Apparatus and method for designing polynomial curve

Abstract

The present invention relates to a polynomial curve design technology and, more specifically, to an apparatus and method for designing a polynomial curve by using a hodograph polygon and an interpolation polygon. According to an embodiment of the present invention, a hodograph and a curve themselves can be simultaneously controlled, and the curve can accurately pass through a designated point to enable local control capable of finely adjusting the curve in a specific section, thereby improving the precision and flexibility of a curve design. In addition, a complex curve can be more efficiently designed by supporting precise data interpolation adjustment even in a high-order polynomial curve, and the shape and length of the curve can be precisely controlled at the same time.

Description

Polynomial curve design apparatus and method

[0001] The present invention relates to a polynomial curve design technique, and more specifically, to an apparatus and method for designing a polynomial curve using radial polygons and interpolation polygons.

[0002]

[0003] In fields such as computer graphics, font design, and computer-aided design / manufacturing (CAD / CAM), various types of curves are used to represent geometric shapes or the movement trajectories of objects. Among these curves, the most fundamental are polynomial curves, and the most widely used representation of polynomial curves in the design process is the Bezier curve. Bezier curves have the advantage of a simple mathematical structure while enabling intuitive design using Bezier control polygons. On the other hand, as the shape of the curve to be designed becomes more complex, more control points are required, which consequently increases the degree of the curve. In high-order polynomial curves, Bezier control polygons can become excessively large relative to the curve, presenting a disadvantage where curve design becomes increasingly difficult. To address these issues, the Gauss-Legendre (GL) curve was proposed as a polynomial curve representation method utilizing hodograph control polygons. This method has the advantage of enabling the design of high-order polynomial curves by precisely designing the shape of the curve based on nodes and weights. While the Gauss-Legendre curve maintains the endpoint interpolation characteristic, it has limitations in that it is difficult to finely adjust the curve in specific sections because the direct influence of the hodograph polygon is limited only to the hodograph of the curve, thereby restricting control over intermediate curve points.

[0004] Therefore, improved polynomial curve design technology is required that enables local control and fine-tuning while maintaining the advantages of the Gauss-Legendre curve.

[0005] The background technology of the present invention is disclosed in Korean Registered Patent No. 10-262060.

[0006]

[0007] The present invention provides a polynomial curve design apparatus and method that defines the basic shape of a curve using hodograph polygons and enables detailed adjustment of the curve using interpolation polygons.

[0008] The technical problems that the present invention aims to solve are not limited to those mentioned above, and other unmentioned technical problems will be clearly understood by those skilled in the art to which the present invention belongs from the description below.

[0009] According to one aspect of the present invention, a polynomial curve design device is provided.

[0010] A polynomial curve design device according to one embodiment of the present invention may include a node determination unit for determining a node, a weight determination unit for determining a weight, a polynomial calculation unit for calculating a polynomial basis using the node and the weight, a radiograph polygon generation unit for setting a curve tangent vector at the node using an input point and generating a radiograph polygon, an interpolation polygon generation unit for setting a point that the curve must pass through using an input point and generating an interpolation polygon, and a polynomial curve generation unit for generating a curve using the radiograph polygon, the interpolation polygon, and the polynomial basis.

[0011] According to another aspect of the present invention, a method for designing a polynomial curve and a computer program for executing the same are provided.

[0012] A method for designing a polynomial curve according to one embodiment of the present invention and a computer program for executing the same may include the steps of determining a node, determining a weight, calculating a polynomial basis using the node and the weight, setting a tangent vector of the curve at the node using an input point and generating a radial polygon, setting a point that the curve must pass through using an input point and generating an interpolation polygon, and generating a curve using the radial polygon, the interpolation polygon, and the polynomial basis.

[0013]

[0014] According to an embodiment of the present invention, the hodograph and the curve itself can be controlled simultaneously, and local control is enabled so that the curve can accurately pass through a designated interpolation point, allowing for fine adjustment of the curve in a specific section, thereby improving the precision and flexibility of the curve design.

[0015] In addition, it supports precise data interpolation adjustments even for high-order polynomial curves, enabling more efficient design of complex curves and precise simultaneous control of the curve's shape and length.

