4. Interpolation surfaces

Let <sup>S</sup>Int : <sup>R</sup><sup>2</sup> ! <sup>R</sup><sup>3</sup> be a two-parameter mapping which represents a given surface. If structured data, i.e., tensor product data, needs to be interpolated, one may expect to come up with tensor product surfaces as well, where two parameters ð Þ ξ; η allow covering two different directions associated with the surface. In the computational space, i.e., in the plane ð Þ ξ; η , the domain, Ω ¼ ξ0; ξLu � � � <sup>η</sup>0; <sup>η</sup>Lv h i, is a rectangle, and its image is the surface in 3D as shown in Figure 15, where L<sup>ξ</sup> and L<sup>η</sup> are the number of curves in their respective directions.

B-spline tensor product surfaces allow interpolating structured data, and they are defined as the tensor product of two families of curves C<sup>k</sup> <sup>i</sup>ð Þ <sup>ξ</sup> and <sup>D</sup><sup>l</sup> j ð Þ η , which is

$$\underline{\mathfrak{C}}^{\text{Int}}\left(\underline{\xi}\_{i},\eta\_{j}\right) = \underline{\mathfrak{x}}\_{\dot{\eta}^{\prime}} \cdot \underline{\mathfrak{S}}^{\text{Int}}\_{\text{dil}}(\xi,\eta) = \underline{\mathcal{C}}^{k}\_{i}(\xi) \otimes \underline{\mathcal{D}}^{l}\_{\dot{\eta}}(\eta) ; \quad (\xi,\eta) \in \overline{\Omega} \tag{34}$$

2. Create interpolation curves with constant values of η, so-called ξ-interpolants (see right side

Figure 17. The last two steps to interpolate structured data are depicted: η-interpolants (left) and resulting bicubic

3. Proceed accordingly with previous step, interpolation curves with constant values of ξ; so-

4. Compute the tensor product between ξ- and η-interpolants in order to get bicubic patches

The right-hand side in Figure 17 shows typical bicubic patches as a chessboard surface emphasizing that we deal with a piecewise continuous entity. The computational cost associated with the above algorithm is reasonable because the most expensive part is computing the interpolants

These surfaces are again a two-parameter mapping, <sup>σ</sup><sup>T</sup> : <sup>R</sup><sup>2</sup> ! <sup>R</sup>3, but their construction procedure is simpler than interpolation surfaces; see, for instance, [5, 8, 9]. The idea here is just

This idea became very popular in CAGD systems long time ago. Those systems usually support a command which allows extruding a geometrical entity, for instance, a cylinder can be easily created by extruding a circle along a straight line. Figure 18 shows the above procedure applied to an aircraft wing where an NURBS airfoil profile was translated or extruded along a straight line accordingly. We interpolated an NACA 65 polyline with a

<sup>σ</sup><sup>T</sup>ð Þ¼ <sup>ξ</sup>; <sup>η</sup> α ξð Þþ β ηð Þ: (35)

Scalar and Parametric Spline Curves and Surfaces http://dx.doi.org/10.5772/intechopen.74929 17

called η-interpolants are created this time (see left side of Figure 17).

patches are shown. The de Boor's control polygon is highlighted in red lines.

literally translating a curve α along another curve β, which yields

NURBS curve as we mentioned in Section 3.4.

of Figure 16).

(see Section 3).

(see right side of Figure 17).

5. Translational surfaces

In applications of practical interest, usually cubic piecewise continuous curves are preferred because they provide a global C<sup>2</sup> representation that is smooth enough, called a bicubic surface [5, 8, 9].

#### 4.1. Creating a surface of interpolation

The following steps describe creating a surface of interpolation:

1. An input control polygon, whose points are in R3, is provided. They correspond to data that is structured and ordered, which is usually a matrix-type array of points (see left side of Figure 16). For simplicity, points in the i-direction are associated with the ξ parameter while j 0 s are associated with η.

Figure 15. We depict the mapping between physical and computational spaces.

Figure 16. The first two steps to interpolate structured data are depicted: input control polygon (left) and ξ-interpolants (right) are shown.

Figure 17. The last two steps to interpolate structured data are depicted: η-interpolants (left) and resulting bicubic patches are shown. The de Boor's control polygon is highlighted in red lines.

2. Create interpolation curves with constant values of η, so-called ξ-interpolants (see right side of Figure 16).

3. Proceed accordingly with previous step, interpolation curves with constant values of ξ; socalled η-interpolants are created this time (see left side of Figure 17).

4. Compute the tensor product between ξ- and η-interpolants in order to get bicubic patches (see right side of Figure 17).

The right-hand side in Figure 17 shows typical bicubic patches as a chessboard surface emphasizing that we deal with a piecewise continuous entity. The computational cost associated with the above algorithm is reasonable because the most expensive part is computing the interpolants (see Section 3).
