6(b), the initial curve acquired by the initial curve acquisition unit 11 may include a point P whose curvature is not continuous.

[0055] In this case, a common method is to determine the presence of a point where the angle difference between two curvature vectors is greater than a threshold, as seen in the curvature discontinuity point P in Fig. 6(a), and perform segmentation at that point. However, as seen in the curvature discontinuity point P in Fig. 6(b), segmentation is not possible when the angle difference between the curvature vectors is smaller than the threshold. Therefore, in application example 1 of this embodiment, segmentation is performed by utilizing gaps in α values ​​in α-LCG or α graph.

[0056] Fig. 7 is a block diagram showing an example of the functional configuration of an aesthetic curve generating device 2 according to application example 1. In Fig. 7, components having the same reference numerals as those shown in Fig. 1 have the same functions, and therefore repeated explanations will be omitted here. As shown in Fig. 7, the aesthetic curve generating device 2 according to application example 1 further includes an alpha gap detection unit 13 as a functional configuration. Moreover, the aesthetic curve generating device 2 according to application example 1 includes a curve generating unit 12' instead of the curve generating unit 12.

[0057] The α gap detection unit 13 generates an α-LCG or α graph from the initial curve acquired by the initial curve acquisition unit 11, and detects points on the initial curve where the gap in the α value is equal to or greater than a threshold. Here, the α gap detection unit 13 receives as input a curve including the curvature discontinuity point P shown in Fig. 6(b) and determines whether or not there is a gap in the α value at the point P. Fig. 8 is a diagram for explaining the processing of the α gap detection unit 13, Fig. 8(a) shows the initial curve including the curvature discontinuity point P, and Figs. 8(b) to (d) show the LCG, α-LCG, and α graph of the initial curve, respectively.

[0058] 8(c), at curvature discontinuity point P, there is a large gap between the α value of the right end point of segment S1 to the left of point P and the α value of the left end point of segment S2 to the right of point P. Note that since the horizontal axis of α-LCG is -log κ, the part marked with a circle is the α value at point P. The α gap detection unit 13 is capable of detecting point P where the gap in the α value in the α-LCG is equal to or greater than a threshold value.

[0059] 8(d), there is a large gap between the α value of the right end point of segment S1 on the left side of point P and the α value of the left end point of segment S2 on the right side of point P. The α gap detection unit 13 is capable of detecting point P in the α graph where the gap in the α values ​​is equal to or greater than a threshold value.

[0060] When a gap in the α value is detected by α gap detection unit 13, curve generation unit 12' divides the initial curve at that point and performs a curve generation process for each divided segment. In the example shown in Fig. 8(a), curve generation unit 12' performs a curve generation process for two segments, segment S1 and segments S2 and S3.

[0061] <Application Example 2 (Splitting extreme points of curvature)> The reference curves constituting the initial curve acquired by the initial curve acquisition unit 11 are not necessarily monotonic curvature curves, and may have points that are extreme values ​​of curvature (hereinafter, referred to as curvature extreme points). At these curvature extreme points, the values ​​on the vertical axis of the LCG, α-LCG, and α graph are not defined and become infinite, making it impossible to perform mathematically rigorous processing. Figure 9 shows an example of an initial curve (Figure 9(a)) with point R as the curvature curvature point, and Figures 9(b) to (d) show the LCG, α-LCG, and α graph of the initial curve, respectively.

[0062] In the case of the example of the initial curve shown in Fig. 9(a), the section from point Q to point S is represented as a single segment B-spline curve (i.e., a Bezier curve) S2, and has a curvature extreme point R at the center of the segment. At the curvature extreme point R, the value on the vertical axis of each graph becomes infinite, as shown in Figs. 9(b) to (d). In this way, α curve approximation cannot be performed on an initial curve that has a curvature extreme point within a segment, i.e., has a non-monotonic curvature.

[0063] Therefore, α-curve approximation is performed on a curve obtained by removing the curvature extreme points from the initial curve. In order to remove the curvature extreme points while maintaining the curvature tendency of the initial curve, the curve may be divided at the points. However, simply dividing the original curve at the points will result in the curvature values ​​near the division points remaining large. Therefore, applying a fitting approximation using an n-th degree function (e.g., a quadratic curve) where n≧2 to reconstruct the divided curve within the allowable error range is an effective practical method.

