JP4952322B2 - Curved surface generator - Google Patents

Description

  The present invention relates to a curved surface generation apparatus that is used in a CAD system or the like and generates a curved surface in a predetermined area.

  Conventionally, a CAD system is often used for the design of various structures, and various developments have been made on curved surface generation devices that are used in this CAD system to express the complex shape of the structure. ing.

  For example, Patent Document 1 discloses an apparatus for generating a curved surface in a predetermined region including a plurality of patches surrounded by a plurality of mesh lines by generating a Coons curved surface for each patch. Yes. Specifically, the transverse vector inside the patch is calculated by a cubic Bezier function equation using the tangent vector at each corner of the patch. In this cubic Bezier function equation, as a boundary condition, the differential value of the transverse vector at each corner, that is, the value of the twist vector at each corner is all set to zero.

According to this apparatus, since the values of the twist vectors at each corner are all 0 as described above, the transverse vector does not change between adjacent patches at each corner. The curved surface can be prevented from being distorted.
JP-A-11-126262

  However, if the value of the twist vector at each corner is set to 0 as in the above-described apparatus, distortion may occur in a portion other than the connection portion of each patch, that is, in each patch. That is, in the device, even when the crossing vector of the connection portion between the patches is changed on the original curved surface, the value of the twist vector, that is, the amount of change of the crossing vector is set to 0, so The curved surface cannot be correctly expressed, and the effect appears inside the patch.

  An example in which a curved surface is generated with the value of the twist vector set to 0 as described above is shown in FIG. In this figure, a region for generating a curved surface is constituted by patches P1 to P6 surrounded by mesh lines m1 to m4 and n1 to n3. As shown in this figure, when the twist vector is 0, a smooth curved surface is generated at the patch connection portion, but there is a problem that distortion occurs in each patch, for example, P2 and P3.

  In view of such circumstances, an object of the present invention is to provide a curved surface generating apparatus that can more reliably generate a smooth curved surface over a predetermined region.

The invention according to claim 1 for solving the above-described problem is a curved surface generation apparatus that generates a curved surface in an area composed of one or more patches surrounded by a plurality of mesh lines in the u direction and a plurality of mesh lines in the v direction. The tangent vector in the u direction at each corner of the patch is calculated based on the shape of the mesh line in the u direction, and the tangent vector in the v direction at each corner is calculated based on the shape of the mesh line in the v direction. Based on the tangent vector calculation means to be calculated and the tangent vector in the u direction calculated at each corner, the trajectory along the mesh line in the v direction of at least one of the end point or the start point of the tangent vector in the u direction is estimated. In addition, based on the tangent vector in the v direction calculated at each corner, the v direction A tangential vector trajectory estimating means for estimating a trajectory along a mesh line in the u direction at least one of an end point or a start point of the line vector; a trajectory along the v direction mesh line of the estimated u tangent vector; Twist vector calculation means for calculating a twist vector at each corner based on a locus along the mesh line in the u direction of the tangent vector, the tangent vector in the u direction, the tangent vector in the v direction, and the twist vector at each corner. based on the bets, possess a curved surface calculating means for calculating the curved surface of the region, the twist vector calculating means, the tangent vector locus v direction of mesh lines along the tangent vector of the estimated u direction in the estimation means V-direction twist at each corner based on the trajectory of A uv direction twist vector calculating means for calculating a vector and calculating a twist vector in the u direction at each corner based on a locus along the mesh line in the u direction of the estimated tangent vector in the v direction; And a uv direction averaging means for calculating a uv average twist vector by averaging the twist vector in the u direction and the twist vector in the v direction at each corner, and calculating the uv average twist vector based on the uv average twist vector. A curved surface generation apparatus characterized by calculating a twist vector at a corner is provided.

  According to this apparatus, the twist vector calculating means calculates a twist vector based on the shape of the mesh line and the tangent vector at each corner of the patch, and the curved surface calculating means based on the twist vector and the tangent vector. Therefore, it is possible to calculate a smooth curved surface reflecting the characteristics of the mesh line, that is, the characteristics of the curved surface derived from the mesh line. That is, since the twist vector is given as a control amount of the curved surface without forcibly setting the twist vector to 0 as in the prior art, it is possible to suppress the occurrence of distortion on the generated curved surface. .

  Specifically, the tangent vector calculation unit calculates a tangent vector in the uv direction at each corner based on the shape of the mesh line, and the tangent vector locus estimation unit calculates the tangent vector in the uv direction at each corner. The trajectory along the mesh line at the start point or the end point is estimated based on the tangent vector along the mesh line in each direction, and the shape of each mesh line can be reflected in the trajectory of the tangent vector. Then, since the twist vector in the u direction and the twist vector in the v direction at each corner are calculated based on the estimated locus of the tangent vector, the shape of the mesh line is also reflected in the twist vector in each direction. It becomes possible. Accordingly, since the twist vector reflecting the shape of the mesh line and the entire shape of the curved surface can be calculated, the smooth curved surface can be calculated more accurately by calculating the curved surface based on the twist vector. It becomes.

Further, according to this apparatus, the uv direction average twist vector that equally reflects the u direction twist vector and the v direction twist vector is calculated at each corner, and each corner is calculated based on the uv direction average twist vector. Since the twist vector is calculated, it is possible to obtain a twist vector that matches any mesh line shape in the uv direction.

Further, in the present invention, the twist vector calculating means may calculate each of the corners when the region includes a plurality of patches and the predetermined corners of the patches are adjacent to each other around a predetermined intersection of the mesh lines. by the curved surface at said intersection point around the uv average twist vector calculated respectively mutually interpolated to a smooth shape for, preferably calculates the twist vector at each corner (claim 2).

  According to this configuration, the uv-direction average twist vector calculated for each patch with respect to a predetermined corner is interpolated so that the curved surface around the intersection has a smooth shape. The connection between patches can be made smooth while maintaining the characteristic of the curved surface in each patch while maintaining the value.