[0016] The effects of the present invention are not limited to the effects described above, and should be understood to include all effects that can be inferred from the composition of the invention described in the description or claims of the present invention.

[0017]

[0018] FIG. 1 is a drawing for explaining a polynomial curve design device according to one embodiment of the present invention.

[0019] FIG. 2 is an exemplary diagram showing the form of a polynomial basis function of a Gauss-Legendre-Lagrange curve (GLL) defined according to an embodiment of the present invention.

[0020] FIG. 3 is an example diagram showing the difference between a Gauss-Legendre curve generated based on a hodograph polygon according to an embodiment of the present invention and a Gauss-Legendre-Lagrange curve generated based on a hodograph polygon and an interpolation polygon.

[0021] FIG. 4 is an exemplary drawing showing a change in the shape of a curve that occurs when the control points of a Gauss-Legendre-Lagrange curve are modified according to an embodiment of the present invention.

[0022] FIG. 5 is an exemplary diagram illustrating a method for approximating a rational Bezier curve using a Gauss-Legendre-Lagrange curve according to an embodiment of the present invention.

[0023] FIG. 6 is an exemplary drawing showing various Gauss-Legendre-Lagrange curves obtained by fine-tuning a high-order polynomial curve, the Gauss-Legendre curve, using a polynomial curve design device according to one embodiment of the present invention.

[0024] FIG. 7 is a drawing for explaining a polynomial curve design method according to an embodiment of the present invention.

[0025]

[0026] The present invention is susceptible to various modifications and may have various embodiments; therefore, specific embodiments are illustrated in the drawings and described in detail. However, this is not intended to limit the present invention to specific embodiments, and it should be understood that the invention includes all modifications, equivalents, and substitutions that fall within the spirit and scope of the invention. In describing the present invention, detailed descriptions of related prior art are omitted if it is determined that such detailed descriptions would unnecessarily obscure the essence of the invention. Furthermore, singular expressions used in this specification and claims should generally be interpreted to mean "one or more" unless otherwise stated.

[0027] Throughout the specification, when it is stated that a part is "connected (connected, in contact, combined)" with another part, this includes not only cases where they are "directly connected," but also cases where they are "indirectly connected" with other members interposed between them. Furthermore, when it is stated that a part "includes" a certain component, this means that, unless specifically stated otherwise, it does not exclude other components but rather allows for the inclusion of additional components.

[0028] The terms used herein are merely for describing specific embodiments and are not intended to limit the invention. Singular expressions include plural expressions unless the context clearly indicates otherwise. In this specification, terms such as “comprising” or “having” are intended to indicate the presence of the features, numbers, steps, actions, components, parts, or combinations thereof described in the specification, and should be understood as not precluding the existence or addition of one or more other features, numbers, steps, actions, components, parts, or combinations thereof.

[0029] The present invention will be described below with reference to the attached drawings. However, the present invention may be implemented in various different forms and is not limited to the embodiments described herein. Furthermore, in order to clearly explain the present invention in the drawings, parts unrelated to the explanation have been omitted, and similar parts throughout the specification have been given similar reference numerals.

[0030] FIG. 1 is a drawing for explaining a polynomial curve design device according to one embodiment of the present invention.

[0031] Referring to FIG. 1, the polynomial curve design device (10) includes a node determination unit (100), a weight determination unit (110), a polynomial calculation unit (130), a radial polygon generation unit (150), an interpolation polygon generation unit (170), and a polynomial curve generation unit (190).

[0032] The node determination unit (100) determines a node. At this time, the node determination unit (100) can determine the node according to a numerical quadrature such as the Gauss-Legendre quadrature or the Gauss-Lobatto quadrature. The node determination unit (100) can use n different nodes (where n is a natural number) given within the domain range of the curve. For example, the node determination unit (100) can determine the definite integral of the function f(t) in the interval [a,b] of the polynomial curve design device (10). n nodes used when calculating numerically It can be selected. In addition, the node determination unit (100) assumes the integration interval is [-1,1] and the definite integral of f(t) You can select a node for the Gauss-Legendre numerical quadrature or the Gauss-Lovato numerical quadrature used when calculating.

