JP2014191487A - Curvature computing device, curvature line writing device, and curvature computing method, and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To accurately determine the curvature of a curvature line.SOLUTION: The curvature computing device includes a curvature computing unit for calculating the curvature of a curvature line by solving a simultaneous equation, on the basis of an equation in which a second-order differential equation related to the arc length when the curvature line is assumed to be a space curve is equal to a second-order differential equation related to the arc length when the curvature line is assumed to be a curve on a curve surface, and an equation obtained by using the arc length as a parameter and differentiating the equation showing the curvature line on the curve surface with respect to the arc length.

Description

  The present invention relates to a curvature calculation device, a curvature line writing device, a curvature calculation method, and a program.

  In shipbuilding, in order to obtain complex curved shapes of the bow and stern parts, a flat member cut out from a steel plate is subjected to cold bending with a press machine, etc., and further heated linearly with a gas torch etc. The method of quenching and bending in the other direction is generally used. The work of bending the steel sheet in this way is called goyo iron, and is a craftsmanship that requires skill.

  On the other hand, in Non-Patent Document 1, the curvature lines, which are two sets of curves on the curved surface that continuously follow the direction of the maximum curvature and the minimum curvature on the curved surface (main direction), are developed base line and goyo iron. We have proposed a method of expanding the curvature line as the construction line in the work. Since the curvature line can be expanded on a plane by geodesic expansion (removing the normal curvature component while maintaining the geodetic curvature and expanding the actual length), in the curvature line expansion method, the expansion process is performed by the curvature line. It is said that strict development can be made without making assumptions.

Kohei Matsuo, 1 other person, “Development of a New Outboard Deployment System that Supports the Manufacture of Curved Outboards for Ships”, Transactions of the Japan Society of Mechanical Engineers (C), November 2010, Vol. 76, No. 771, p. 2797-2802

  In the curvature line expansion method described in Non-Patent Document 1, it is necessary to obtain the curvature of the curvature line when the curvature line is developed on a plane, but Non-Patent Document 1 shows a specific method for obtaining the curvature of the curvature line. It has not been. In order to develop a curvature line on a plane with high accuracy, it is required to accurately obtain the curvature of the curvature line.

  The present invention has been made in view of such circumstances, and an object thereof is to provide a curvature calculation device, a curvature line writing device, a curvature calculation method, and a program capable of accurately obtaining the curvature of the curvature line. It is in.

  The present invention has been made to solve the above-described problems, and a curvature calculation apparatus according to an aspect of the present invention includes a second-order differential equation related to an arc length when a curvature line is regarded as a space curve, and the curvature line. On the basis of the formula obtained by equalizing the second-order differential formula with respect to the arc length as a curve on the curved surface, and the formula obtained by differentiating the formula indicating the curvature line of the curved surface by the arc length with the arc length as a parameter, A curvature calculating unit that solves simultaneous equations and calculates the curvature of the curvature line is provided.

  The curvature calculation apparatus according to an aspect of the present invention is the above-described curvature calculation apparatus, wherein an arc length is calculated by using a differential equation related to an arc length when the curvature line is regarded as a space curve and the curvature line as a curved line. Based on the equation set to be equal to the equation differentiated with respect to and the equation obtained by differentiating the equation of the curved surface of the curved surface by the arc length with the arc length as a parameter, the simultaneous equations are solved and the curvature of the curvature line is solved. A differential operation unit that calculates at least one of the differential, the torsion, and the torsional derivative.

  Further, the curvature line writing apparatus according to one aspect of the present invention performs a second-order differentiation with respect to the arc length using the equation of the second-order differentiation with respect to the arc length when the curvature line is regarded as a space curve and the curvature line as a curve on the curved surface. Calculate the curvature of the curvature line by solving the simultaneous equations based on the expression that is equal to the expression and the expression obtained by differentiating the expression of the curved surface of the curved surface by the arc length with the arc length as a parameter. A curvature calculating unit; and a curvature line writing unit that writes a curvature line on the planar member based on the curvature calculated by the curvature calculating unit.