Also, in the present invention, the twist vector calculation means calculates the uv average twist vector calculated for each corner adjacent to each other around the predetermined intersection, respectively, as the u-direction component, the v-direction component, and the perpendicularity. A normalizing means for decomposing the component into direction components, and converting these component values into values per unit tangent vector based on the values of the tangent vector in the u direction and the tangent vector in the v direction at each corner; Each component value per unit tangent vector calculated for each corner is averaged between the corners for each component to calculate a unit twist vector, and this unit twist vector is converted to a tangent vector in the u direction at each corner and v By correcting based on the value of the tangent vector in the direction, the twist base at each corner Preferably it has a correction means for calculating the torque, respectively (claim 3).

  In this configuration, the normalization unit converts each component value of the uv direction average twist vector into a unit tangent vector value, thereby extracting the slope of the uv direction average twist vector of each corner and supplying the correction unit to the correction unit. Then, the inclination is averaged between the corners, and the unit twist vector having the averaged inclination is corrected in accordance with the values of the tangent vector in the u direction and the tangent vector in the v direction at each corner. Is calculated as a vector that has the same inclination and reflects the shape of each patch. In other words, according to this configuration, since the inclination of the twist vector at a corner around a predetermined intersection can be made constant while reflecting the characteristics of each patch, a smooth curved surface is generated at the connection portion of the patch around the intersection. It becomes possible to do. In addition, as described above, the values of the tangent vector in the u direction and the tangent vector in the v direction at each corner are reflected in the twist vector at each corner, so that distortion of the curved surface inside each patch can be more reliably suppressed. It becomes possible. That is, according to the present configuration, it is possible to generate a curved surface in which the patch connection portion and the shape inside the patch are smooth while maintaining the characteristics of the curved surface.

In the present invention, a twist vector at a predetermined point on the curved surface is calculated based on the curved surface calculated by the curved surface calculating means, and at least one of the magnitude of the twist vector or the amount of change of the twist vector is calculated in advance. comparing the set predetermined value, the curved surface on the basis of the comparison result preferably has a judging means for judging whether a predetermined shape (claim 4).

  In this way, since it is determined whether the curved surface has a predetermined shape based on the magnitude or amount of the twist vector representing the change in shape of the curved surface, it is possible to more easily determine the shape of the curved surface. It becomes possible to do.

Further, in the present invention, it is preferable to have a notifying means for notifying that when the determining means determines that the curved surface does not have a predetermined shape (Claim 5 ).

  In this way, the user can easily recognize whether or not the curved surface has a predetermined shape by the notification means, and convenience is improved.

In the present invention, the determination unit calculates the magnitude of the twist vector or the amount of change of the twist vector in the local coordinate system in the u direction and the v direction, and the magnitude of the twist vector on the local coordinate. It is preferable to compare the height and the amount of change with a predetermined value (claim 6 ).

  In this way, the validity of the curved surface shape can be determined based on the magnitude or amount of change of the twist vector with respect to the surface surrounded by the mesh line, so that the smoothness of the curved surface can be determined more accurately. Is possible.

  As described above, according to the present invention, it is possible to provide a curved surface generation apparatus that can generate a smooth curve by a region surrounded by a predetermined boundary line.

  Hereinafter, preferred embodiments of the present invention will be described with reference to the drawings.

  FIG. 5 is a schematic block diagram of the curved surface generation apparatus according to the embodiment of the present invention, and FIGS. 2 to 4 are flowcharts showing calculation procedures in the curved surface generation apparatus. 6 to 19 are diagrams for explaining a specific calculation method of the curved surface generation apparatus.

  The curved surface generation apparatus 10 is used in a CAD system driven on a computer (not shown). That is, the curved surface generation apparatus 10 executes various calculations by a CPU in the computer based on an operation of a keyboard or a mouse connected to the computer, and outputs the calculation results to a display or the like connected to the computer. It is comprised so that.

  As shown in FIG. 5, the curved surface generation apparatus 10 includes a mesh line storage unit 20, a tangent vector calculation unit 30, a tangent vector locus estimation unit 40, a twist vector calculation unit 50, a curved surface calculation unit 60, a determination unit 70, and a notification unit 80. It has. The curved surface generation apparatus 10 includes a plurality of u-direction mesh lines Mu [n] (u) (n = 0 to n0) and a plurality of v-direction mesh lines Mv [m] as shown in FIG. (V) For generating a curved surface in a region composed of a plurality of patches P [m, n] respectively surrounded by (m = 0 to m0). In this embodiment, n0 = 3 and m0 = 4.

  The mesh line storage means 20 is for storing the mesh line Mu [n] (u) in the u direction and the mesh line Mv [m] (v) in the v direction. A function of each mesh line calculated in advance is stored. Here, each mesh line is obtained in advance along the curved surface generated in the region.

  The mesh lines Mu [n] (u) and Mv [m] (v) are given by interval polynomials U (u) and V (v) in the u and v directions, respectively, and each patch P [m, n], as shown in FIG. 6B, mesh lines U [m, n] (u) and U [m, n + 1] (u) in the u direction and mesh line V [m in the v direction, respectively. , N] (v) and V [m + 1, n] (v). Each patch P [m, n] has four corners C1 [m, n], C2 [m, n], C3 [m, n], and C4 [m, n]. The intersections of the mesh lines at the corners C1 to C4 are denoted as Q [m, n], Q [m + 1, n], Q [m, n + 1], Q [m + 1, n + 1], respectively.

  The tangent vector calculation unit 30 calculates a tangent vector at each corner C of each patch P based on the mesh lines U and V output from the mesh line storage unit 20. The tangent vector trajectory estimating means 40 estimates the trajectories in the u and v directions of the calculated tangent vectors of the respective corners C. The twist vector calculating unit 50 is configured to convert the twist vector at each corner C based on the tangent vector at each corner C calculated by the tangent vector calculating unit 30 and the locus of the tangent vector estimated by the tangent vector locus estimating unit 40. Is calculated. The curved surface calculation means 60 generates a curved surface in the region based on the calculated tangent vector and twist vector at each corner C.

  The determination unit 70 calculates a twist vector based on the curved surface generated by the curved surface calculation unit 60, and determines whether or not the curved surface has a smooth shape using the twist vector. is there. The notification means 80 notifies the user or the like when the determination means 70 does not have a smooth curved surface.

  Hereinafter, specific configurations and operations of these means will be described with reference to the flowcharts of FIGS.