[0033] The weight determination unit (110) determines weights that reflect the contribution of f(t) at each node. At this time, the weight determination unit (110) can determine weights according to the Gauss-Legendre numerical quadrature, the Gauss-Lovato numerical quadrature, etc. The weight determination unit (110) can determine n weights (where n is a natural number) corresponding to n different nodes (where n is a natural number) given within the domain range of the curve. The nodes thus determined and weights Using this, the definite integral of a function f(t) defined on the interval [a,b] It can calculate the definite integral of f(t) by assuming the integration interval is [-1,1]. For example, the weight determination unit (110) can calculate the definite integral of f(t) by assuming the integration interval is [-1,1]. The weights used when calculating the Gauss-Legendre numerical quadrature or the Gauss-Lovato numerical quadrature of the following mathematical formula 1 can be obtained. At this time, if the weights obtained from the weight determination unit (110) are used, an approximate value can be obtained even in situations where it is difficult to calculate the integral value of the function f. You can obtain.

[0034] [Mathematical Formula 1]

[0035]

[0036] In this equation, n is the number of nodes, and the node is the roots of the n-th order Gaussian-Legendre polynomial P_n listed in order from smallest to largest. The weight corresponding to each node is determined by mathematical formula 2.

[0037] [Mathematical Formula 2]

[0038]

[0039] Here, is of the n-th degree Gaussian-Legendre polynomial It is the derivative value at, and is of the n-1st order Gaussian-Legendre polynomial It means the function value at.

[0040] The polynomial calculation unit (130) calculates a polynomial basis using the node determined by the node determination unit (100) and the weight determined by the weight determination unit (110). For example, the polynomial calculation unit (130) Let be an n-degree Gaussian-Legendre polynomial basis or a Gaussian-Lovato polynomial basis. can calculate, Let be a Gaussian-Legendre-Lagrange polynomial basis of degree 2n. It can calculate.

[0041] The process of calculating a Gaussian-Legendre-Lagrange polynomial basis of degree 2n in the polynomial calculation unit (130) The process may include calculating a polynomial that satisfies a specific condition in which only one of the values ​​is 1 and all others are 0. First, the polynomial calculation unit (130) can calculate the polynomial f_0 as a polynomial of degree 2n given by mathematical formula 3. and the rest and All values ​​of can include a polynomial f_0 where f_0 becomes 0.

[0042] [Mathematical Formula 3]

[0043]

[0044] The polynomial calculation unit (130) For k such that, when a polynomial G_k of degree 2n-1 is equal to Equation 4, the polynomial If the given is a polynomial of degree 2n, then and the rest , and , (step, Igo A polynomial where all values ​​of ) are 0 It may include.

[0045] [Mathematical Formula 4]

[0046]

[0047] The polynomial calculation unit (130) For k and polynomial G_k, the polynomial g_k If we let be a polynomial of degree 2n given by, and the rest , and , (step, Igo All values ​​of ) can include a polynomial g_k such that 0.

[0048] Additionally, the polynomial calculation unit (130) may include generating a polynomial basis for generating a curve having two polygons A and B as a hodograph polygon and an interpolation polygon, respectively, for given two polygons A and B. For example, having n sides (where n is a natural number). Polygons in space and having n points Polygons in space A 2n-degree Gaussian-Legendre-Lagrange polynomial basis such that for , a polynomial curve p of degree 2n or less having as a Gaussian-Legendre polygon and an interpolation polygon, respectively, is determined as in Equation 5 It may include generating.

[0049] [Mathematical Formula 5]

[0050]

[0051] In this case, a basis of 2n-degree Gaussian-Legendre-Lagrange polynomials It can be created as follows.

[0052] First, for the problem in a one-dimensional space, ordered pairs of n+1 real numbers and ordered pairs of n real numbers For (where n is a natural number) , and (where i is A polynomial h(t) with a degree of 2n or less that satisfies all conditions of (all natural numbers) is a polynomial defined in the polynomial calculation unit (130). h(t) can be expressed as Equation 6 using a linear combination of .

