1. Introduction

A bicubic surface is formed by bicubic portions

,

(1)

bounded by the cells of a rectangular grid on the xy plane of the Cartesian coordinate system Oxyz. Portions (1) are interconnected and have a preset degree of smoothness.

The equation of each portion contains 16 coefficients, which are determined based on the incidence, smoothness, and boundary conditions. The surface on the grid consists of mn bicubic portions of form (1), each of which is determined by its own set of 16 coefficients a_ij. The calculation of the coefficients is reduced to solving the system of linear algebraic equations with a 16 mn square characteristic matrix. For example, a system of 64 linear equations must be solved to calculate the coefficients of the equation of a surface formed by four bicubic portions.

Scientific novelty. The paper proposes an algorithm which can nearly halve (from 16 mn to 9 mn) the size of the characteristic matrix. As opposed to known algorithms, the constructed surface is considered as a set of longitudinal (elongated along the x-axis) bicubic tapes connected in the transverse direction with C² smoothness (with continuous changes in curvature when crossing the common boundary of the two tapes). Each tape is formed from sequentially connected bicubic portions. We formulated and proved the algebraic conditions of C²-smooth bicubic tape connection.

Practical significance. Composite surfaces are often modeled in modern architecture when designing structures with sufficient spatial freedom. In particular, the search for new non-linear forms has led to the appearance of “tent architecture” [1] and “fold architecture” [2], which use overlaps with complex curvilinear outlines. If a constructed surface has no large gradients relative to a base plane xy, bicubic polynomials in the scalar values x, y can be effectively used to model it [3, 4, 5].

2. Problem statement

A rectangular grid is marked on the xy plane. Points with different elevations are indicated in the nodes of this grid. The angular and boundary points have gradients (the slope angles of the constructed surface to the xy plane) in the longitudinal (along the x axis) and transverse (along the axis) directions. It is required to form a rectangular C²-smooth surface with given gradients passing through the specified points. The C² smoothness means a continuous (without “jumps”) change in the surface curvature at any point and in any direction [6, 7].

We will construct a surface from bicubic portions of form (1) bounded by the cells of the rectangular grid , where m, n are positive integers. At m=n =1, we obtain a bicubic portion. At m ≥2 and n =1, we obtain a bicubic band. At m=n =2, we obtain the simplest compound bicubic surface.

Note. A bicubic band is a rectangular-plan surface elongated along the x axis, formed by a set of bicubic portions interconnected by transverse joints (with C² smoothness).

3. Bicubic portion

A rectangular cell is marked on the xy plane of the Cartesian coordinate system xyz, where h_x = x₁ - x₀, h_y = y₁ - y₀ (Fig. 1).

 

Fig. 1. Boundary conditions

 

The angular points A(x₀, y₀, z_A), B(x₁, y₀, z_B), C(x₁, y₁, z_C), D(x₀, y₁, z_D) are specified and the equations of boundary curves (cubic parabolas) are given:

(2)

The free term of the cubic parabola equation is hereinafter denoted by letter α, and the coefficients at the increasing degrees of the argument are denoted by letters β, γ, δ, respectively. The subscripts indicate the boundary points of the relevant parabolas. It is required to find the equation of the bicubic surface (portion) Ô(x, y) = ABCD “stretched” along the given boundary curves.

The expanded equation of bicubic portion (1) is:

(1à)

Assuming that y = y₀, we isolate the equation of boundary curve AB from (1a):

(3)

Equating the coefficients included in the first equation from (2) and the coefficients included in equation (3), we obtain:

(4)

Similarly, assuming that x = x₀, we isolate the equation of boundary curve AD from (1a) and equate the coefficients at the identical degrees of the variable y:

(5)

Assuming that x = x₁, we isolate the equation of boundary curve BC from (1a). Assuming that y = y₁, we isolate the equation of boundary curve DC from (1a). Equating the coefficients of the obtained equations with the relevant coefficients from (2) and taking into account (4), (5), we obtain the system of five linearly independent equations

(6)

with respect to nine coefficients a₁₁, a₁₂, a₁₃, a₂₁, a₂₂, a₂₃, a₃₁, a₃₂, a₃₃.