  First, a plurality of mesh lines U [m, n] (u) in the u direction and a plurality of mesh lines V [m, n] (v) in the v direction are read from the mesh line storage unit 20 (step S11). .

  Next, in the tangent vector calculation means 30, the tangent vectors f1 [m, n] to f4 [m in the u direction at the respective corners C1 to C4 of the respective patches P [m, n] based on the mesh lines U and V. , N] and the tangent vectors g1 [m, n] to g4 [m, n] in the v direction, respectively.

  Here, as shown in FIG. 7, the tangent vectors f1 and f4 are tangent vectors in the u direction starting from intersections Q [m, n] and Q [m, n + 1] at corners C1 and C4, respectively. The tangent vectors f2 and f3 are tangent vectors in the u direction that end at the intersections Q [m + 1, n] and Q [m + 1, n + 1] at the corners C2 and C3, respectively. The tangent vectors g1 and g2 are v-direction tangent vectors starting from the intersections Q [m, n] and Q [m + 1, n] at the corners C1 and C2, respectively. The tangent vectors g3 and g4 are the corners C3. , C4, tangent vectors in the v direction, each having an intersection Q [m + 1, n + 1] and Q [m, n + 1] as end points.

  In the tangent vector calculation means 30, the u-direction mesh lines U [m, n] surrounding the patch P [m, n] are obtained by using the u-direction tangent vectors f1 to f4 as shown in the equations (1) to (4). ] (U) and U [m, n + 1] (u) are respectively differentiated. Similarly, the v-direction tangent vectors g1 to g4 are differentiated from the v-direction mesh lines V [m, n] (v) and V [m + 1, n] (v) surrounding the patch P [m, n], respectively. To calculate. (Step S12).

  The u [x] is the u coordinate of the intersection point Q [x, n].

  The above calculation is performed for all patches P [m, n], and the tangent vectors at the corners C1 to C4 of each patch P [m, n] are calculated. Here, in patches that are adjacent to each other and share a predetermined mesh line and intersection, the tangent vectors on the shared mesh line at the intersection are the same. For example, as shown in FIG. 7, in the patch P [m, n-1] adjacent to the patch P [m, n] in the v direction, the corner C4 [m, n- The tangent vector f4 [m, n-1] in the u direction of 1] is the same as the tangent vector f1 [m, n] in the u direction at the corner C1 [m, n] in the patch P [m, n].

  Next, the tangent vector locus means 40 estimates the locus along each mesh line of each tangent vector. For the tangent vectors (f1, f4, g1, g2) starting from each intersection Q among the tangent vectors, the trajectory along the mesh line of the end point is estimated, and each intersection Q of the tangent vectors is the end point. For the tangent vector (f2, f3, g3, g4), the locus along the mesh line of the starting point is estimated. For the tangent vectors f1 to f4 in the u direction, the trajectory along the mesh line in the v direction is estimated, and for the tangent vectors g1 to g4 in the v direction, the trajectory along the mesh line in the u direction is estimated.

  Here, a method for estimating the locus along the mesh line Mu [1] in the u direction of the tangent vector g in the v direction will be described with reference to FIGS. 8A and 8B.

  First, from each tangent vector calculated by the tangent vector calculation means 30, as shown in FIG. 8A, each intersection point Q [0,1] to Q [4,1] on the mesh line Mu [1]. ] Starting tangent vectors g1 [0,1], g1 [1,1] (= g2 [0,1]), g1 [2,1] (= g2 [1,1]), g1 [3,1] (= g2 [2,1]) and g2 [3,1] are extracted. Then, the end points of these tangent vectors are spline-interpolated to calculate the trajectory Ge [1] (u, v) along the mesh line Mu [1] in the u direction of these end points.

  On the other hand, the trajectory Gs [0] along the mesh line Mu [1] of the tangential vector in the v direction that ends at each intersection Q [0,1] to Q [4,4] on the mesh line Mu [1]. As shown in FIG. 8B, (u, v) is a tangent vector g4 [0,0], g4 [1,0], g4 [2,0], g4 [3,0], g3 [3 , 0] is calculated by spline interpolation.

  In this way, the trajectory Ge [n] (u, v) along the mesh line in the u direction at the end point of the tangent vector g in the v direction starting from each intersection point, and the tangent vector in the v direction starting from each intersection point. The trajectory Gs [n] (u, v) along the mesh line in the u direction at the start point of g is calculated. Similarly, the trajectory Fe [m] (u, v) along the mesh line in the v direction at the end point of the tangent vector f in the u direction starting from each intersection point and the tangent vector in the u direction starting from each intersection point. A trajectory Fs [m] (u, v) along the mesh line in the v direction of the starting point is calculated (step S13). These trajectories represent how the tangent vector of the curved surface changes along the mesh line, and represent the changing tendency of the curved surface shape.

  Here, as shown in FIG. 9A, the tangent vectors at the respective corners C1 to C4 of the respective patches P [m, n] can be set such that all the intersections are the starting points, for example. However, if it is set in this way, as shown in FIG. 9B, even if the tangent vector at the corner C1 of the same patch Px, its value changes when the uv direction is reversed. Probability is high. When the value of each tangent vector changes in this way, the shape of the locus of these tangent vectors changes from Gx to Gx ′, and a curved surface generated when a curved surface is generated using the locus of the tangent vector as will be described later. May change the shape. That is, if the tangent vector at each corner of each patch is set as described above, the shape of the curved surface depends on the uv direction, which is not preferable. On the other hand, in this embodiment, as described above, the tangent vectors in the u direction and the v direction are used for the corners of the patch that extend from the corner to the + side in the u direction and the v direction. Each intersection is set as a vector starting from the intersection, and for a corner where the patch extends from the corner to the minus side in the u direction and the v direction, the tangent vectors in the u direction and the v direction are respectively used as vectors having the intersection as an end point. It is configured so that the tangent vector of each patch is calculated inside the patch. In this way, even if the uv direction is reversed, it is possible to avoid changing the locus of the start point and end point of the tangent vector, and to prevent the curved surface shape from depending on the calculation direction. can do.

