A key problem for all 3D printers is to convert a defined solid into a set of horizontal slices (planes) through the solid. Each slice will represent one scan-pass of the printer head for the 3D printer as it builds up the solid, one layer at a time.
Each slice through the solid should be defined as a binary 2D surface array, with a binary one in the array indicating that specific point on the plane is inside the solid, and a binary zero in the array indicating that the point in the plane is outside the solid. A one in the scan plane array indicates that the printer should deposit some material at this point in the plane, and a zero indicates that the printer should not deposit any material at this point in the plane. For this problem, we will define a solid using a set of points in 3D space. The points define the vertices on the surface of the solid. The edges of the solid are defined as the lines between the points. The faces of the solid are defined by the set of points which are in the same plane. In the general case, the faces of the solid's surface can be made as numerous and as small as required, to form the required shape. Our solids will be defined in a 3 dimensional space, with points in the space represented by an x,y,z triple. Thus a solid tetrahedron can be represented by four points in the space. The four points 0,0,0 100,0,0 0,100,0 and 0,0,100 define a simple right tetrahedron. This will be the first solid that we want to print with our 3D printer. We define a horizontal scan surface for our printer as the plane defined by the three points 0,0,0 100,0,0 0,100,0. There is no reason to define the scan surface array any larger than the solid itself, so the scan surface array can be a 100 x 100-point binary array. Each point in a scan-surface array must indicate whether it is inside or outside the solid shape. If the point is inside the solid, a binary one will tell the printer to deposit material at this point. If the point is outside the solid, a binary zero will tell the printer to not deposit any material there. For our 3D printer, we need to generate 100 of these binary scan surface arrays, one for each resolution step as the printer moves up through the solid space. These 100 scan surface arrays define where the 3D printer will deposit material on each layer, to form our tetrahedron. The 3D printer will start with the lowest layer of the scan arrays, and build the material up from there. The problem is to define a J verb that will generate the one hundred 100x100 2D binary arrays (essentially building a 100x100x100 3D binary matrix), which represents the 100 horizontal slices through the solid. That binary matrix will indicate to the 3D printer where to place the material to create the solid. Once you have generated the 100x100x100 binary array for the tetrahedron, test your function out by generating a similar scan surface array for the solid created by the intersection of two tetrahedrons: Tetrahedron 1: 50,0,0 0,50,0 50,50,0 25,25,100 Tetrahedron2: 0,50,0 0,0,50 0,50,50 100,25,25 For further challenges, see if your J function can generate the set of 3D printer scan surface arrays for the four other Platonic solids besides the tetrahedron. Here are the J definitions for all the Platonic solids: http://tinyurl.com/bry2yxc You probably will want to scale these solids up by a factor of 10 or so. In general, your J function should be able to generate binary scan surface arrays for a 3D printer given any arbitrary solid, defined by a set of points in 3-space. Actually, besides the points, for complex solids you may need a vector indicating which side of each surface or face is the inside of the solid. Skip ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