Note. Isolating the equations of the cubic parabolas BC, AD from (1a) and equating the coefficients of these equations to the relevant coefficients from (2) while taking into account (4) and (5), we obtain not five, but six linear equations. We can show that any of these six equations results from the five remaining equations so one equation, namely , is neglected.

Four boundary conditions should be specified to determine the nine unknown coefficients included in (6). The “plane angles” conditions can be taken as additional boundary conditions: the first mixed derivatives of function (1a) are equal to zero at the angular points of the constructed bicubic portion [8]. Differentiating (1a) and equating the first mixed derivatives to zero, we obtain:

(7)

We find the coefficients a₁₁, a₁₂, a₁₃, a₂₁, a₂₂, a₂₃, a₃₁, a₃₂, a₃₃ from the system of equation (6), (7). The remaining seven coefficients of equation (1a) are calculated according to (4), (5). The problem is solved.

Note. Whereas bicubic portion is set by the angular points A, B, C, D and the gradients (the slope angles of the tangents at the angular points), equations (2) of the segment boundaries are determined by a simple calculation [9, 10]. For example, if the slope angles α ^x _A, α ^x _B of the tangents are set at the finite points of boundary curve AB(see Fig. 1), the coefficients γ_AB, δ_AB included in the equation of this curve are calculated from the system of equations

(8)

where α_AB = z_A, β_AB = tgα ^x _A. The equations of other boundaries are determined similarly.

Example 1

We are given the coordinates of the angular points A(0; 0; 2,5), B(10; 0; 5), C(10; 10; 12,5), D(0; 10; 7,5). Gradients are fixed at the angular points (see Fig. 1):

The boundary plane angles conditions (the first mixed derivatives are equal to zero at the angles of the portion) are additionally accepted. We must find equation (1a) of the bicubic portion satisfying the conditions of the problem.

Solution

Substituting the values h_x = h_y =10 and the coordinates of points A, B in (8), we find the coefficients of the equation of boundary curve AB. Similarly, calculating the coefficients of the equations of boundary curves AD, BC and DC, we obtain:

(9)

Substituting the coefficients of equations (9) into (4), (5), we obtain:

The remaining coefficients included in (1a) are found from the system of equations (6), (7):

We determined all the coefficients of equation (1a). Figure 2 shows the grid of generators of the bicubic portion BCD constructed according to (1a).

 

Fig. 2. Bicubic portion

4. Bicubic band

A bicubic band consists of series-connected (with C² smoothness) bicubic portions. The bicubic portions ABCD and BMNC must be connected along the joint BC(Fig. 3).

 

Fig. 3. Connection of bicubic portions (Theorem 1)

 

Let us show that C² smoothness is achieved if the longitudinal boundaries of the band are C²-smooth and the first and second mixed derivatives at the junction points B, C of the connected portions are equal.

Theorem 1. To achieve C²-smooth connection of the bicubic portions Ô₁ (x, y)= ABCD and Ô₂ (x, y)= BMNC along the transverse joint BC, it is sufficient to ensure that the longitudinal boundaries ABM and DCN are C²-smooth and that the first and second mixed derivatives are equal at points B, C:

,	(10)
.	(11)

Proof. The requirement for C²-smoothness of the band Ô1+Ô2 means that at any point of the joint BC the following equalities should be met:

,	(12)
.	(13)

Let us consider condition (12). The cubic functions included in (12) are uniquely determined by their values at the boundary points of the joint B, C, as well as by the values of the first derivatives at these points. According to (10), these values coincide; therefore, the functions coincide along the joint BC. Condition (12) is satisfied.