  Next, the twist vector calculation means 50 calculates the twist vector at each corner C1 to C4 of each patch P [m, n]. The twist vector calculation unit 50 includes an uv direction twist vector calculation unit 51, an uv direction averaging unit 52, a normalization unit 53, and a correction unit 54.

  The uv direction twist vector calculation means 51 is based on the tangent vector trajectory calculated by the tangent vector trajectory estimation means 40 and the v direction twist vector and v of corners C1 to C4 in each patch P [m, n]. The direction twist vector is calculated. The uv averaging means 52 calculates the u-direction and v-direction twist vectors at the corners C1 to C4 of each patch P [m, n] calculated by the uv direction twist vector calculation means 51 for each corner C1 to C4. And the uv average twist vector is calculated. When the corners are adjacent to each other around a predetermined intersection, the standard means 53 calculates a unit twist vector to be described later for the adjacent corners based on the uv average twist vector of each corner. The correction means 54 is a means for calculating the final twist vector of each corner C1 to C4 of each patch based on this unit twist vector.

  First, the uv direction twist vector calculation means 51 is based on the trajectories Fe [n], Fs [n], Ge [n], and Gs [n] of each tangent vector calculated by the tangent vector locus estimation means 40. The twist vectors tw1_u [m, n] to tw4_u [m, n] in the u direction and the twist vectors tw1_v [m, n] to tw4_v in the u direction at the corners C1 to C4 of each patch P [m, n]. [M, n] is calculated (step S14).

  Here, as an example, a method for calculating the twist vectors tw1_u [m, n] and tw1_v [m, n] in each direction at the corner C1 will be described. First, as shown in FIG. 10, the tangent vector g1 [m, n] is derived from Ge [n] (u, v) which is a locus along the mesh line in the u direction of the tangential vector g1 [m, n] in the v direction. ] Change vector q1 [m, n] with respect to the u direction. Specifically, the change vector q1 [m, n] is calculated by differentiating Ge [n] (u, v) as shown in Equation (5).

  This change vector q1 [m, n] represents how the tangent vector in the v direction changes along the u direction in the vicinity of the corner C1, and the curved surface of the patch P [m, n] It shows that the twist is made toward the end point Qz of the change vector q1 as it proceeds in the u direction. Here, when the tangent vector g1 in the v direction does not change along the u direction in the vicinity of the corner C1, that is, when the curved surface does not twist with respect to the u direction, the end point of the tangent vector in the v direction is the point Q0. Become. Therefore, the twist vector tw1_u [m, n] in the u direction indicating the twist in the u direction of the curved surface at the corner C1 can be calculated as a vector from the Q0 to the Qz, as shown in Expression (6). It can be expressed as

tw1_u [m, n] = Qz−Q0 = (f1 + q1) − (g1 + f1) = q1−g1 Formula (6)
Similarly, as shown in FIG. 11, Fe [m] (u, v), which is the locus along the mesh line in the v direction of the tangent vector f1 [m, n] in the u direction at the corner C1, is used as follows. The change vector r1 [m, n] with respect to the v direction of the tangent vector f1 [m, n] is calculated from the equation (7), and the twist vector tw1_v [m, n] in the v direction is calculated based on this r1 [m, n]. ] Is calculated from equation (8).

tw1_v [m, n] = Rz−R0 = (g1 + r1) − (g1 + f1) = r1−f1 Formula (8)
Next, in the uv direction averaging means 52, a twist vector tw1_u [m, n] in the u direction and a twist vector tw1_v [m, n] in the v direction at the corner C1 of the patch P [m, n] are shown. 12 (a) and FIG. 12 (b), the uv average twist vector Tw1 [m, n] at the corner C1 of the patch P [m, n] is averaged using the equation (9). Calculate (step S15).

Tw1 [m, n] = (tw1_u [m, n] + tw1_v [m, n]) / 2 Formula (9)
The above calculation is performed for the other corners C2 to C4 and is performed for all the patches P [m, n], and the uv average twist vector Tw1 at each corner C1 to C4 of each patch P [m, n]. [M, n] to Tw4 [m, n] are calculated.

  When the tangent vectors f1 to f4 in the u direction, the tangent vectors g1 to g4 in the v direction, and the uv average twist vectors Tw1 to Tw4 are calculated in the respective corners C1 to C4 of the patches P [m, n] in this way. The curved surface S [m, n] (u, v) of each patch P [m, n] can be calculated using these values. In particular, in the present embodiment, the uv average twist vectors Tw1 to Tw4 representing the twist tendency of the curved surface are given to the corners C1 to C4 so as to reflect the shape of each mesh line and the curved surface. [M, n] can be calculated as having a smooth shape assumed from the mesh line.

  However, when the curved surface S [m, n] of each patch P [m, n] is obtained using the uv average twist vectors Tw1 to Tw4, the curved surface in each patch becomes a smooth shape, but the adjacent patches The connecting part may not have a smooth shape.

  Therefore, in the present apparatus 10, when a plurality of patches are adjacent to each other around a predetermined intersection, and a plurality of corners are adjacent to each other around a predetermined intersection, the connecting portion of each patch has a smooth shape. Thus, the normalization means 53 and the correction means 54 interpolate the uv average twist vectors at the respective corners (step S16).

  Here, as shown in FIG. 13, four patches P [m, n], P [m−1, n], P [m−1, n−1], which surround the intersection Q [m, n], P [m, n−1] are adjacent to each other, and the corners C1 [m, n], C2 [m−1, n], C3 [m−1, n−1], and C4 [m, n− of these patches P are present. 1] are adjacent to each other. The mutual interpolation procedure will be described with reference to the flowchart of FIG.

