Laser cutting is one of the main modern technologies in sheet metal product manufacturing industry. The use of laser cutting brings many benefits, such as minimizing the cutting width and reducing thermal deformation . A typical production process consists of designing parts, nesting the parts on metal sheets, and cutting the parts from the sheets . Although it seems pretty neat, there are many cutting processes to consider between nesting parts and cutting plan, like piercing, bridging and common cut. In recent years, many researches in the field of laser cutting have considered the influence of cutting processes. Huang et al.  studied the layout and path planning of rectangular parts with the influence of piercing. Rao et al.  designed a system on parts layout and path planning for irregular shape parts. Raggenbass et al.  and Pan et al.  also studied on layout and path planning system for irregular shape parts. Fig. 1 illustrates an example of nesting parts on a metal sheet. As shown in Fig. 1, a part is defined as an outer contour with possibly a set of inner contours forming holes in the part. A contour itself is composed of a set of geometry elements which can be line segments, polylines, circles, arcs, ellipses, B-spline curves etc. Each element is defined by its end nodes and can be cut in both directions (unless explicitly specified otherwise). To minimize the material used on the metal sheet, nesting algorithms attempt nesting parts within parts (circular parts nested inside ring parts in Fig. 1). Most researches on laser cutting focus on cutting path planning after nesting, while few consider the influences of piercing, bridging and common cut .
Piercings are operations where the laser penetrates the metal sheet. This is required when the laser starts to cut in an area of the metal sheet where it has not cut before. To ensure the quality of the processed parts, piercings should be arranged in the position on metal sheet where no parts have been arranged . Fig. 2 shows an example of cutting path with piercing. The complete cutting path for part in Fig. 2 can be represented as τ (P1, P2, P3, P4, P5, P6, P2, P7, P8). P1 is the piercing point as well as the start point of this cutting path. The distance from P2 to P7 is the excessive cutting segment to make sure this cutting path been cut completely. The time of piercing is related to the thickness of the metal sheet and laser power. For thicker metal sheet, piercing usually takes more time. In order to reduce piercing operations and processing time, two approaches, common cut and bridging are often used, as shown in Fig. 3.
Suppose part1 and part2 are the two parts on the metal sheet that need to be cut, as shown in Fig. 3. Fig. 3a shows the conventional approach. The laser beam begins to cut from point P1, goes around the shape contour, and ends at point P2. Then the beam jumps to the piercing point P3, goes around part2 and ends at point P4. The second approach in Fig. 3b adopts common cut. Unlike the first approach, the laser beam does not need to pierce again after it reaches point P2. Instead, it takes advantage of a common edge shared by part1 and part2 (p3-p4) and cuts only three rectangular edges of part2. Fig. 3c shows another approach which completes two parts in one cutting path by a bridge. Although common cut and bridging can reduce piercing, these two methods are still inadequate.
For the common cut approach, on the one hand, it limits the positions of the jump points (only P3 or P4 is allowed in Fig. 3b); on the other hand, it also limits the cutting sequence between parts (complete contour with pierce must be cut first, and residual detached contour later). These two limitations make the jump distance of the common cut longer than that of other cutting approaches, and also make the cutting path planning task more difficult.
For the bridging approach, it is realized by connecting adjacent separate cutting paths to reduce the piercing operations, as well as the number of jumps and the air move. Rather than adding restriction like common cut, bridging will simplify the cutting path planning. However, as shown in Fig. 4, bridging may damage the contour characteristics of the parts. Fig. 4a and Fig. 4b demonstrate the situations that how bridge can be harmful to the geometry precision of the parts when it is connected to the corner or curve of the part contour. Fortunately, the problem can be solved by optimizing the location of bridges .
Although piercings are very expensive, air moves are considerably cheaper than cut moves. It follows that a bridge cannot be longer than a certain length before a combination of piercing and air move becomes cheaper . Generally, in order to reduce the processing time and the part stress caused by the bridge, the bridge should better be short. Based on this, finding the location to build bridge between parts is similar to calculate the shortest distance between contours. Manber et al.  used bridge to minimize the number of pierce point for contour groups connected through common cuts. Dewil et al. [10–13] made a lot of contributions in laser cutting field and proposed that a bridge can be used to connect two sub-regions to reduce the number of piercing. Although the bridging concept is popular in laser cutting, the bridging algorithm itself is seldom researched. To the best of our knowledge, no public literature can be found on this aspect.
In this work, an efficient bridging algorithm based on Delaunay triangulation is proposed and implemented in the laser cutting application. Specifically, the part contours that have been previously nested are discretized into a point cloud. With the help of Delaunay triangulation, the point cloud is triangulated into multiple triangles, then some invalid triangles are screened out to decrease computational complexity, and the shortest distance between adjacent parts are calculated by the remaining triangles. Eventually, the target bridge is built from selected shortest distance segments. The whole bridging algorithm proceeds quickly since the time complexity of Delaunay triangulation is near linear. It can bridge all parts in O (n log n) time, in which n is the size of the discretized point. Consider that bridging may damage the contour characteristics of the parts, some limitations are set during discretizing part contours and screening out invalid triangles. Besides, the problem of bridging into a ring (bridge loop) is solved while calculating the shortest length of all bridges.
The remainder of this paper is organized as follows. In Section 2, the structure of the bridge and the algorithm input are introduced. Section 3 introduces the method of discretizing nested parts into a point cloud. In Section 4, constructing triangles and building bridges by selected triangles are presented. Comparison of bridge effect with commercial software and actual cutting experiments are given in Section 5. The last section concludes this paper.