[0053] [Mathematical Formula 6]

[0054]

[0055] polynomial h Equation 6 can be rearranged into Equation 7 to be expressed in the form of a barycentric combination.

[0056] [Mathematical Formula 7]

[0057]

[0058] In this equation, i=1, For i such that ,n Igo am.

[0059] To simplify Equation 7, first, the function constituting the 2n-degree Gaussian-Legendre-Lagrange polynomial basis ( ) can be defined as mathematical formula 8.

[0060] [Mathematical Formula 8]

[0061] , ,

[0062] 2n-degree Gaussian-Legendre-Lagrange polynomial basis Using this, the polynomial h of mathematical equation 7 can be expressed as mathematical equation 9.

[0063] [Mathematical Formula 9]

[0064]

[0065] At this time, a mistake class For , the necessary and sufficient condition for a polynomial h with a degree less than or equal to 2n to be given by Equation 9 is , and (where i is 1 i It satisfies all natural numbers of n, and in this case, h(1)=c_n. Therefore, having n edges (where n is a natural number). Polygons in space and having n points Polygons in space For , a polynomial curve p of degree 2n or less that has as a Gaussian-Legendre polygon and an interpolation polygon, respectively, is a 2n-degree Gaussian-Legendre-Lagrange polynomial basis It is determined as in mathematical formula 5 using .

[0066] Also, the Gaussian-Legendre-Lagrange polynomial basis in the polynomial calculation unit (130) forms a basis for the vector space P_2n of polynomials of degree 2n or less, and has the unit partition property It satisfies the unit partitioning property. The unit partitioning property plays a very important role in the process of generating polynomial curves from control polygons, because if the polynomial bases do not have the unit partitioning property, the polynomial curve generated using them will not maintain its shape and will be deformed as the control polygon is translated.

[0067] The hodograph polygon generation unit (150) uses points input by the user to specify the tangent vector of the curve at the node and determines the hodograph polygon that serves as a sketch in the curve design. Specifically, the hodograph polygon of the curve is a polygon determined by the node and weights and is constructed from the tangent vector information of the curve at the node. For example, defined in the interval [a,b] For a curve p in space, the nodes and weights are respectively class The radiograph polygon of curve p when is the point determined by mathematical equation 10 A polygon with n sides obtained by connecting them in order am.

[0068] [Mathematical Formula 10]

[0069]

[0070] For example, the hodograph polygon generating unit (150) is a polygon having n sides (where n is a natural number). It can generate. Here, the curve p generated based on this is a given polygon About , , that is, it means that mathematical equation 10 must be satisfied. In this case, since p(b)=p_n is satisfied, the curve generated through the hodograph polygon The tangent vector at the point and the position of the curve at both endpoints can be adjusted. At this time, the hodograph polygon generated by the hodograph polygon generator (150) may be a Gauss-Legendre polygon, which is a hodograph polygon using nodes and weights of the Gauss-Legendre numerical quadrature, or a Gauss-Lovato polygon, which is a hodograph polygon using nodes and weights of the Gauss-Lovato numerical quadrature. The hodograph polygon generator (150) can freely adjust the shape of the polygon by utilizing a polynomial curve design device to allow the user to change the position of each point of the hodograph polygon. Ultimately, this allows the hodograph of the curve at the node and the position of the curve at both endpoints to be adjusted.

[0071] The interpolation polygon generation unit (170) uses points input by the user to set points that the curve must pass through and determines the interpolation polygon. Specifically, the interpolation polygon of the curve is a polygon determined by the nodes and is constructed from the position of the curve at the nodes. Specifically, defined in the interval [a,b] For a curve p in space, the node When m of these (where m is m natural number n) node The interpolation polygon for is a polygon with m points is. For example, the interpolation polygon generation unit (170) is generated by the node determination unit (100). n points (where n is a natural number) based on the interpolation condition at A polygon composed of It can generate. Here, point q_i is (where, The curve p generated A point that must be passed, that is This means that it must satisfy [the condition]. The interpolation polygon generation unit (170) can use a polynomial curve design device to set or change the position of the interpolation point that the user must pass through, thereby finely adjusting the curve and designing a curve of the desired shape. In addition, the interpolation polygon generation unit (170) can use a polynomial curve design device to move the interpolation point while the curve is drawn, thereby precisely designing the shape of the entire curve by reflecting the position of the new interpolation point while maintaining the slope at the interpolation point.