  First, in the normalization means 53, the uv average twist vectors Tw1 [m, n], Tw2 [m-1, n], Tw3 [m-1, n-1], Tw4 [ m, n−1] are decomposed into u-direction component, v-direction component and perpendicular direction component, and these component values are converted into values per unit tangent vector based on the tangent vector values at the corners C1 to C4. (Step S21). For example, the value t1_u [m, n] per unit vector of the u-direction component Tw1 [m, n] _u of the uv average twist vector Tw1 [m, n] is represented by a corner C1 as shown in the schematic diagram of FIG. The u-direction tangent vector f1 [m, n] and the v-direction tangent vector g1 [m, n] in [m, n] can be calculated by the equation (10), and the v-direction component Tw1 [m, n ] _V per unit vector value t1_v [m, n] can be calculated by equation (11), and the value t1_h [m, n] per unit vector of the perpendicular component Tw1 [m, n] _h And can be calculated from the equation (12).

t1_u [m, n] = | Tw1 [m, n] _u | / (| f1 [m, n] | × | g1 [m, n] |) Equation (10)
t1_v [m, n] = | Tw1 [m, n] _v | / (| f1 [m, n] | × | g1 [m, n] |) Equation (11)

t1_h [m, n] = | Tw1 [m, n] _h | / (| f1 [m, n] | × | g1 [m, n] |) Equation (12)
In this way, as shown in FIG. 14, each component value of the uv average twist vector at each corner is converted into a value per unit vector.

  Next, the correction means 54 averages each component value of each uv average twist vector for each component. For example, the average value t_u_ave [m, n] of the u direction component is calculated from the equation (13).

t_u_ave [m, n] = (t1_u [m, n] + t2_u [m−1, n] + t3_u [m−1, n−1] + t4_u [m, n−1]] / 4 Equation (13)
Here, t2_u [m−1, n] is a u direction component per unit tangent vector of the uv average twist vector Tw2 [m−1, n] at the corner C2 [m−1, n], and t3_u [ m−1, n−1] is a u direction component per unit tangent vector of the uv average twist vector Tw3 [m−1, n−1] at the corner C3 [m−1, n−1], and t4_u. [M, n-1] is a u direction component per unit tangent vector of the uv average twist vector Tw4 [m, n-1] at the corner C4 [m, n-1].

  Similarly, the average value t_v_ave [m, n] of the v direction component per unit tangent and the average value t_h_ave [m, n] of the perpendicular direction component per unit tangent are calculated. From these average values, the unit twist vector t_ave [m, n] is t_u_ave [m, n], t_v_ave [m, n], and t_h_ave [m, n] in the u, v, and perpendicular directions. As the following vector (step S22).

t_ave [m, n] = (t_u_ave [m, n], t_v_ave [m, n], t_h_ave [m, n])
Next, by correcting the unit twist vector t_ave [m, n] with the size of each patch, as shown in FIG. 16, final corner twist vectors T1 to T4 at each corner of each patch are calculated (steps). S23). Specifically, the unit twist vector t_ave is calculated by Expressions (14) to (17) by multiplying the tangent vector in the u direction and the magnitude of the tangent vector in the v direction at each corner of each patch.

T1 [m, n] = t_ave × (| f1 [m, n] | × | g1 [m, n] |) Equation (14)
T2 [m−1, n] = t_ave × (| f2 [m−1, n] | × | g2 [m−1, n] |) Equation (15)
T3 [m−1, n−1] = t_ave × (| f3 [m−1, n−1] | × | g3 [m−1, n−1] |) Equation (16)
T4 [m, n−1] = t_ave × (| f4 [m, n−1] | × | g2 [m, n−1] |) Equation (17)
These final corner twist vectors T1 to T4 are vectors having the same direction and sizes corresponding to the sizes of the patches. Therefore, if the curved surface is calculated using the final corner twist vectors T1 to T4, the change tendency of the surface around the intersection Q [m, n] can be made the same, and the area around the intersection Q [m, n] can be made the same. The patch connection can be smoothed. Further, since each of the final corner twist vectors T1 to T4 has a size corresponding to the size of the patch, the characteristics of each patch are also maintained at the connection portion.

  Here, as shown in FIG. 6A, the corner C1 [0,0] of the patch P [0,0] corresponding to the corner of the entire area and the corner C2 [3,0] of the patch P [3,0]. , Patch P [0,2] corner C4 [0,2] and patch P [3,2] corner C3 [3,2] do not have adjacent corners, so The uv direction average twist vector Tw calculated in the above is output as the final corner twist vector T.

  Next, in the curved surface generating means 60, final corner twist vectors T1 [m, n] ˜ at the respective corners C1 [m, n] ˜C4 [m, n] of the respective patches P [m, n] calculated. Based on T4 [m, n], u-direction tangent vectors f1 [m, n] to f4 [m, n] and v-direction tangent vectors g1 [m, n] to f4 [m, n] at each corner, A curved surface S [m, n] is calculated for each patch P [m, n] (step S17 in the flowchart of FIG. 2). In the present embodiment, each curved surface S [m, n] is calculated from Equation (18) using a cubic Hermite function.

  Here, S (0,0), S (1,0), S (0,1), S (1,1) are intersection points Q [m, n], Q [m + 1, n], Q [m, respectively. , N + 1] and Q [m + 1, n + 1].

Further, S _u (u, v) is S _u (u, v) = ∂S (u, v) / ∂u and represents a tangent vector in the u direction. That is,
S _u (0,0) is the tangent vector f1 [m, n],
S _u (1, 0) is the tangent vector f2 [m, n],
S _u (0,1) is the tangent vector f4 [m, n],
S _u (1,1) is the tangent vector f3 [m, n].

S _v (u, v) is S _v (u, v) = ∂S (u, v) / ∂v, and represents a tangent vector in the v direction. That is,
S _v (0,0) is the tangent vector g1 [m, n],
S _v (1,0) is the tangent vector g2 [m, n],
S _v (0,1) is the tangent vector g4 [m, n],
S _v (1,1) is the tangent vector g3 [m, n].

S _uv (u, v) is S _uv (u, v) = ∂2S (u, v) / ∂u∂v, and represents the twist vector at each corner. That is,
S _uv (0,0) is the final corner twist vector T1 [m, n],
S _uv (1, 0) is the final corner twist vector T2 [m, n],
S _uv (0,1) is the final corner twist vector T4 [m, n],
S _uv (1,1) is the final corner twist vector T3 [m, n].

  In this manner, the apparatus 10 generates the curved surface S [m, n] for each patch P [m, n] based on each tangent vector and each final corner twist vector, and is generated for each patch. Based on the curved surface S [m, n], a curved surface St is generated in an area formed by these patches P [m, n]. An example of the curved surface St thus calculated is shown in FIG. In this figure, curved surfaces are generated in six patches surrounded by 4 × 3 mesh lines. As shown in this figure, by using the present apparatus 10, it is possible to smoothly connect between patches and generate a curved surface having no distortion inside each patch.