  Further, the curvature calculation method according to one aspect of the present invention includes a second-order differential equation regarding the arc length when the curvature line is regarded as a space curve, and a second-order differential equation regarding the arc length using the curvature line as a curve on the curved surface. Is calculated based on an equation that is equal to each other and an equation obtained by differentiating the equation representing the curvature line of the curved surface by the arc length using the arc length as a parameter and calculating the curvature of the curvature line by solving simultaneous equations It comprises a step.

  In the program according to one aspect of the present invention, the second-order differential expression regarding the arc length when the curvature line is regarded as a space curve is equal to the second-order differential expression regarding the arc length using the curvature line as a curved line. And a curvature calculation step for calculating the curvature of the curvature line by solving simultaneous equations based on the expression obtained by differentiating the expression representing the curvature line of the curved surface by the arc length with the arc length as a parameter. This is a program to be executed.

  According to the present invention, the curvature of the curvature line can be obtained with high accuracy.

It is a schematic block diagram which shows the function structure of the curvature calculating apparatus in the 1st Embodiment of this invention. In the same embodiment, it is a flowchart which shows the process sequence which a curvature calculating device calculates and outputs a curvature. It is a schematic block diagram which shows the function structure of the curvature calculating apparatus in the 2nd Embodiment of this invention. In the same embodiment, it is a flowchart which shows the process sequence which a curvature calculating device calculates and outputs a curvature.

<First Embodiment>
Embodiments of the present invention will be described below with reference to the drawings.
In the description of the specification, bold notation indicating vectors or matrices is omitted. Further, the “curve” indicates a superordinate concept including a straight line or a line segment.
FIG. 1 is a schematic block diagram showing a functional configuration of a curvature calculation apparatus according to the first embodiment of the present invention. In the figure, the curvature calculation device 100 includes a curved surface data acquisition unit 110, a curvature line acquisition unit 120, a curvature calculation unit 130, and a calculation result output unit 190.

  The curvature calculating apparatus 100 acquires a curved line of curvature (Line Of Curvature or Curvature Line) and calculates the curvature of the curved line. Here, the curvature line is a line unique to the curved surface, and includes a maximum curvature line and a minimum curvature line. The maximum curvature line is a curve obtained by continuously following the direction in which the curvature of a cross section (normal curvature) is maximum on a curved surface. The minimum curvature line is a curve obtained by continuously following the direction in which the curvature of the cross section is minimum on the curved surface.

For example, when the curvature calculation device 100 calculates and displays a desired curved surface curvature line and the curvature at the curvature line when the steel plate is loaded, the user of the curvature calculation device 100 develops the curvature line on the steel plate. It can be used as a reference for Tekyo iron.
However, the application range of the curvature calculation apparatus 100 is not limited to go iron. For example, the curvature calculation device 100 can be used for various purposes that refer to curvature, such as evaluation of a curved surface shape.

The curved surface data acquisition unit 110 acquires data indicating a curved surface (hereinafter referred to as “curved surface data”) such as CAD (Computer Aided Design) data.
The curvature line acquisition unit 120 acquires the curvature line of the curved surface indicated by the curved surface data acquired by the curved surface data acquisition unit 110. For example, the curvature line acquisition unit 120 calculates the coordinates of points constituting the curvature line from the curved surface data acquired by the curved surface data acquisition unit 110. However, the method by which the curvature line acquisition unit 120 acquires the curvature line is not limited to this, and data indicating the curvature line calculated by another device may be acquired.
The curvature calculation unit 130 calculates the curvature of the curvature line acquired by the curvature line acquisition unit 120.