Let us consider condition (13). The cubic functions included in (13) are uniquely determined either by their values and first derivatives at points B and C, or by their values and second derivatives at these points. Therefore, to fulfill requirement (13), in addition to the equality of the functions at points B and C, we should additionally ensure either the equality of the first derivatives , or the equality of the second derivatives at these points. According to (11), the second derivatives at points B, C coincide. According to the C²-smoothness condition for boundary curves ABM and DCN, the values of the functions at points B and C also coincide. Therefore, the functions coincide along the joint BC. Condition (13) is satisfied. The theorem is proven.

Let us construct a two-section C²-smooth bicubic band with the fixed transverse guides AD, BC, MN and with the given gradients in the longitudinal direction (see Fig. 3). According to the condition of Theorem 1, the longitudinal boundary ABM formed by the cubic parabolas AB and BM should be a C²-smooth compound curve (cubic spline). The same requirement applies to the longitudinal boundary DCN. Let us consider an algorithm for constructing a cubic spline with fixed end points (fixed tangents at the finite points).

4.1. Cubic spline with fixed end points

Points A(x₀, y₀, z₀), B(x₁, y₀, z₁), M(x₂, y₀, z₂) are indicated in the vertical plane y = y₀ of the Cartesian coordinate system xyz. A composite C²-smooth curve formed by the cubic parabolas f₁ (x)= AB and f₂ (x)= BM(cubic spline) must be drawn through these points. The slope angles of the tangents to the constructed curve (“fixed end points”) are indicated at boundary points A and M.

The condition for the C²-smooth connection of the parabolas f₁ (x) and f₂ (x) has the form [11, 12]

,

(14)

where S₀, S₁, and S₂ are the values of the second derivatives of the functions z = f₁ (x) and z = f₂ (x) at the nodes A, B, and M. The designations h_{1 x} = x₁ - x₀ and h_{2 x} = x₂ - x₁ are used hereinafter.

Condition (14) should be supplemented with the fixed end points conditions:

(15)

We find the values of S₀, S₁, S₂ from the system of equations (14), (15) and substitute them into the equations

(16)

The computational algorithm of (14), (15), (16) makes it possible to find equations of a cubic spline with fixed end points. If the spline is formed from m segments (m-1 junction points), the specified algorithm will contain m-1 smoothness conditions of form (14) and m equations of form (16).

4.2. Algorithm for calculating a bicubic tape

Let us construct a band consisting of m bicubic portions ABCD, BMNC, MKLN, … (Fig. 4).

 

Fig. 4. Transverse guides of the bicubic band

 

We will assume that the equations of the frame transverse lines AD, BC, MN, KL, … are either preset or found according to (8). To solve the problem, 16 m coefficients included in equations (1) of connected portions must be calculated.

Step 1. We find the equations of the longitudinal boundaries of the band formed by composite C²-smooth m-sectional cubic curves (cubic splines) using algorithm (14) ... (16).

Step 2. We create a system of 5 m equations with (6). We supplement this system of equations with four boundary conditions with (7) plane angles and 4(m-1) smoothness conditions of (10) and (11) (see Theorem 1). We obtain a system of 9 m linear equations, from which we find 9 m coefficients.

Step 3. Using direct calculation by formulas (4) and (5), we find 7 m coefficients included in the equations of portions. Jointly with the previously found 9 m coefficients, we obtain 16 m coefficients included in the equations of connected portions. The problem is solved.

Example 2

Let us construct a C²-smooth bicubic band passing through fixed transverse guides

(17)

Longitudinal gradients are set in the angular points (see Fig. 3).

Solution

We will find the equation of the portion Ô₁ = ABCD in form (1a). We will find the equation of the portion Ô₂ = BMNC in the form

(1b)

where x₀ = y₀ =0, x₁ = y₁ =10, x₂ =25.