  Next, it is determined using the determination means 70 whether or not the calculated curved surface St has a smooth shape as a whole (step S18). Here, the determination is performed for each curved surface S [m, n] of the patch P [m, n]. That is, whether the curved surface generated for the patch P [m, n] is, for example, a desired curved surface Side [m, n] having a smooth shape as shown in FIG. As shown in FIG. 18B, it is determined whether or not the surface is an abnormal curved surface Sf [m, n] whose shape changes greatly.

  Here, in the curved surface generating apparatus 10 according to the present invention, each final corner twist vector T1 to T4 is calculated based on the shape of the mesh line, and the patch P [m, n] is shown in FIG. 18B. The generation of such an abnormal curved surface Sf [m, n] can be sufficiently suppressed, but the curved surface S [m, n] generated for each patch P [m, n] by the determination means 70. By determining the appropriateness, it is possible to more reliably avoid the generation of an abnormal curved surface due to, for example, the mesh line not having an appropriate shape. The determination procedure in the determination means 70 will be described with reference to the flowchart of FIG.

  As shown in FIG. 18B, the fact that the shape of the inside of the patch P [m, n] has changed greatly means that the twist inside the generated curved surface Sf [m, n] is large. Or it means that this twist is changing drastically. Therefore, the determination means 70 determines the shape of the curved surface S [m, n] using the twist vector representing the twist of the curved surface. Here, in the present embodiment, the determination means 70 calculates the twist vectors in the u and v local coordinate systems, respectively, and performs the determination using the twist vectors based on the local coordinates.

  First, the corner twist vector verification means 72 verifies whether or not the final corner twist vectors T1 to T4 at the respective corners C1 to C4 of the patch P [m, n] are within a predetermined range.

  The corner twist vector verification means 72 first calculates a curved surface Stra [m, n] when each final corner twist vector T1 to T4 is set to 0 using a conventional method (step S31). Then, as shown in FIG. 19, each mesh line U [m, n] (u), U [m, n + 1] (u), V [m, n] (v) surrounding the patch P [m, n]. , V [m + 1, n] (v), twist vectors tw_m1_tra (u, v) to tw_m4_tra (u, v) at the respective points are calculated. Here, the twist vector tw (u, v) at the point Qx on the curved surface S (u, v) can be calculated by the equation (19).

  Next, for the corners C1 to C4, the existence range of the twist vector on the two mesh lines forming each corner is calculated, and the allowable range of the twist vector is calculated from this range (step S32).

  The corner C1 [m, n] will be described. First, the twist vector tw_m1_tra (u, v) at each point on the mesh line U [m, n] (u) is calculated using the equation (19). Then, the twist vector is decomposed into the u-direction component, the v-direction component, and the perpendicular direction, respectively, and the maximum for each component of the twist vector tw_m1_tra (u, v) on the mesh line U [m, n] (u). Find the value and the minimum value. Similarly, the maximum value and the minimum value of each component of the twist vector tw_m4_tra (u, v) on the mesh line V [m, n] (v) are obtained. Next, the maximum / minimum value of each component of the twist vector tw_m1_tra (u, v) and the maximum / minimum value of each component of the twist vector tw_m4_tra (u, v) are compared to extract a common range of each component, This common range is obtained as the allowable ranges d1_u, d1_v, d1_h of the components of the twist vector at the corner C1.

  That is, as shown in the schematic diagram of FIG. 19, the maximum value tm1_u_max of the u direction component of the twist vector tw_m1_tra (u, v) on the mesh line U [m, n] (u) and the mesh line V [m, n ] Of the maximum value tm4_u_max of the u-direction component of the twist vector tw_m4_tra (u, v) on (v) and the twist vector tw_m1_tra (u, u) on the mesh line U [m, n] (u) The minimum value tm1_u_min of the u direction component in v) and the larger one (tm1_u_min) of the minimum value tm4_u_min of the u direction component of the twist vector tw_m4_tra (u, v) on the mesh line V [m, n] (v) are extracted. , The range from tm1_u_min to tm4_u_max Determined as an allowable range d1_u in the u direction component in. The permissible ranges d1_v and d1_h are similarly calculated for the v-direction component and the perpendicular direction component.

  Similarly for the other corners C2 to C4, the allowable range of each component of the twist vector at each corner is calculated. Here, this allowable range represents the width of change around each corner of the curved surface calculated by the conventional method, and the final corner twist vectors T1 to T4 calculated by the apparatus 10 are equal to or less than this change width. If so, it can be said that the curved surface generated by the apparatus 10 is as smooth as the curved surface calculated by the conventional method around the corner.

  Therefore, the corner twist vector determining means 72 calculates the component values of the final corner twist vectors T1 to T4 at the corners C1 to C4 calculated by the apparatus 10 (step S33), and these component values are calculated as described above. It is determined whether it is within each allowable range, and the validity of these final corner twist vectors T1 to T4 is determined (step S34). Here, if all the component values of each final corner twist vector T1 to T4 are all within the allowable range (YES in step S34), it is determined that these final corner twist vectors T1 to T4 are appropriate values. Proceed to S35. On the other hand, if any one of the component values exceeds the allowable range, the final corner twist vectors T1 to T4 are abnormal, and it is determined that a desired smooth curved surface has not been generated. Alert the user.

  Next, the validity of the curved surface is determined by the twist vector calculation means 74 other than the corner based on the twist vector at the point other than the corner.

  First, the magnitudes of the twist vectors on the mesh lines calculated by the conventional method and the respective change widths A and B are calculated. (Step S35). Next, with respect to the curved surface S [m, n] generated by the present apparatus 10, twist vectors tw_m1 [m, n] (u, v) to tw_m4 [m, n] (u, v) is calculated from the equation (19) (step S36). The change widths x and y of the twist vector magnitude and change amount are larger than the change widths A and B of the twist vector magnitude and change amount on the same mesh line calculated by the conventional method. It is determined whether it is small (step S37).