The calculation result output unit 190 outputs the curvature calculated by the curvature calculation unit 130. For example, the calculation result output unit 190 has a display screen such as a liquid crystal panel, displays the curvature line acquired by the curvature line acquisition unit 120 on a diagram showing a desired curved surface, and further calculated by the curvature calculation unit 130. The curvature is displayed as numerical data or the length of an arrow.
However, the method by which the calculation result output unit 190 outputs the calculation result is not limited to the method of displaying the calculation result on the screen. For example, the calculation result may be output by a method other than the screen display, such as the calculation result output unit 190 transmitting the calculation result to another device.

Next, the calculation of the curvature line performed by the curvature line acquisition unit 120 and the calculation of the curvature performed by the curvature calculation unit 130 will be described.
First, a curvature line can be grasped as a space curve. The space curve here is a curve included in a three-dimensional space. The following properties are obtained for the spatial curve.
The curve (spatial curve) is expressed as c (s) = (x (s), y (s), z (s)) with the arc length (path on the curve) s as a parameter.

Further, the unit tangent vector (Unit Tangent Vector) of the curve c (s) is t, the main normal vector (Principal Normal Vector) is n, and the subnormal vector (Binormal Vector) is b. t, n, and b form a right-handed orthonormal basis called a Frenet frame in this order.
Further, the relationship of equation (1) is established, where κ is the curvature.

However, prime (') shows the differentiation with respect to the arc length s.
Further, the relationship of the formula (2) is established, where τ is the torsion.

  Furthermore, the relationship of Formula (3) is established.

Formulas (1) to (3) represent the Frenetele formula (Frenet-Serret Formulas).
Further, the relationship shown in the equation (4) holds for the first-order differential c ′ (s) of the curve c (s).

  Further, the relationship shown in the equation (5) holds for the second-order differential c ″ (s) of the curve c (s).

Here, k represents a curvature vector.
Equation (5) is differentiated by arc length s to obtain equation (6).

  Moreover, Formula (7) can be obtained from Formula (5).

However, “·” indicates an inner product. In addition, “(s)” is omitted from c ″.
Further, when n ′ in the formula (6) is replaced by using the above-mentioned Fresnelee formula second formula (formula (2)), formula (8) is obtained.

Differentiating equation (8) and substituting t ', n' and b 'using the Fresnel's formula, the fourth-order derivative c ⁽⁴⁾ (s) of c (s) becomes as shown in equation (9) Indicated.

Here, C ₄ ^t , C ₄ ⁿ , and C ₄ ^b are each represented by the formula (10).

  In general, the m-th order derivative of c (s) is expressed as in Expression (11).

Here, C _m ^t , C _m ⁿ , and C _m ^b are each composed of terms including κ, τ, and derivatives thereof.
On the other hand, the curvature line is a curve obtained by continuously following the direction in which the curvature of the cross section is maximum or minimum on the curved surface, so it is also possible to grasp the curvature line as a curve on the curved surface (Curve On Surface) .
First, using u and v (0 ≦ u ≦ 1, 0 ≦ v ≦ 1) as parameters, when the point (x, y, z) moves in a rectangular region on the uv plane, it is expressed as in equation (12). .

  A unit surface normal vector N of the curved surface is expressed as in Expression (13).

Here, R _u represents a partial differential with respect to u, and R _v represents a partial differential with respect to v. “×” indicates an outer product, and “||” indicates a norm of the vector.
Further, the relationship between the unit tangent vector t of the curve c (s) included in the curved surface R (u, v) and the unit normal vector n of the curve is expressed as shown in Expression (14).

Here, as described above, s is an arc length, and k is a curvature vector of the curve c (s). Also, _{k n} is Hokyokuritsu shows vectors (Normal Curvature Vector), _{k g} denotes geodesic curvature vectors (Geodesic Curvature Vector). Normal curvature vector k _n is the curvature vector k, which is a component of a direction perpendicular to the curved surface. The geodesic curvature vector k _g is the U direction component of the curvature vector k. However, it is defined as U = N × t (“×” indicates an outer product).
Further, κ _n represents a normal curvature, and κ _g represents a geodesic curvature.
The normal curvature κ _{n at the} point P is expressed as shown in Equation (15).