Step 1. According to (14), (15), and (16), we find the equations for the segments of the longitudinal boundary ABM connected at point B with Ѳ smoothness:

Similarly, we find the equations of the segments of boundary curve DCN connected at point C with Ѳ smoothness:

Step 2. We make a system of ten equations of form (6) with respect to the unknown coefficients a_ij, b_ij included in equations (1a), (1b) of the required bicubic portions:

(18)

Here, h_y = y₁ - y₀ =10, h_{1 x} = x₁ - x₀ =10, h_{2 x} = x₂ - x₁ =15.

We write down the plane angles condition:

(19)

The system of equations (18) and (19) is supplemented by the requirements for a C²-smooth connection of the bicubic portions Ô₁, Ô₂ along the joint BC(see Theorem 1):

(20)

We obtained a system of eighteen equations (18), (19), and (20). We find 18 coefficients from this system of equations:

Step 3. We find the remaining 14 coefficients according to (4) and (5):

We determined all the coefficients of equations (1a), (1b) of the bicubic segments Ô₁ and Ô₂ with a Ѳ-smooth connection. Figure 5 shows the grid of generators of the bicubic band Ô₁+Ô₂ constructed according to (1a) and (1b).

 

Fig. 5. Ñ2-smooth band (Example 2)

 

Example 3

Let us attach the section PADR with a horizontal guide PR located at a height of z =2.5 to the two-section band ABCD + BMN considered in example 2 (Fig. 6a).

 

Fig. 6. Three-section Ñ2-smooth band (Example 3): à - boundary conditions; b - grid of generators

 

We will assume we are given the coordinates of the grid nodal points: x₀ = y₀ =0, x₁ =y₁ =10, x₂ =20, x₃ =35. The equations for the transverse guides AD, BC, MN are shown in example 2. The equation for the guide line PR degenerates into the equation z_PR =2.5. The longitudinal gradients are set at the angular points P, M, N, R of the constructed band.

Solution

Let us find the equations of the bicubic portions Ô₁ = PADR and Ô₂ = ABCD with (1à) and (1b), assuming that x₀ = y₀ =0, x₁ = y₁ =10, x₂ =20 and the equation of the portion Ô₃ = BMNC in the following form

(1c)

assuming that x₃=35.

Step 1. According to algorithm (14) ... (16), we find the equations of the segments of the longitudinal boundary PABM. We obtain a set of C²-smoothly connected cubic parabolas satisfying the “fixed end points” conditions:

(21)

Using the same algorithm, we find the equations for the segments of the longitudinal boundary RDCN:

(22)

Step 2. We make a system of 5 m =15 equations of form (6) with respect to a_ij, b_ij, and c_ij:

(23)

The values included in (23) are determined according to (17), (21), and (22). For example, it follows from (17) that . Line PR is a horizontal segment, so

We supplement the system of equations (23) with the conditions of form (7) (plane angles):

(24)

We write down 4(m-1)=8 smoothness conditions: the conditions for the equality of the first mixed derivatives

(25)

and the conditions for the equality of the second mixed derivatives

(26)

at the junction points A, D, B, and C(see Theorem 1).

We find 27 coefficients from the system of 27 linear equations (23) ... (26):

Step 3. We find the remaining coefficients using the formulas of (4), (5):

We determined all 48 coefficients of equations (1a), (1b), and (1c) of the bicubic portions Ô₁, Ô₂, and Ô₃. Figure 6b shows the grid of generators of the bicubic band Ô=Ô₁ +Ô₂ +Ô₃.

5. Bicubic surface

Let us construct a bicubic surface consisting of mn bicubic portions: m portions in the longitudinal direction (along the x axis) and n portions in the transverse direction (along the y axis). The longitudinal and transverse frame lines of the surface are formed by cubic splines. We will assume that the constructed surface consists of n bicubic bands, each of which consists of m bicubic portions. The bicubic bands should be C²-smoothly interconnected (along the longitudinal lines of the frame). Let us consider the conditions for the smooth connection of bicubic bands.