  This determination is YES, that is, the magnitude of the twist vector tw_m calculated from the curved surface generated by the apparatus 10 and the change width of the amount of change are the magnitude of the conventional twist vector tw_m_tra obtained from the curved surface calculated by the conventional method. If it is smaller than each change width of the height and the change amount, it is determined that the twist of the curved surface is small, that is, the curved surface is smooth on the mesh line, and the process proceeds to step S38. On the other hand, if the determination is NO, it is determined that the value of the twist vector tw_m is not valid, and the process proceeds to step S41, where the notification means 80 issues a warning to the user.

  Next, the validity of the twist vector at a point other than the points on the mesh line is determined, and the twist inside the curved surface is verified.

  Specifically, the twist vector twx (u, v) at the predetermined point Qx of the curved surface S [m, n] is calculated from the equation (19), and the magnitude L of the twist vector twx (u, v) is calculated. Then, the change amount D is calculated (step S38). Then, the magnitude L and the change amount D of the twist vector twx are respectively compared with predetermined values (step S39). Here, if it is determined that the magnitude L and the change amount D of the twist vector twx are not less than each predetermined value and not more than each predetermined value, it is determined that the curved surface is smooth even inside the patch P, and the curved surface S [m, n] is output (step S40). On the other hand, if the determination is NO, it is determined that the curved surface is highly twisted or severely twisted inside the patch P, and a warning is given to the user by the notification means 80, and the process is terminated (step S41). ).

  As described above, according to the curved surface generation device 10, the tangent vector calculation means 30 calculates the tangent vectors f and g in the uv direction at each corner C based on the shape of the mesh line, and estimates the tangent vector locus. The means 40 interpolates the start point or the end point of the uv tangent vectors f and g at each corner C along the mesh lines in each direction, whereby the trajectories Fs, Since Fe, Gs, and Ge are estimated and the twist vector at each corner C is calculated using this trajectory, the twist vector reflecting the shape of each mesh line, that is, the shape of the entire curved surface can be calculated. It becomes possible.

  Then, by calculating a curved surface based on the tangent vectors f and g and the twist vector, it is possible to calculate a smoother curved surface reflecting the characteristics of the mesh line, that is, the characteristics of the curved surface derived from the mesh line. It becomes. That is, since the curved surface is calculated without compulsorily setting the twist vector to 0 as in the prior art, it is possible to prevent the entire curved surface from being distorted.

  Further, as described above, the trajectories Fs, Fe, Gs, and Ge of the tangent vectors f and g are calculated by performing spline interpolation on the start and end points of the tangent vectors f and g along the mesh lines in the respective directions. In this way, the locus can be easily calculated and the locus can be made to have a smoother shape.

  Further, as described above, in the uv direction averaging means 52, the twist vectors tw1_u [m, n] to tw4_u [m, n], tw1_v [m] in the uv direction at each corner C based on the trajectory in each direction. , N] to tw4_v [m, n], and averaging them, the twist vectors Tw1 [m, n] to Tw4 [m, n] of each corner C are calculated. Thus, it is possible to calculate a twist vector that uniformly reflects the twist vector, and the shape of the curved surface at the corner C can be a shape that smoothly changes in any direction.

  As described above, when the corners C are adjacent to each other around a predetermined intersection, if the twist vectors of the corners C are mutually interpolated, a twist vector as a control amount is set in each corner C. It is possible to make the connection between adjacent patches P smooth while giving.

  In particular, the standard means 53 converts each component value of the uv average twist vector Tw1 [m, n] to Tw4 [m, n] averaged in the uv direction into a value per unit tangent vector, and the correction means. In 54, each component value per unit tangent vector for each corner C adjacent to each other is averaged to calculate a unit twist vector, and the unit twist vector is corrected by the size of each patch P, thereby correcting each corner C. If the final corner twist vectors T1 [m, n] to T4 [m, n] are configured to be calculated, the twist vectors greatly differ between the corners C while maintaining the characteristics of the curved surface generated in each patch P. It is possible to maintain connectivity between the patches P by suppressing the occurrence of such a situation.

  Further, as described above, there is provided a determination means for calculating a twist vector at each point from the generated curved surface, and determining whether the magnitude and change amount of the twist vector are within a predetermined range set in advance. In this case, it can be determined relatively easily whether the shape of the generated curved surface is a smooth shape, so that a smooth curved surface can be generated more reliably.

  In particular, if the determination unit 70 determines that the curved surface does not have a smooth shape, a notification unit 80 that notifies the fact is provided, so that a user or the like can easily recognize the abnormality of the curved surface. As a result, convenience is improved and work efficiency is improved.

  In addition, if the determination unit 70 determines the magnitude and the amount of change of the twist vector in the local coordinate system, how the curved surface changes with respect to the surface surrounded by the mesh line. Therefore, it is possible to more accurately determine whether a curved surface is smoothly continuous even when the curved surface changes greatly when viewed in an orthogonal coordinate system, for example.

  Here, in the above-described embodiment, a case where a region for generating a curved surface is divided into a plurality of patches P is shown, but one patch may be used.

  Further, the interpolation method of the end point and start point of the tangent vector is not limited to the spline interpolation, and any method can be used as long as each end point and start point can be smoothly interpolated.

  In the above embodiment, as a method of interpolating twist vectors at adjacent corners, a method of calculating a unit twist vector and correcting the unit twist vector with the size of each patch has been shown. For example, the twist vectors of adjacent corners may be averaged.

  Further, the method of calculating the curved surface from the tangent vector and the twist vector is not limited to the above, and any method may be used as long as these vectors are given as control amounts.