Here, λ = dv / du indicates the tangential direction of the curve c (s) at the point P. E, F, and G indicate coefficients in the first basic form (First Fundamental Form), and L, M, and N indicate coefficients in the second basic form (Second Fundamental Form).
The extreme value (Extreme Value) of the normal curvature κ _n is obtained by setting dκ _n / dλ = 0 in Equation (15), and is expressed as Equation (16).

  Expression (17) is obtained from Expression (15) and Expression (16).

Therefore, the extreme value of the normal curvature κ _n satisfies the simultaneous equations shown in the equation (18).

  The simultaneous equations of equation (18) are homogeneous (Homogeneous) linear equations with respect to du and dv, and the necessary and sufficient condition that the simultaneous equations have a non-trivial solution (Nontrivial Solution) is As shown.

  Here, det || represents a determinant. Alternatively, Expression (20) can be obtained from Expression (19).

When the Gaussian curvature is represented by K and the mean curvature is represented by H, Equation (20), which is a quadratic equation of the normal curvature κ _n , is expressed as Equation (21).

  Solving equation (21) yields the solution shown in equation (22).

Here, κ ₁ indicates a maximum principal curvature (Maximum Principal Curvature), and κ ₂ indicates a minimum principal curvature (Minimum Principal Curvature). The direction in which the normal curvature κ _n takes the maximum value or the minimum value on the tangential plane is called a principal direction.

  A curve on the curved surface is expressed as c (s) = R (u (s), v (s)), and a first order derivative c ′ (s) of the curve c (s) is used by using a chain rule. Is shown as in equation (23).

  Further, the second-order differential c ″ (s) of the curve c (s) is expressed as shown in Expression (24).

However, subscript u of R, such as R _uu and R _uv , indicates a partial differential with respect to u, and subscript v of R indicates a partial differential with respect to v.
Further, the third-order differential c ′ ″ (s) of the curve c (s) is expressed as shown in the equation (25).

Further, the fourth-order derivative c ⁽⁴⁾ (s) of the curve c (s) is expressed as in Expression (26).

In general, the m-th order derivative c ^(m) (s) of the curve c (s) is expressed as in Expression (27).

Here, alpha _m is, u ', u'', ···, u (m-1) and v', v '', ··· , v the total sum of terms containing ^(m-1) Show.

Any principal curvature direction vector satisfies the above-described equation (18). Therefore, from the first equation of equation (18), equation (28) is obtained when the normal curvature κ _n is either the main curvature κ ₁ or κ ₂ .

  Here, η is a non-zero constant and is determined by normalization of the first basic form shown in Equation (29).

Equation (29) shows normalization with an arc length of 1.
Expression (30) is obtained from Expression (29) and Expression (28).

  Since the main curvature direction vector also satisfies the second equation of equation (18), equation (31) is obtained.

  Here, μ is a non-zero constant, and is expressed as in Expression (32) as in the case of η.

  The curvature line can be grasped as a curve included in the curved surface or as a space curve. Therefore, Expression (33) can be obtained from Expression (5) and Expression (24).

However, (alpha) ₂ is shown like Formula (34).

The values of u ′ and v ′ are obtained from Expression (28) or Expression (31).
Equation (35) is obtained by _applying the inner product of Ru to both sides of equation (33).

Further, the resulting equation (36) is reacted with the inner product of both sides to _{R v} of formula (33).

The expression (35) and formula (36), u '', v 'contains three unknowns' and kappa _g. Therefore, the third equation is required to solve the simultaneous equations. This equation is obtained by differentiating equation (18) as equation (37).

Here, β ₁ and β (bar) ₁ are expressed as in Equation (38).

  The simultaneous equations shown in the equation (39) can be obtained from the first equation of the equations (35), (36), and (37).