Let the bicubic surface consist of two bands (Fig. 7).

 

Fig. 7. Bicubic surface frame

 

Band Ô_AB formed by the bicubic portions is bounded by the cubic splines A₀ A₁ … A_m and B₀ B₁ … B_m. B and Ô_BC formed by the bicubic portions is bounded by the cubic splines B₀ B₁ … B_m and C₀ C₁ … C_m.

Theorem 2. To achieve C²-smooth connection of the bicubic bands Ô_AB, Ô_BC along the longitudinal joint B₀ … B_m, it is sufficient to ensure the equality of the first and second mixed derivatives at the initial B₀ and finite B_m points of the longitudinal joint in addition to the C²-smoothness of the frame lines:

,	(27)
.	(28)

Proof . The requirement for the C²-smooth connection of bicubic bands means that the following equalities are met at any point of the joint B₀ … B_m

,	(29)
.	(30)

Let us consider condition (29). The cubic Ѳ-smooth composite functions included (29) are determined by their values at nodal points B₀, B₁, …, B_m and the values of the first derivatives at finite points B₀, B_m of the longitudinal joint B₀ … B_m. The numerical values of the functions at points B₀, B₁, …, B_m are equal to the tangents of the slope angles of the tangents to the transverse lines of the frame. Due to the smoothness of the transverse lines of the frame, the values of the functions at these points coincide. According to condition (27), the values of the derivatives of these functions at the finite points B₀ and B_m also coincide; therefore, the functions coincide along the joint B₀ …. B_m. Condition (29) is satisfied.

Let us consider condition (30). The cubic Ѳ-smooth composite functions included in (30) are determined by their values at points B₀, B₁, …, B_m and the values of the second derivatives at boundary points B₀, B_m of the longitudinal joint B₀ … B_m. The values of the functions at points B₀, B₁, …, B_m are proportional to the curvature of the transverse joints at these points. Due to the C²-smoothness of the transverse lines of the frame, these values coincide. According to condition (28), the values of the second derivatives of these functions at boundary points B₀, B_m also coincide; therefore, the functions coincide along the joint B₀ …. B_m. Condition (30) is satisfied. The theorem is proven.

5.1. Algorithm for calculating a bicubic surface

The calculation is reduced to calculating 16 mn coefficients of the equations of the bicubic portions forming the constructed surface. We assume that the surface consists of n longitudinal bicubic bands, and each band consists of m bicubic portions. Each portion is described by an equation of form (1).

Step 1. Using the algorithm from (14), (15), and (16), we find the equations of the longitudinal and transverse lines of the frame (cubic splines).

Step 2. We make a system of 5 m equations of form (6) for each bicubic band. We add 4(m-1) smoothness conditions for the band (see Theorem 1). We obtain 9 m-4 equations. We obtain n(9 m-4) equations for n bands. We supplement the resulting equations with 4(n-1) conditions (27), (28) for the smooth band connection (see Theorem 2), as well as 4 “plane angles” conditions. We obtain the system of 9 mn linear equations used to find 9 mn coefficients included in the equations of bicubic portions of the constructed surface.

Step 3. Using direct calculation of formulas (4) and (5), we find 7 mn coefficients included in the equations of portions. Along with the previously found 9 mn coefficients, we obtain 16 mn coefficients included in the equations of connected portions. The problem is solved.

Example 4

Let us construct a C²-smooth bicubic surface passing through points

The gradients in the longitudinal and transverse directions are fixed at the angular points A₀, A₂, C₀, C₂:

The longitudinal gradients are given at boundary points B₀, B₂. The transverse gradients are given at boundary points A₁, C₁ (Fig. 8à).

 

Fig. 8. Bicubic surface (Example 4): a - fixed frame; b - grid of generators

 

Solution

Step 1. Per (14), (15), and (16) we find the equations of the frame lines satisfying the conditions of the problem.