It is a schematic perspective view which shows the example of the curved surface produced | generated by the conventional method. It is a flowchart which shows the calculation point of the curved surface production | generation apparatus which concerns on embodiment of this invention. It is a flowchart which shows the detail of the mutual interpolation method of a twist vector among the calculation points shown in FIG. It is a flowchart which shows the detail of the curved surface abnormality determination method among the calculation points shown in FIG. It is a schematic block diagram of the curved surface production | generation apparatus which concerns on this invention. (A) It is a figure which shows the area | region which produces | generates a curved surface with the curved surface production | generation apparatus based on this invention. (B) It is the elements on larger scale of (a). It is a figure which shows the tangent vector in a predetermined patch. (A) It is explanatory drawing about the method of calculating the locus | trajectory along the mesh line of a predetermined | prescribed tangent vector. (B) It is explanatory drawing about the method of calculating the locus | trajectory along the mesh line of a predetermined | prescribed tangent vector. (A) It is explanatory drawing about the other method of calculating the locus | trajectory along the mesh line of a tangent vector. (B) It is explanatory drawing about the other method of calculating the locus | trajectory along the mesh line of a tangent vector. It is explanatory drawing about the method of calculating the twist vector of u direction in a predetermined | prescribed corner. It is explanatory drawing about the method of calculating the twist vector of v direction in a predetermined | prescribed corner. (A) It is explanatory drawing about the method of averaging the twist vector of u direction and the twist vector of v direction in a predetermined | prescribed corner. (B) It is the elements on larger scale of (a). It is explanatory drawing which shows the twist vector in the corner around a predetermined | prescribed intersection. It is explanatory drawing about the method of mutually interpolating the twist vector in the corner around a predetermined | prescribed intersection. It is explanatory drawing about the calculation method of a unit twist vector. It is a figure which shows the last corner twist vector of each corner calculated in this curved surface production | generation apparatus 10. FIG. It is a schematic perspective view which shows the example of a curved surface produced | generated by this curved surface production | generation apparatus 10. FIG. (A) It is a figure which shows the ideal curved surface produced | generated by a predetermined patch. (B) It is a figure which shows the curved surface which has the abnormal shape produced | generated by the predetermined patch. It is explanatory drawing about the determination method of each twist vector performed with this curved surface production | generation apparatus 10. FIG.

Explanation of symbols

10 Curved surface generator
30 Tangential vector calculation means
40 Tangential vector locus estimation means
50 Twist vector calculation means
51 uv direction twist vector calculation means
52 uv direction averaging means
53 Standardization means
54 Correction means
60 Curved surface calculation means
70 judgment means
72 Corner twist vector judging means
74 Twist vector judgment means other than corner
80 Notification means
P patch
C1-C4 corner
f1-f4 u direction tangent vector
g1-g4 v-direction tangent vector
Trajectory of tangent vector end point in Feu direction
Trajectory of tangent vector starting point in Fsu direction
Trajectory of end point of tangent vector in Ge v direction
Gsv direction tangent vector start point trajectory tw1_u to tw4_u u direction twist vector tw1_v to tw4_v v direction twist vector
Tw1-Tw4 uv average twist vector
t_ave unit twist vector
T1-T4 final corner twist vector
S curved surface

Claims (6)

In a curved surface generating apparatus that generates a curved surface in an area composed of one to a plurality of patches surrounded by a plurality of mesh lines in the u direction and a plurality of mesh lines in the v direction,
A tangent vector in the u direction at each corner of the patch is calculated based on the shape of the mesh line in the u direction, and a tangent vector in the v direction at each corner is calculated based on the shape of the mesh line in the v direction. Tangent vector calculation means;
Based on the tangent vector in the u direction calculated at each corner, the trajectory along the mesh line in the v direction of at least one of the end point or the start point of the tangent vector in the u direction is estimated and calculated at each corner. tangential vector trajectory estimation means for estimating a trajectory along a mesh line in the u direction at least one of an end point or a start point of the tangential vector in the v direction based on the tangent vector in the v direction;
Twist vector calculation means for calculating a twist vector at each corner based on the locus along the mesh line in the v direction of the estimated tangent vector in the u direction and the locus along the mesh line in the u direction of the tangent vector in the v direction. When,
Wherein based on the u direction of the tangent vector and the v direction of the tangent vector and the twist vector at each corner, it has a surface and a curved surface calculating means for calculating the curved surface of the region,
The twist vector calculating means includes:
Based on the trajectory along the mesh line in the v direction of the tangent vector in the u direction estimated by the tangent vector trajectory estimation means, the twist vector in the v direction at each corner is calculated, and the estimated tangent in the v direction Uv direction twist vector calculating means for calculating a twist vector in the u direction at each corner based on a locus along the mesh line in the u direction of the vector;
Uv direction averaging means for calculating the uv average twist vector by averaging the twist vector in the u direction and the twist vector in the v direction at each corner calculated for each corner;
A curved surface generating apparatus that calculates a twist vector at the corner based on the uv average twist vector .   The curved surface generation apparatus according to claim 1,
  The twist vector calculation means calculates each of the corners when the region includes a plurality of patches and the predetermined corners of the patches are adjacent to each other around a predetermined intersection of the mesh lines. A curved surface generating apparatus characterized by calculating a twist vector at each corner by interpolating the uv average twist vector so that the curved surface around the intersection has a smooth shape.   The curved surface generation apparatus according to claim 2,
  The twist vector calculating means includes:
  The uv average twist vector calculated for each corner adjacent to each other around the predetermined intersection is decomposed into a u-direction component, a v-direction component, and a perpendicular direction component, respectively, Normalization means for converting into values per unit tangent vector based on the values of the tangent vector in the u direction and the tangent vector in the v direction at the corner;
  Each component value per unit tangent vector for each corner calculated by the normalization means is averaged between the corners for each component to calculate a unit twist vector, and this unit twist vector is calculated in the u direction at each corner. And a correction unit that calculates a twist vector at each corner by performing correction based on the values of the tangent vector and the tangent vector in the v direction.   In the curved-surface production | generation apparatus in any one of Claims 1-3,
  Based on the curved surface calculated by the curved surface calculation means, a twist vector at a predetermined point on the curved surface is calculated, and at least one of the magnitude of the twist vector or the amount of change of the twist vector and a predetermined value set in advance A curved surface generation apparatus comprising: a determination unit that determines whether or not the curved surface has a predetermined shape based on the comparison result.   The curved surface generation apparatus according to claim 4,
  A curved surface generating apparatus characterized by comprising notifying means for notifying that when the determining means determines that the curved surface does not have a predetermined shape.   The curved surface generation apparatus according to claim 4 or 5.
  The determination means calculates the magnitude of the twist vector or the change amount of the twist vector in the local coordinate system in the u direction and the v direction, and the magnitude and change amount of the twist vector on the local coordinate A curved surface generation apparatus characterized by comparing with a predetermined value.