  Further, simultaneous equations shown in the equation (40) can be obtained from the second equation of the equations (35), (36), and (37).

| L + κE | ≧ | N + κG | case, by solving equation (39), resulting geodesic curvature kappa _g. Otherwise By solving equation (40), resulting geodesic curvature kappa _g.
As described above, Expression (39) and Expression (40) are obtained based on Expression (33) and Expression (37). Expression (33) is a second-order differential expression (expression (5)) regarding the arc length when the curvature line is regarded as a space curve, and an expression obtained by second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface. It is obtained when (Equation (24)) is equal. The expression (37) corresponds to an example of an expression obtained by differentiating an expression indicating a curved surface curve with the arc length using the arc length as a parameter.

The curvature line acquisition unit 120 calculates u by integrating u ′ shown in Expression (28) or Expression (31), and calculates v by integrating v ′.
For example, the curvature line acquisition unit 120 sets initial positions at equal intervals on the boundary of the curved surface, and uses parameters such as the Runge-Kutta method from each initial position to indicate the points included in the curvature line passing through the initial position. u and v are obtained in order. By specifying the parameters u and v, the curvature line acquisition unit 120 specifies the point R (u, v) included in the curvature line, and acquires the curvature line by connecting the obtained points.
The curvature line acquisition unit 120 acquires a curvature line for each of κ ₁ and κ ₂ . maximum curvature line is obtained by kappa _1, the minimum curvature ray by kappa ₂ is obtained.

The curvature calculation unit 130 solves the simultaneous equations shown in the equation (39) or the equation (40) for each point of the curvature line specified by the curvature line acquisition unit 120 (a point included in the curvature line), and at each point, to calculate the geodesic curvature κ _g.

Next, the operation of the curvature calculation apparatus 100 will be described with reference to FIG.
FIG. 2 is a flowchart showing a processing procedure in which the curvature calculation device 100 calculates and outputs the curvature. When the curvature calculation device 100 receives a user operation for instructing calculation of the curvature, the curvature calculation device 100 performs the processing shown in FIG.
In the process of FIG. 2, first, the curved surface data acquisition unit 110 acquires curved surface data (step S101). Next, the curvature line acquisition unit 120 acquires the curvature line of the curved surface indicated by the curved surface data acquired by the curved surface data acquisition unit 110 (step S102). Moreover, the curvature calculating part 130 calculates the curvature in the curvature line which the curvature line acquisition part 120 acquired (step S103). Then, the calculation result output unit 190 outputs the curvature calculated by the curvature calculation unit 130 (step S104).

Here, the Hakachi line curvature kappa _g of curves included in a developable curved (Developed Surface, curved deployable just a plane bending without expansion and contraction), the curvature of the curve projecting in the tangent plane (Tangent Plane) kappa Are known to be equal. Therefore, the user of the curvature calculation device 100 (hereinafter, the user of the curvature calculation device is simply referred to as “user”) can obtain the geodesic curvature κ _g of the curvature line on the curved surface by developing the curvature line in a plane. By using it as the curvature of the curve, the curvature line can be developed on a plane.

Specifically, the user first sets an initial position at a position corresponding to the initial position of the curvature line set by the curvature line acquisition unit 120 at the end of the flat member cut out from the steel plate. The user can obtain a curvature line by following the planar member in the direction indicated by the curvature displayed by the calculation result output unit 190.
And the processor who performs go iron can perform cold bending along the minimum curvature line, and can perform bending by linear heating along the maximum curvature line. Since the minimum curvature line and the maximum curvature line are perpendicular to each other except for a singular point (Singular Point), the operator performs a cold bending process along the minimum curvature line, so that the maximum curvature line direction orthogonal to the minimum curvature line is obtained. Large bending can be performed by cold bending. And a processor should just perform the small bending of the minimum curvature line direction orthogonal to a maximum curvature line by the bending process by the linear heating performed along a maximum curvature line.
In addition, when the bending process is performed, the processor may generate a bend corresponding to the curvature with reference to the curvature calculated by the curvature calculation device 100.

