Cylindrical Surface Pasting

What is surface pasting?

Surface pasting is a surface composition technique that is used to construct surfaces with varying levels of detail. A feature is placed on top of an existing surface to provide a region of increased detail. The base surface can be a simple tensor product surface or can be a composite surface produced by previous surface pasting operations. After pasting, the base surface remains unchanged and the feature surface combines characteristics of its unpasted shape and the underlying surface. See the Surface Pasting page for more details on how surface pasting works.

Cylindrical pasting is a parametric-blending method that creates a smooth transition surface between a pair of B-spline surfaces that do not originally intersect. This blending surface is a deformed cylinder, and its creation is based on the surface pasting composition method, which adds detailed features to base surfaces by means of an efficient displacement method. In cylindrical pasting, a transition cylinder can be pasted on a NUBS surface or onto a NUBS cylinder. A displacement scheme is used to locate the control points of the blending cylinder to achieve approximate $C^1$ continuity between the boundaries of the base surfaces and the edges of the cylinders.

The main idea in surface pasting is the mapping of the feature control points to get the feature surface to lie in the appropriate place relative to the base surface. There are three types of control points to map, each of which requires a different mapping technique. The first control points are those along the boundary of the feature. These should be mapped to achieve approximate $C^0$ continuity. The second layer of control points are mapped to achieve approximate $C^1$ continuity. And the remaining interior control points are mapped to achieve the desired feature shape. Here we focus on the mapping of the first two layers, as their mapping is the pasting process.

Mapping the domain

In the first case, the base surface is a normal NUBS surface with a rectangular domain. Only one of the two edges of the feature cylinder will be pasted on the base, as shown in Figure 1 (left). We locate the position of the edge of the feature on the base surface through a domain association. The edge of the feature domain corresponding to the edge of the feature surface that is to lie on the base surface is mapped to circle in the base domain as shown in Figure 1 (right). By default, we initially locate the domain for the feature cylinder at the center of the base domain with a predefined radius; the user may scale and translate this circle within the base domain. The second circle (dotted) in this figure is used for mapping the derivatives, as discussed in the next section.

Figure 1: A blending cylinder on a cylindrical NUBS surface.

In the second case, both the base and features surfaces are NUBS cylinders. Again, only one of the feature cylinder's edges is pasted on the base as illustrated in Figure 2 (left), with the top cylinder as the base. To locate the edge of the feature surface on the base surface, we again map an edge of the feature's domain into the base domain. As shown in Figure 2 (right), the mapping of this edge is different. Since the base is a cylinder, we map the edge of the domain to a line that spans the base domain. Since the two sides of the base domain represent the seam of the cylinder, we have mapped the closed curve of the edge of the feature surface to a closed curve on the base surface. The arrow in this figure is used to map the derivatives, as discussed in the next section.

Figure 2: A blending cylinder on a cylindrical NUBS surface.

Control point displacement scheme

We mapped the L0 layer of the feature cylinder in the same manner as standard pasting. Each L0 point is located at the Greville point in the embedded feature domain, giving a zero displacement vector relative to its local frame, and the standard pasting procedure is used to paste these points. In other words, the boundary control points of the feature cylinder map to points on the base surface. This procedure is done for both edges of the feature cylinder. The result is that the boundary of the pasted cylinder will lie close, but not directly on, the base surface. If the C0 discontinuity is too high, it can be reduced by performing knot insertion on the feature cylinder, as is done for standard pasting.

Next, we map the L1 layer to approximate C1 continuity, which in standard pasting required the displacement vectors for the L1 layer control points to be set to zero. With cylindrical pasting, we only have the boundaries of the cylinder's domain associated with the base domain. So in our initial attempt, for pasting a cylinder on a NUBS surface we placed a second circle in the base domain, with the same center but smaller radius than the first circle (the dotted circle of Figure 1). And to paste a cylinder on another cylinder, we associated the L1 layer with the base domain by using a vector perpendicular to the location of the feature domain edge within the base domain (the arrow in Figure 1). In both cases, once the identification of the L1 layer of the feature with the base domain was made, we mapped the L1 layer of the feature in the same manner as standard pasting.