[0072] The polynomial curve generation unit (190) generates a polynomial curve using the generated hodograph polygon, the interpolation polygon, and the polynomial basis. For example, the polynomial curve generation unit (190) Let be an n-degree Gaussian-Legendre polynomial basis Any polygon with n sides in space For this, polynomial curves of degree n or less that have this polygon as a Gaussian-Legendre polygon are determined as in Equation 11 (where n is a natural number).

[0073] [Mathematical Formula 11]

[0074]

[0075] At this time, For a polynomial curve p of degree n or less in space, the necessary and sufficient condition for it to be equal to Equation 11 is And, in this case, p(1)=p_n.

[0076] As another example of a polynomial curve generation unit (190) generating a polynomial curve using a generated hodograph polygon, an interpolation polygon, and a polynomial basis, Let be a Gaussian-Legendre-Lagrange polynomial basis of degree 2n, and a curve is generated according to Equation 5. To explain in more detail, it has n edges Polygons in space and n points Polygons in space For , this polygon has a Gaussian-Legendre polygon and an interpolation polygon, respectively. A Gauss-Legendre-Lagrange (GLL) curve p with a spatial degree of 2n or less is generated according to Equation 5. At this time The necessary and sufficient condition for a polynomial curve p with a spatial degree of 2n or less to satisfy Equation 5 is , It satisfies the condition, and when this is satisfied, p(1)=p_n.

[0077]

[0078] FIG. 2 is an example diagram showing the form of a function constituting a Gaussian-Legendre-Lagrange polynomial basis defined according to one embodiment of the present invention.

[0079] Referring to FIG. 2, (a), (b), and (c) of FIG. 2 are polynomial basis functions These are diagrams for showing graphs that change differently depending on the degree of a polynomial basis function according to the value of n (n=1, 2, 3). It can be seen that the polynomial curve design device (10) can generate higher-dimensional polynomials and curves by increasing n according to the user's needs. First, when n=1, a polynomial with a degree of 2 , , The graph of can be represented as (a). When n=2, a polynomial of degree 4 , , , , The graph of can be represented as (b). When n=3, a polynomial of degree 6 , , , , , , The graph of can be represented as (c).

[0080]

[0081] FIG. 3 is an example diagram showing the difference between a Gauss-Legendre curve generated based on a hodograph control polygon according to one embodiment of the present invention and a Gauss-Legendre-Lagrange curve generated based on a hodograph polygon and an interpolation polygon.

[0082] Referring to FIG. 3, FIG. 3(a) represents a Gaussian-Legendre curve generated based on a hordograph control polygon, FIG. 3(b) represents a Gaussian-Legendre-Lagrange curve generated using a hordograph polygon and an interpolation polygon, and FIG. 3(c) is a diagram showing curve (a) and curve (b) together for comparison. Specifically, for points p_0=(-2,0), p_1=(-1,2), p_2=(1,2), and p_3=(2,0), the curve in (a) is in space P is a cubic Gauss-Legendre curve p that has the polygon [p_0p_1p_2p_3] with 3 sides as a Gauss-Legendre polygon. The curve in (b) is a Gauss-Legendre-Lagrange curve of degree 6 that has the polygon [p_0p_1p_2p_3] as a Gauss-Legendre polygon, but has the polygon [q_1q_2q_3] as an interpolation polygon for the points q_1=(-1.3,1), q_2=(0,2), and q_3=(1.3,1). .... Comparing the two curves (a) and (b) in (c), since the two curves share a hodograph polygon with the same three sides, the tangent vector at each node and You can see that it matches. However, the points at each node and is different, and this This is the result of accurately passing through point q_i of the interpolation polygon [q_1q_2q_3] by reflecting the condition.

[0083]

[0084] FIG. 4 is an example diagram showing a change in the shape of a curve that occurs when the control points of a Gauss-Legendre-Lagrange curve are modified according to one embodiment of the present invention.