Alternatively, the curvature calculation device 100 (for example, the curvature calculation unit 130) may develop the curvature line on a plane. For example, the calculation result output unit 190 may superimpose and display the flattened curvature line and the curvature calculated by the curvature calculation unit 130 on the diagram showing the planar member. In this case, the user should just copy the curvature line which the calculation result output part 190 shows on a plane member as it is. The burden on the user can be reduced in that the user does not have to develop the curvature line on a plane.
Furthermore, the curvature calculation device 100 may be configured as a curvature line writing device including a curvature line writing unit that writes a curvature line on a planar member based on the curvature calculated by the curvature calculation unit 130. In this case, the user does not need to write a curvature line on the planar member, and the burden on the user can be further reduced in this respect.
After step S104, the process of FIG.

As described above, in the curvature calculation unit 130, the expression of the second-order differentiation with respect to the arc length when the curvature line is regarded as a space curve is equal to the expression of second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface. And the equation obtained by differentiating the expression of the curved surface of the curved line with the arc length as a parameter, the simultaneous equation is solved to calculate the curvature of the curvature line.
Thereby, the curvature calculating part 130 can calculate a curvature at the time of seeing a curvature line as a space curve. The calculation result output unit 190 displays the curvature calculated by the curvature calculation unit 130, so that the user can develop the curvature line on a plane as described above. And the processor who performs go iron can perform cold bending along the minimum curvature line, and can perform bending by linear heating along the maximum curvature line.

<Second Embodiment>
FIG. 3 is a schematic block diagram showing a functional configuration of the curvature calculation apparatus according to the second embodiment of the present invention. In the figure, a curvature calculation device 200 includes a curved surface data acquisition unit 110, a curvature line acquisition unit 120, a curvature calculation unit 130, a differential calculation unit 240, and a calculation result output unit 290.
In the same figure, portions having the same functions corresponding to the respective portions in FIG. 1 are denoted by the same reference numerals (110, 120, 130), and description thereof is omitted.

The differential calculation unit 240 calculates at least one of the differentiation of the curvature of the curvature line, the torsion, or the torsional derivative. The differentiation calculated by the differential operation unit 240 may be a first-order differentiation or a second-order or higher differentiation.
Similar to the calculation result output unit 190 (FIG. 1), the calculation result output unit 290 outputs the curvature calculated by the curvature calculation unit 130. In addition, the calculation result output unit 290 outputs the curvature differentiation, the torsion, and the torsional differentiation calculated by the differentiation calculation unit 240.
As in the case of the calculation result output unit 190, the calculation result output unit 290 may display the calculation result (curvature, derivative of curvature, or derivative of torsion) on the screen, or the calculation result on another device. The calculation result may be output by a method other than the screen display, such as transmitting.

Next, the differentiation of curvature and torsion performed by the differential calculation unit 240 will be described.
As in the case of the curvature described above with reference to Expression (33) to Expression (40), Expression (41) is obtained from Expression (8) and Expression (25).

Here, C ₃ ^t , C ₃ ⁿ , C ₃ ^b, and α ₃ are expressed as in Expression (42).

Equation (43) is obtained by _applying the inner product of Ru to both sides of equation (41).

Further, the resulting equation (44) is reacted with the inner product of both sides to _{R v} of formula (41).

  Further, the inner product of n is allowed to act on both sides of the equation (41) to obtain the equation (45).

  Equations (43), (44), and (45) contain four unknowns u ″ ″, v ″ ″, κ ′, and τ. By performing second order differentiation on the equation (18), the equation (46) is obtained as the fourth equation.

Here, β ₂ and β (bar) ₂ are expressed as in Equation (47).

  From the equations (43), (44), (45) and (46), the simultaneous equations shown in equation (48) can be obtained.

  Further, simultaneous equations shown in the equation (49) can be obtained from the second equation of the equations (43), (44), (45), and (46).