[0064] 9(a), α curve approximation is performed directly on segment S1, which has a monotonic curvature from point P to point Q. Meanwhile, segment S2 from point Q to point S is divided into two segments at curvature extreme point R (segment S2-1 from point Q to point R and segment S2-2 from point R to point S). Curve approximation by fitting is performed for each of the two divided segments S2-1 and S2-2, and then α curve approximation is performed on segment S2 from point Q to point S.

[0065] FIG. 10 is a block diagram showing an example of the functional configuration of an aesthetic curve generating device 3 according to application example 2. In FIG. 10, components having the same reference numerals as those shown in FIG. 1 have the same functions, and therefore repeated explanations will be omitted here. As shown in FIG. 10, the aesthetic curve generating device 3 according to application example 2 further includes, as functional components, a curvature extreme point detecting unit 14 and an approximation processing unit 15. Moreover, the aesthetic curve generating device 2 according to application example 2 includes a curve generating unit 12″ instead of the curve generating unit 12.

[0066] The curvature extreme point detection unit 14 detects curvature extreme points in the initial curve acquired by the initial curve acquisition unit 11. For example, the curvature extreme point detection unit 14 finds points where the first derivative of the curvature is zero by bisection. Alternatively, the curvature extreme point detection unit 14 may generate an α-curvature logarithmic graph as shown in FIG. 9(c) or an α graph as shown in FIG. 9(d) for the initial curve, and detect points where the α value on the vertical axis is equal to or greater than a threshold as curvature extreme points. This is also a kind of segmentation using the α value.

[0067] When a curvature extreme point is detected by the curvature extreme point detection unit 14, the approximation processing unit 15 divides the initial curve at the point, and performs curve approximation by fitting using an n-th degree function for each divided segment. For example, the approximation processing unit 15 fits the curve of each divided segment with a quadratic curve (parabola). In the example shown in Fig. 9, the approximation processing unit 15 divides the segment S2 from point Q to point S into a segment S2-1 from point Q to point R and a segment S2-2 from point R to point S, and performs curve approximation by a quadratic curve for each of the segments S2-1 and S2-2.

[0068] For a segment in which a curvature extreme point has been detected by curvature extreme point detection unit 14, curve generation unit 12" performs a curve generation process using α curve approximation based on the curve curved-approximated by approximation processing unit 15, for each of the multiple segments divided by approximation processing unit 15 or for the entire multiple segments as a processing unit. On the other hand, for a segment in which a curvature extreme point has not been detected by curvature extreme point detection unit 14, curve generation unit 12" performs a curve generation process using α curve approximation directly on that segment.

[0069] Figures 11 and 12 are figures showing examples of a curve generated by approximation processing unit 15 and a curve generated by curve generation unit 12" based on the initial curve shown in Figure 9, along with each graph. Figure 11(a) shows a curve generated by performing α curve approximation on segment S1 from point P to point Q, and performing curve approximation by fitting on segments S2-1 and S2-2 divided into two between point Q and point S. Figures 11(b) to (d) respectively show the LCG, α-LCG, and α graph of the curve shown in Figure 11(a). Figure 12(a) shows a curve generated by further performing α curve approximation on the approximation curve generated for segment S2 between point Q and point S. Figures 12(b) to (d) respectively show the LCG, α-LCG, and α graph of the curve shown in Figure 12(a).

[0070] As shown in Figure 9, for the initial curve created by the CAD expert, segment S1 from point P to point Q suggests that the LCG is a parabola. On the other hand, for segment S2 from point Q to point S, as mentioned above, the vertical axis values ​​in the α-LCG and α graph diverge to infinity at the curvature extreme point R (mathematically, LCG also diverges to infinity, but this is the upper limit due to the floating-point representation of the computer). Furthermore, the α-LCG and α graph are significantly wavy, suggesting that there is a problem with the quality of the curve in this part (in fact, we have confirmed that there is a problem).

[0071] As shown in Figures 11 and 12, with regard to the α curve approximation to the initial curve, the LCG of the segment S1 from point P to point Q is a parabola and the α-LCG is a straight line, satisfying the conditions of the α curve.

[0072] On the other hand, as shown in FIG. 11, the segment S2 from point Q to point S has no extreme curvature value at point R as a result of performing curve approximation with a quadratic curve (parabola) on the initial curve. Here, FIG. 11(c)(d) shows that α=2 / 3. A log-aesthetic curve with α=2 / 3 is a parabola, but the result of performing curve approximation with a quadratic curve is reflected in this α value. Also, as shown in FIG. 12, as a result of performing α curve approximation on the approximation curve, α-LCG shows a straight line, and the α value shows a monotonous change in the α graph.