[0085] Referring to Fig. 4, one can observe how the shape of a Gauss-Legendre-Lagrange curve of degree 6 or less, generated by the Gauss-Legendre polygon [p_0p_1p_2p_3] and the interpolation polygon [q_1q_2q_3], changes when point p_i or point q_j is modified. Fig. 4(a) shows the change in the curve when only the interpolation point q_3 of the interpolation polygon is modified. Fig. 4(a) shows a Gauss-Legendre curve of degree 3 or less having the polygon [p_0p_1p_2p_3] as the Gauss-Legendre polygon. (400) appears. At this time, the interpolation point q_i Let (i=1,2,3), and a Gauss-Legendre-Lagrange curve of degree 6 or less having the polygon [p_0p_1p_2p_3] as a Gauss-Legendre polygon is exactly a curve It is identical to. (a) shows a Gauss-Legendre-Lagrange curve (410) generated by continuously moving only the interpolation point q_3 to the right on this Gauss-Legendre-Lagrange curve (400). When moving the interpolation point q_3 of the Gauss-Legendre-Lagrange curve to the right in (a), the parameter t In the region close to it, the curve shifts to the right, but In regions far from [the point], the curve points hardly change. FIG. 4(b) shows the change in the curve when only the control point p_2 of the Gaussian-Legendre polygon is modified. In (b), the X-point dashed line (430) is the same curve as the Gaussian-Legendre-Lagrange curve (400) in (a), and (b) shows the Gaussian-Legendre-Lagrange curve (450) when only the control point p_2 is moved to points c and d from the curve of the X-point dashed line (430), and the Gaussian-Legendre curve (470) when only the control point p_2 is moved to points c and d from the curve of the X-point dashed line (430). In (b), if one point p_2 of the Gaussian-Legendre polygon is continuously moved to the lower left, in the case of the Gaussian-Legendre-Lagrange curve (450), the parameter t In the range, the curve is deformed to the lower left, but in other ranges, the curve hardly changes. In contrast, the Gauss-Legendre curve (470) changes very severely overall regardless of the range of the parameter t. That is, the polynomial curve design device (10) is designed to modify the control points of the Gauss-Legendre-Lagrange curve so as not to affect the entire curve, but to affect a locally limited range, thereby allowing only specific parts to be modified while maintaining the overall shape of the curve.

[0086]

[0087] FIG. 5 is an exemplary diagram illustrating a method for approximating a rational Bezier curve using a Gauss-Legendre-Lagrange curve according to one embodiment of the present invention.

[0088] Referring to FIG. 5, FIG. 5(a) shows the Bézier curve b and the rational Bézier curve This is a diagram representing. The two curves are defined by the same Bézier control polygon [b_0b_1b_2b_3], which is a rational Bézier curve. is a curve generated with weights (1,1,1.5,1). In (b), the Gaussian-Legendre curve p is a curve generated from the Gaussian-Legendre control polygon of the Bézier curve b, and curve p is exactly the same as curve b. The Gaussian-Legendre-Lagrange curve in (b). has the Gaussian-Legendre control polygon of curve b as a Gaussian-Legendre polygon, A rational Bézier curve is a 6th-order curve that has as an interpolation polygon. is an approximate curve. As can be seen in Fig. 5, the shape of a rational Bézier curve can be approximated using higher-order polynomial curves through the Gauss-Legendre-Lagrange curve. (c) is a rational Bézier curve and the Gauss-Legendre-Lagrange curve This is a drawing that compares shapes by overlaying them, which confirms the high approximation accuracy of the Gauss-Legendre-Lagrange curve. In other words, it can be seen that fine-tuning is easy using the Gauss-Legendre-Lagrange curve.

[0089]

[0090] FIG. 6 is an exemplary diagram illustrating the process of fine-tuning a high-order polynomial curve into a Gaussian-Legendre-Lagrange curve using a polynomial curve design device according to one embodiment of the present invention.