  In the case of | L + κE | ≧ | N + κG |, the curvature κ ′ and the torsion τ can be obtained by solving the equation (48). In other cases, the curvature κ ′ and the torsion τ can be obtained by solving the equation (49).

Next, the equations (41) to (49) are generalized in the case of the m-th order derivative c ^(m) (s) of the curve c (s).
Expression (50) is obtained from Expression (11) and Expression (27).

Also, from the equation (11), it can be seen the coefficient of the highest order differential of the curvature κ in c ^{(m) (s)} is 1 exists in terms of C _m ^n. On the other hand, it can be seen the coefficient of the highest order differential of torsion of a curve τ in c ^{(m) (s)} is present in the section of C _m ^b kappa.
Thus, defining the coefficients C _(bar) ^{m n} by the equation (51).

Also, define the coefficients C _(bar) ^{m b} as in equation (52).

Sides is reacted with the inner product of _{R u} of formula (50), obtained from equation (51) and (52) Equation (53).

Further, both sides is reacted with the inner product of _{R v} of formula (50), obtained from equation (51) and (52) Equation (54).

  Further, an inner product of n is allowed to act on both sides of the equation (50), and the equation (55) is obtained from the equations (51) and (52).

Expressions (53), (54), and (55) include four unknowns u ^(m) , v ^(m) , κ ^(m−2), and τ ^(m−3) . By performing (m−1) th order differentiation on Expression (18), Expression (56) is obtained as the fourth equation.

Here, β _m−1 and β (bar) _m−1 are expressed as in Equation (57).

  From the equations (53), (54), (55) and (56), the simultaneous equations shown in equation (58) can be obtained.

  The simultaneous equations shown in the equation (59) can be obtained from the second equation of the equations (53), (54), (55) and (56).

In the case of | L + κE | ≧ | N + κG |, the curvature κ ^(m−2) and the torsional differential τ ^(m−3) can be obtained by solving the equation (58). In other cases, the curvature derivative κ ^(m−2) and the torsional derivative τ ^(m−3) can be obtained by solving the equation (59).
As described above, Expression (58) and Expression (59) are obtained based on Expression (50) and Expression (56). Expression (50) is an expression (expression (11)) of a derivative (m-order derivative) related to the arc length when the curvature line is regarded as a space curve, and a derivative ((11)) of the curvature line as a curve on the curved surface ( It is obtained when the equation (formula (27)) obtained by m-order differentiation is equal. Expression (56) corresponds to an example of an expression obtained by differentiating an expression indicating a curvature line of a curved surface by the arc length using the arc length as a parameter.
The differential calculation unit 240 solves the equation (58) or the equation (59) to calculate the derivative of the curvature κ, the torsion τ or the derivative thereof.

Next, the operation of the curvature calculation apparatus 200 will be described with reference to FIG.
FIG. 4 is a flowchart showing a processing procedure in which the curvature calculation device 200 calculates and outputs the curvature. When the curvature calculation device 200 receives a user operation for instructing calculation of curvature, the curvature calculation device 200 performs the processing shown in FIG.
Steps S201 to S203 in FIG. 4 are the same as steps S101 to S103 in FIG.

After step S203, the differential operation unit 240 calculates at least one of the differentiation of the curvature of the curvature line, the torsion, or the torsional differentiation (step S204).
Then, the calculation result output unit 290 outputs the curvature calculated by the curvature calculation unit 130, the curvature differentiation calculated by the differentiation calculation unit 240, the torsion, and the torsional differentiation (step S205).
Thereafter, the process of FIG. 4 is terminated.