The equation of the longitudinal boundary line A₀ A₁ A₂:

(31)

The equation of the longitudinal line B₀ B₁ B₂:

(32)

The equation of the longitudinal boundary line C₀ C₁ C₂:

(33)

(31) … (33) take into account that x₀ = y₀ =0, x₁ =10, x₂ =25.

Similarly, we find the equations of the transverse lines of the frame. The equation of the transverse boundary line A₀ B₀ C₀:

(34)

The equation of the transverse line A₁ B₁ C₁:

(35)

The equation of the transverse boundary line A₂ B₂ C₂:

(36)

The surface frame is fixed.

Step 2. The constructed surface consists of two bicubic bands Ô_AB and Ô_BC smoothly interconnected along the joint B₀ B₁ B₂. The band Ô_AB is formed by the portions

(37)

The band Ô_BC is formed by the portions

(38)

We make a system of 10 equations of form (6) for each band.

For the band Ô_AB we obtain:

(39)

Here, h_{1 x} = x₁ - x₀ =10, h_{1 y} = y₁ - y₀ =10, and h_{2 x} = x₂ - x₁ =15.

For the band Ô_BC we obtain:

(40)

Here, h_2y =10. The values included in (39) and (40) are determined according to (31)…(36). For example, it follows from (34) that

We write down the smoothness conditions for each bicubic band (see Theorem 1). For band Ô_AB, we obtain four conditions for a smooth connection of portions (the equality of the first and second mixed derivatives at points A₁, B₁):

(41)

For band Ô_BC we also obtain four conditions for a smooth connection of portions (the equality of the first and second mixed derivatives at points B₁, C₁):

(42)

We write down the conditions for the smooth connection of the bicubic bands Ô_AB and Ô_BC at the junction points B₀ and B₂ (see Theorem 2):

(43)

We write down the plane angles condition:

(44)

The system of equations (39) … (44) contains 36 equations with respect to 36 coefficients a_ij, b_ij, c_ij, and d_ij included in equations (37) and (38). Solving this system of equations, we obtain:

Step 3. Using direct calculation of formulas (4) and (5), we find the remaining 28 coefficients included in equations (37), (38):

We determined all 64 coefficients included in equations (37) and (38). Figure 8b shows the grid of the generators of the bicubic surface Ô_AB+Ô_BC.

5.2. Private cases of a bicubic surface

The computational algorithm based on preliminary fixing of the surface frame is valid when the frame is formed by a mixed set of cubic splines.

Example 5

The surface frame is set by the straight lines B₀ B₁ B₂ and A₁ B₁ C₁. The surface boundaries are set by the straight lines A₀ A₁ A₂, A₀ B₀ C₀, and A₂ B₂ C₂ and the cubic spline C₀ C₁ C₂ with the longitudinal gradients (Fig. 9à).

 

Fig. 9. Bicubic surface on a frame of straight lines and cubic splines (Example 5):
a - fixed frame; b - grid of generators

 

We are given the coordinates of the following nodal points:

Let us construct a C²-smooth bicubic surface satisfying the conditions of the problem.

Solution

We will assume that the constructed surface consists of the bicubic bands Ô_AB and Ô_BC smoothly interconnected along the straight joint B₀ B₁ B₂. Each band, in turn, consists of bicubic portions (37), (38).

Step 1. We find the equations of the frame lines satisfying the conditions of the problem.