[0091] Referring to FIG. 6, (a) is the Gaussian-Legendre polygon of the curve being designed (b) is a diagram showing a Gauss-Legendre polygon, which has 16 sides. (b) is a diagram showing a Gauss-Legendre curve of degree 16 generated by a Gauss-Legendre polygon. (c) is a diagram showing the Gauss-Legendre polygon and the Gauss-Legendre curve generated therefrom together, showing how the curve follows the shape of the Gauss-Legendre polygon. (d) to (i) are example diagrams showing Gauss-Legendre-Lagrange curves designed using a Gauss-Legendre polygon and an interpolation polygon, showing various forms of curves generated by adjusting the interpolation points while maintaining the Gauss-Legendre polygon. In (d) to (i), the Gauss-Legendre curve of (b) is shown as a dotted line, and each Gauss-Legendre-Lagrange curve modified from it is shown as a solid line to facilitate comparison. The curves indicated by solid lines in (d) through (i) are Gauss-Legendre-Lagrange curves obtained by fine-tuning the Gauss-Legendre curve indicated by dotted line in (b). These curves are generated by fine-tuning them using interpolation points to possess unique detailed features while maintaining a shape similar to the Gauss-Legendre curve. In other words, this example demonstrates that it is easy to fine-tune curves using Gauss-Legendre-Lagrange curves.

[0092]

[0093] FIG. 7 is a drawing for explaining a polynomial curve design method according to one embodiment of the present invention.

[0094] Referring to FIG. 7, in step S710, the polynomial curve design device (10) determines nodes and weights. The polynomial curve design device (10) may use n different nodes (where n is a natural number) given within the domain range of the curve and determine a weight corresponding to each node. Additionally, the polynomial curve design device (10) may determine nodes and weights based on the Gaussian-Legendre numerical quadrature method. The definite integral of f(t) assuming the integration interval is [-1,1] An approximation even in situations where it is difficult to calculate the integral value of function f by calculating it using the Gauss-Legendre numerical quadrature method used to find You can obtain.

[0095] In step S730, the polynomial curve design device (10) can calculate a polynomial basis using the nodes and weights determined in step S710. The polynomial curve design device (10) Let be an n-degree Gaussian-Legendre polynomial basis. can produce, Let be a Gaussian-Legendre-Lagrange polynomial basis of degree 2n. It can produce.

[0096] In step S750, the polynomial curve design device (10) generates a radial polygon using points input by the user and generates an interpolation polygon using points that the curve must pass through. The polynomial curve design device (10) allows the user to freely adjust the shape of the polygon by changing the position of each control point of the radial polygon, and allows the user to fine-tune the curve and design a curve of the desired shape by setting or changing the position of the interpolation point that the user must pass through. Additionally, when the interpolation point is moved while the curve is drawn, the polynomial curve design device (10) can precisely design the shape of the entire curve by reflecting the position of the new interpolation point while maintaining the slope at the interpolation point.

[0097] In step S770, the polynomial curve design device (10) generates a polynomial curve using each control point of the generated radial polygon, the interpolation points of the interpolation polygon, and a polynomial basis. The polynomial curve design device (10) generates a radial polygon having n edges Adjusts the overall shape of the curve and an interpolation polygon with m points A Gauss-Legendre-Lagrange (GLL) curve is generated to pass through the interpolation point. At this time, the polynomial curve design device (10) uses a hodograph polygon to adjust the velocity vector of the curve and interpolates the curve while maintaining the slope.

[0098]

[0099] The polynomial curve design method described above can be implemented as computer-readable code on a computer-readable medium. The computer-readable recording medium may be, for example, a removable recording medium (CD, DVD, Blu-ray disc, USB storage device, removable hard disk) or a fixed recording medium (ROM, RAM, computer-equipped hard disk). The computer program recorded on the computer-readable recording medium can be transmitted to another computing device via a network such as the Internet and installed on the other computing device, thereby allowing it to be used on the other computing device.

[0100] Although it has been described above that all components constituting an embodiment of the present invention are combined or operate as a single unit, the present invention is not necessarily limited to such an embodiment. That is, within the scope of the purpose of the present invention, all components may be selectively combined in one or more ways to operate.

[0101] Although operations are depicted in a specific order in the drawings, it should not be understood that the operations must be executed in the specific order depicted or in a sequential order, or that all depicted operations must be executed to obtain the desired result. In certain situations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various configurations in the embodiments described above should not be understood as a necessary separation, and it should be understood that the described program components and systems can generally be integrated together into a single software product or packaged into multiple software products.