[0073] Note that, although an example has been described here in which the initial curve is divided at a curvature extreme point, curve approximation by fitting or the like is performed for each divided segment, and then α curve approximation is performed, curve approximation by fitting or the like may be omitted. In other words, when a curvature extreme point is detected by the curvature extreme point detection unit 14, the curve generating unit 12'' may divide a segment of the initial curve at the curvature extreme point, and perform α curve approximation for each divided segment.

[0074] In addition, the above-described embodiment and application examples 1 and 2 are merely examples of the embodiment of the present invention, and the technical scope of the present invention should not be interpreted as being limited thereby. In other words, the present invention can be embodied in various forms without departing from the gist or main characteristics of the present invention. [Explanation of symbols]

[0075] 1,2,3 Aesthetic curve generator 11 Initial curve acquisition section 12,12',12” curve generator 13 Alpha gap detector 14 Curvature extreme point detector 15 Approximation processing section

Claims

1. an initial curve acquisition unit for acquiring data of an initial curve to be processed; a curve generating unit that generates a target curve by performing a curve generating process based on the initial curve acquired by the initial curve acquiring unit so as to satisfy the conditions that a logarithmic curvature graph (LCG) is represented by a piecewise smooth function f and that the differential of the function f with respect to the logarithm of the curvature or the logarithm of the radius of curvature changes monotonically. An aesthetic curve generating device characterized by:

2. 2. The aesthetic curve generating device according to claim 1, wherein the function f is a polynomial of degree n (n≧2).

3. The aesthetic curve generating device according to claim 1, characterized in that the curve generating unit generates the objective curve by performing the curve generation process so as to minimize a first objective function for evaluating whether or not an equation indicating that the LCG is a parabola is satisfied, and a second objective function for evaluating whether or not an equation indicating that an α-curvature logarithm graph (α-LCG), which shows the relationship between the slope α of the LCG and the logarithm of the curvature or the logarithm of the radius of curvature, is a straight line.

4. The aesthetic curve generating device according to claim 1, characterized in that the curve generating unit generates the objective curve by performing the curve generation process so as to minimize a first objective function for evaluating whether or not an equation indicating that the LCG is a third-order polynomial is satisfied, and a second objective function for evaluating whether or not an equation indicating that an α-curvature logarithm graph (α-LCG), which shows the relationship between the slope α of the LCG and the logarithm of the curvature or the logarithm of the radius of curvature, is a parabola.

5. An α gap detection unit detects a point on the initial curve where a gap in the value of the slope α is equal to or greater than a threshold value in an α graph showing a relationship between the α-LCG or arc length of the initial curve and the slope α of the LCG, When the α gap detection unit detects a point where the gap is equal to or greater than a threshold, the curve generation unit divides a segment of the initial curve at the point where the gap is equal to or greater than the threshold, and performs the curve generation process for each divided segment.

5. The aesthetic curve generating device according to claim 1, wherein the aesthetic curve generating device is a computer-readable storage device.

6. a curvature extreme point detection unit that detects a curvature extreme point where the curvature becomes an extreme value on the initial curve; an approximation processing unit that divides a segment of the initial curve at the curvature extreme point when the curvature extreme point is detected by the curvature extreme point detection unit, and performs n-th order (n≧2) curve approximation for each divided segment, The curve generating unit performs the curve generating process on the approximation curve generated by the approximation processing unit for each segment or for a whole set of a plurality of segments as a processing unit.

5. The aesthetic curve generating device according to claim 1, wherein the aesthetic curve generating device is a computer-readable storage device.

7. A curvature extreme point detection unit detects a curvature extreme point where the curvature becomes an extreme value in the initial curve, When the curvature extreme point is detected by the curvature extreme point detection unit, the curve generation unit divides a segment of the initial curve at the curvature extreme point and performs the curve generation process for each divided segment.

5. The aesthetic curve generating device according to claim 1, wherein the aesthetic curve generating device is a computer-readable storage device.

8. A first step in which an initial curve acquisition unit of a computer acquires data of an initial curve to be processed; and a second step in which the curve generating unit of the computer performs a curve generating process based on the initial curve acquired by the initial curve acquiring unit to generate a target curve so as to satisfy the conditions that a logarithmic curvature graph (LCG) is represented by a piecewise smooth function f and that the derivative of the function f with respect to the logarithm of the curvature or the logarithm of the radius of curvature changes monotonically. The aesthetic curve generating method according to the present invention is characterized in that:

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