The equation of line A₀ A₁ A₂:

The equation of line B₀ B₁ B₂:

The equation of line A₀ B₀ C₀:

The equation of line A₁ B₁ C₁:

The equation of line A₂ B₂ C₂:

Following the algorithm in (14), (15), and (16), we find the equation of the Ѳ-smooth boundary curve C₀ C₁ C₂ with the gradients :

Step 2. We write down the system of equations (39) ... (44) containing 36 equations for 36 coefficients a_ij, b_ij, c_ij, and d_ij included in equations (37) and (38). In equations (39) and (40) we substitute the values of the coefficients α, β, γ, and δ corresponding to the equations of frame lines. For example, it follows from the equation of boundary line A₀ B₀ C₀ that

Having solved the system of equations (39) ... (44), we find 36 coefficients included in equations (37), (38):

Step 3. Using direct calculation of formulas (4) and (5), we find the remaining 28 coefficients included in equations (37) and (38):

We determined all 64 coefficients included in equations (37) and (38). Figure 9b shows the grid of generators of the bicubic surface Ô_AB+Ô_BC.

Example 6

The surface frame is set by a spatial quadrilateral with the angular points A₀ (0; 0; 5), A₂ (25; 0; 5), C₂ (25; 20; 20), and C₀ (0; 20; 5) and the straight guides B₀ B₁ B₂ and A₁ B₁ C₁ lying in the vertical planes y =10 and õ =10 (Fig. 10à).

 

Fig. 10. Bicubic surface on a straight line frame (Example 6):
à - fixed frame; b - grid of generators

 

Let us construct a bicubic surface “stretched” on a given frame.

Solution

We will assume that the constructed surface consists of the bicubic bands Ô_AB and Ô_BC smoothly interconnected along the straight joint B₀ B₁ B₂. Each band, in turn, consists of bicubic portions (37), (38).

Step 1. We find the equations of the frame lines.

The equation of line A₀ A₁ A₂:

The equation of line B₀ B₁ B₂:

The equation of line Ñ₀ Ñ₁ Ñ₂:

The equation of line A₀ B₀ C₀:

The equation of line A₁ B₁ C₁:

The equation of line A₂ B₂ C₂:

Step 2. We write down the system of equations (39) … (44) substituting values of the coefficients α, β, γ, δ corresponding to the equations of frame lines. For example, it follows from the equation of boundary line A₀ B₀ C₀ that Having solved the system of equations (39) … (44), we find the coefficients of equations (37) and (38):

Step 3. Using direct calculation of the formulas (4) and (5), we find the remaining 28 coefficients included in equations (37) and (38):

We determined all the coefficients included in the equations of bicubic portions (37) and (38). Figure 10b shows the grid of the generators of the bicubic surface Ô_AB+Ô_BC. The resulting surface slightly differs from the oblique plane A₀ A₂ C₂ C₀. For example, at x =6, y =4, the elevation marks of the points on the bicubic surface and on the oblique plane are z =5.623232 and z =5.720, respectively, differing by 1.7%.

6. Software

Matrix calculations (solving the systems of linear equations) were performed using the freely distributed software SMath Studio. The grid of the bicubic surface generators was calculated and visualized in all examples using the AutoLISP programming language in the AutoCAD environment [13]. The transparency of the examples is ensured by indicating the numerical values of all the calculated magnitudes with an accuracy of nine significant figures.

7. Discussion

We aimed to avoid the classical idea of a composite bicubic surface as a set of bicubic portions meeting certain conditions for the border of a surface (incidence of given points, fixed gradients, etc.) and for the smooth interconnection of portions.

Our proposed approach creates a surface frame made of algebraic cubic splines. Constructing a cubic spline that interpolates a given set of points is a trivial task which was fully solved in the second half of 20th century [14, 15]. The work of Kazan mathematicians Kornishin M.S. et. al [16] is noteworthy in Russian literature.

A distinctive feature of the proposed algorithm for calculating a composite C²-smooth bicubic surface consists in the conventional decomposition of the constructed surface into separate bicubic tapes bounded by longitudinal frame lines. Calculating a tape is much easier than calculating a surface [17]. We believe that this approach, which divides the problem into simple calculation blocks, is most consistent with engineering practice.