[0102] The present invention has been described above with reference to its embodiments. Those skilled in the art will understand that the present invention may be embodied in modified forms without departing from the essential characteristics of the invention. Therefore, the disclosed embodiments should be considered in an illustrative rather than a restrictive sense. The scope of the invention is defined by the claims, not by the foregoing description, and all variations within the scope of the claims should be interpreted as being included in the invention.

[0103]

[0104] The modes for carrying out the invention are described together in the best mode for carrying out the invention above.

[0105]

[0106] The present invention relates to a polynomial curve design technology, and more specifically, to a polynomial curve design apparatus and method using radial polygons and interpolation polygons. Since it can be used in various ways, it has industrial applicability.

Claims

1. In a polynomial curve design device, Node determination unit for determining a node; A weight determination unit that determines weights; A polynomial calculation unit that calculates a polynomial basis using the above nodes and weights; A radiograph polygon generation unit that generates a radiograph polygon using input points; An interpolation polygon generation unit that uses input points to set points that a curve must pass through and generates interpolation polygons; and A polynomial curve design device comprising a polynomial curve generation unit that generates a curve using the above-mentioned hodograph polygon, interpolation polygon, and polynomial basis.

2. In Paragraph 1, The above node determination unit and the above weight determination unit The definite integral of f(t) assuming the integration interval is [-1,1] A polynomial curve design device using nodes and weights when calculating with the Gauss-Legendre numerical quadrature.

3. In Paragraph 1, The above node determination unit and the above weight determination unit The definite integral of f(t) assuming the integration interval is [-1,1] A polynomial curve design device using nodes and weights when calculating with the Gauss-Lobatto quadrature.

4. In Paragraph 1, The above hodograph polygon generating part A hodograph polygon with n edges (where n is a natural number) A polynomial curve design device that defines the approximate path a curve must take by defining the slope of the curve at a node.

5. In Paragraph 1, The above interpolation polygon generation unit An interpolation polygon consisting of m points (where m is a natural number) A polynomial curve design device that finely adjusts a curve of a desired shape by setting or changing the position of a new interpolation point while maintaining the slope at the interpolation point.

6. In Paragraph 1, The above polynomial curve generation unit A polynomial curve design device for generating a polynomial Gauss-Legendre-Lagrange (GLL) curve of degree 2n or less.

7. In polynomial curve design methods, Step to determine the node; Step of determining weights; A step of calculating a polynomial basis using the above nodes and weights; Step of generating a hodograph polygon using input points; Steps for setting points that the curve must pass through using input points and generating interpolation polygons; and A polynomial curve design method comprising the step of generating a curve using the above-mentioned hodograph polygon, interpolation polygon, and polynomial basis.

8. In Paragraph 7, The step of determining the above node and the step of determining the weight The definite integral of f(t) assuming the integration interval is [-1,1] A polynomial curve design method using nodes and weights when calculating with the Gauss-Legendre quadrature.

9. In Paragraph 7, The step of determining the above node and the step of determining the weight The definite integral of f(t) assuming the integration interval is [-1,1] A polynomial curve design method using nodes and weights when calculating with the Gauss-Lobatto quadrature.

10. In Paragraph 7, The step of generating a hodograph polygon using input points is A hodograph polygon with n edges (where n is a natural number) A polynomial curve design method that defines the approximate path a curve must take by defining the slope of the curve at a node.

11. In Paragraph 7, The step of setting the points that the curve must pass through using the input points above and generating interpolation polygons An interpolation polygon consisting of m points (where m is a natural number) A polynomial curve design method for finely adjusting a curve of a desired shape by setting or changing the position of a new interpolation point while maintaining the slope of the interpolation point.

12. In Paragraph 7, The step of generating a curve using the above hodograph polygon, interpolation polygon, and polynomial basis is A polynomial curve design method for generating a Gauss-Legendre-Lagrange (GLL) curve of degree 2n or less.

13. A computer program recorded on a computer-readable recording medium that executes any one of the polynomial curve design methods of paragraphs 7 through 12.

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