As described above, the differential calculation unit 240 is an expression in which the differential expression related to the arc length when the curvature line is regarded as a space curve and the expression obtained by differentiating the curvature line as a curve on the curved surface are equal. And the equation obtained by differentiating the curvature line of the curved surface with the arc length as a parameter, and the equation obtained by differentiating the simultaneous equations, the curvature of the curvature line, the torsion or the torsional derivative at least Either one is calculated.
Thereby, the differential calculation part 240 can calculate the differential of the curvature of the curvature line, the torsion, and the torsional derivative. The calculation result output unit 290 displays the calculation result of the differential calculation unit 240, so that the user knows the maximum curvature on the curvature line, where the twist is, and the like, and more appropriately sets the curvature line to a plane. Can be deployed. For example, the user knows the maximum value of κ, and draws the maximum curvature line at the largest bend, such as the maximum curvature line and the maximum curvature line that can be drawn in various ways. And the minimum curvature line can be selected.

A program for realizing all or part of the functions of the curvature calculation devices 100 and 200 is recorded on a computer-readable recording medium, and the program recorded on the recording medium is read into a computer system and executed. Thus, the processing of each unit may be performed. Here, the “computer system” includes an OS and hardware such as peripheral devices.
Further, the “computer system” includes a homepage providing environment (or display environment) if a WWW system is used.
The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Furthermore, the “computer-readable recording medium” dynamically holds a program for a short time like a communication line when transmitting a program via a network such as the Internet or a communication line such as a telephone line. In this case, a volatile memory in a computer system serving as a server or a client in that case, and a program that holds a program for a certain period of time are also included. The program may be a program for realizing a part of the functions described above, and may be a program capable of realizing the functions described above in combination with a program already recorded in a computer system.

  The embodiment of the present invention has been described in detail with reference to the drawings. However, the specific configuration is not limited to this embodiment, and includes design changes and the like without departing from the gist of the present invention.

100, 200 Curvature calculation device 110 Curved surface data acquisition unit 120 Curvature line acquisition unit 130 Curvature calculation unit 190, 290 Calculation result output unit 240 Differential calculation unit

Claims (5)

  When the curvature line is regarded as a space curve, the second-order differential expression is the same as the expression obtained by second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface, and the arc length as a parameter. A curvature calculation device comprising: a curvature calculation unit that calculates a curvature of a curvature line by solving simultaneous equations based on an expression obtained by differentiating an expression indicating a curvature line of the curved surface by an arc length.   The equation of differentiation regarding the arc length when the curvature line is regarded as a space curve and the equation obtained by differentiating the curvature line with respect to the arc length as a curve on the curved surface, and the curvature of the curved surface with the arc length as a parameter Based on the equation obtained by differentiating the equation representing the line by the arc length, a differential operation unit that solves the simultaneous equations and calculates at least one of the derivative of the curvature of the curvature line, the torsion or the torsion of the torsion is provided. The curvature calculation apparatus according to claim 1, wherein: When the curvature line is regarded as a space curve, the second-order differential expression is the same as the expression obtained by second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface, and the arc length as a parameter. Based on an equation obtained by differentiating an equation indicating the curvature line of the curved surface with an arc length, a curvature calculating unit that calculates the curvature of the curvature line by solving simultaneous equations;
A curvature line writing unit that writes a curvature line on a planar member based on the curvature calculated by the curvature calculation unit;
A curvature line writing apparatus comprising: A curvature calculation method of a curvature calculation device,
When the curvature line is regarded as a space curve, the second-order differential expression is the same as the expression obtained by second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface, and the arc length as a parameter. A curvature calculation method comprising: a curvature calculation step of calculating a curvature of a curvature line by solving simultaneous equations based on an expression obtained by differentiating an expression indicating the curvature line of the curved surface with an arc length. In a computer as a curvature calculator,
When the curvature line is regarded as a space curve, the second-order differential expression is the same as the expression obtained by second-order differentiation with respect to the arc length using the curvature line as a curve on the curved surface, and the arc length as a parameter. A program for executing a curvature calculation step of solving a simultaneous equation and calculating a curvature of a curvature line based on an expression obtained by differentiating an expression indicating a curvature line of the curved surface with an arc length.

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