The proposed algorithm was illustrated with 3D surface models. As P. Bezier said, the widespread use of some systems is adversely affected by the fact that, despite the sophistication of applied mathematical methods, users have difficulty in their assimilation [8]. One method to overcome these difficulties is to demonstrate the applied methods and algorithms based on specific examples, as we did in this article.

8. Conclusion

The paper proposes an algorithm for calculating a composite bicubic surface with a continuous change in curvature stretched on a fixed frame. In the proposed algorithm, the problem is divided into two stages: first, the frame equations are found, and then the coefficients included in the equations of the bicubic portions forming the bicubic surface are calculated. According to the specified boundary conditions, the frame lines are described by cubic splines with fixed end points.

This approach to modeling a bicubic surface reduces the size of the characteristic matrix of the system of linear equations with respect to the coefficients included in the bicubic surface equation. The matrix size is reduced from 16 mn to 9 mn, where m and n are the number of bicubic portions along the x, y axes. Surface visualization is reduced to building a grid of longitudinal and transverse generators, the equations of which are formed from the bicubic surface equation by substituting y =const (longitudinal generators) or x =const (transverse generators).

References

1. Udler E. M., Tostov E. Design of tent shells // CADmaster #1(6) / 2001, pp. 43−47.

2. Kirichkov I.V. Refraction of the fold category through the prism of architecture // Architecture and Design. - 2018. – # 3. – P. 1–11. DOI: 10.7256/2585-7789.2018.3.29422

3. Jarke J.V. Bicubic patches for approximating non-rectangular control-point meshes / / Computer Aided Geometric Design. - 1986. - Vol. 3, # l. - P. 456-459.

4. Levner G., Tassinari P., Marini D. Simple general methods for ray tracing bicubic surfaces // Theoretical Foundations of Computer Graphics and CAD. – New York: Springer-Verlag, 1988. – P. 805–820.

5. Fox A., Pratt M. Computational geometry. Application in design and production. Moscow: Mir, 1982. 304 p.

6. Panchuk, K. Spline Curves Formation Given Extreme Derivatives / K. Panchuk, T. Myasoedova, E. Lyubchinov. – Mathematics 2021, 9(1), 47. https://doi.org/10.3390/math9010047

7. Gallier, J. Curves and Surfaces in Geometric Modeling: Theory and Algorithms; University of Pennsylvania: Philadelphia, PA, USA, 2018; P. 61–114.

8. Bezier P. Geometric methods // Mathematics and CAD. Vol. 2. Moscow: Mir, 1989. P. 96–257.

9. Korotkiy V.A. Irregular curves in engineering geometry and computer graphics // Scientific Visualization, 2022. Vol. 14, No. 1. P. 1–17. DOI: 10.26583/sv.14.1.01

10. Korotkiy V.A. Cubic curves in engineering geometry // Geometry and Graphics. – 2020. Vol. 8, No. 3. – P. 3–24. – DOI: 10.12737/2308-4898-2020-3-24

11. Shikin E.V., Plis A.I. Curves and surfaces on a computer screen. User guide to splines. Moscow: Dialog-MEPhI, 1996. 240 p.

12. Golovanov N.N. Geometric modeling. Moscow: DMK-Press, 2020. 406 p.

13. Gotovtsev A.A. Autodesk alias: where to start? // CADmaster #5 (66) / 2012, P. 42–44.

14. 318 ñ. Ahlberg J., Nilson E., Walsh J. The theory of splines and their applications. Moscow: Mir, 1972. 318 p.

15. C. de Boor. A practical guide to spline. Moscow: Radio and Communication, 1985. 303 p.

16. Kornishin M.S., Paimushin V.N., Snigirev V.F. Computational geometry in problems of shell mechanics. Moscow: Nauka, 1989. 208 p.

17. Korotkiy V.A, Usmanova E.A., A bicubic tape surface // Omsk Scientific Bulletin. 2023. #. 2 (186). pp. 19–27. DOI: 10.25206/1813-8225-2023-186-00-00.