BACKGROUND

[0001] In three-dimensional (3D) printing, one technique to create an object is with successive layers of material. The successive layers of material may be applied under computer control. The object may be formed from a variety of sources. For example, the object may be formed from a 3D model. The object may be formed of virtually any geometry. Further, the object may be formed of any material, including ceramics, metals, polymers, and composites.

BRIEF DESCRIPTION OF DRAWINGS

[0002] Features of the present disclosure are illustrated by way of example and not limited in the following figure(s), in which like numerals indicate like elements, in which:

[0003] Figure 1 illustrates an architecture of an apparatus for density rank matrix generation for three-dimensional printing, according to an example of the present disclosure;

[0004] Figure 2 illustrates an environment including the apparatus 100, according to an example of the present disclosure.

[0005] Figure 3 illustrates a high level flowchart of density rank matrix generation, normalization, and rendering for the apparatus of Figure 1 , according to an example of the present disclosure;

[0006] Figure 4A represents a pattern formed by packing spheres with a base that forms a triangle, and Figure 4B represents a model of the underlying lattice suggested by the arrangement of Figure 4A, according to an example of the present disclosure;

[0007] Figure 5A represents a pattern formed by packing spheres with a base that forms a square, and Figure 5B represents a model of the underlying lattice suggested by the arrangement of Figure 5A, according to an example of the present disclosure;

[0008] Figure 6A illustrates a portion of a triangle pyramid lattice with a horizontal base, and Figure 6B illustrates an equilateral triangle with perpendicular bisectors, according to an example of the present disclosure;

[0009] Figure 7 illustrates a model of a rectangular period used to form a square pyramid lattice, according to an example of the present disclosure;

[0010] Figure 8 illustrates a line skeleton of a rectangular period, according to an example of the present disclosure; [0011] Figure 9 illustrates details of a density rank matrix generator of the apparatus of Figure 1 , according to an example of the present disclosure;

[0012] Figure 10 illustrates a unit skeleton line set for a tetrahedral-octahedral lattice defined by pairs of end points, according to an example of the present disclosure;

[0013] Figure 11 illustrates a scaled skeleton line set, according to an example of the present disclosure;

[0014] Figure 12 illustrates an example of a matrix point and its associated line determined by the shortest distance, according to an example of the present disclosure;

[0015] Figure 13 illustrates a technique for finding the closest distance from a point to a line, according to an example of the present disclosure;

[0016] Figures 14A-14D illustrate 2x2x2 period output for different densities, according to an example of the present disclosure;

[0017] Figure 15 illustrates a radial cross section of a density map of a sphere, according to an example of the present disclosure;

[0018] Figure 16 illustrates one octant of an output object of a variable density sphere input, according to an example of the present disclosure;

[0019] Figure 17 illustrates a flowchart of a method for density rank matrix generation for three-dimensional printing, according to an example of the present disclosure;

[0020] Figure 18 illustrates another flowchart of a method for density rank matrix generation for three-dimensional printing, according to an example of the present disclosure;

[0021] Figure 19 illustrates a further flowchart of a method for density rank matrix generation for three-dimensional printing, according to an example of the present disclosure; and [0022] Figure 20 illustrates a computer system, according to an example of the present disclosure.

DETAILED DESCRIPTION

[0023] For simplicity and illustrative purposes, the present disclosure is described by referring mainly to examples. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. It will be readily apparent however, that the present disclosure may be practiced without limitation to these specific details. In other instances, some methods and structures have not been described in detail so as not to unnecessarily obscure the present disclosure.

[0024] Throughout the present disclosure, the terms "a" and "an" are intended to denote at least one of a particular element. As used herein, the term "includes" means includes but not limited to, and the term "including" means including but not limited to. The term "based on" means based at least in part on.

[0025] With respect to the manufacture of 3D objects, reducing an overall amount of material used for the manufacture of a 3D object may be desirable from a weight reduction perspective. Reducing fused material in a core of an otherwise solid object may also be desirable from a thermal management perspective.

However, reducing the amount of overall material used and/or reducing fused material in a core of an otherwise solid object may reduce the object's structural strength and/or the object's ability to accommodate stresses.

[0026] In order to address the aforementioned technical challenges with respect to the manufacture of 3D objects to maintain structural strength and stress accommodation properties, an apparatus for density rank matrix generation for 3D printing and a method for density rank matrix generation for 3D printing are disclosed herein.

[0027] For the apparatus and method disclosed herein, a density rank matrix may be generated. The density rank matrix may be normalized to generate a density threshold matrix. The density threshold matrix may be analyzed with respect to details of a continuous density 3D input object to generate a bistate output object. Accordingly, the density rank matrix may be used in a 3D object manufacturing process, where the 3D object that is manufactured includes variable density, and includes a specified structural strength, specified stress

accommodation properties, and other such properties.

[0028] The attribute of density may be described as solidness or its inverse sparsity. A density of a 3D object may range from 0%, which may be referred to as empty space, to 100%, which may be referred to as solid.

[0029] The density rank matrix may be described as a periodic 3D matrix of size X by Y by Z (i.e., (Χ,Υ,Ζ)), where each element is an integer indicating the order that the element will turn on a voxel with increasing density. The density rank matrix may be referred to herein as a rectangular periodic matrix, where the density rank matrix provides a rectangular period that tiles all of three-dimensional space. The density rank matrix may provide for the rendering of a variable density 3D object. The density rank matrix may be applied to objects for which the interior needs less material or weight, while still maintaining the object strength.

[0030] The density threshold matrix may be described as a matrix for which the rank values in the density rank matrix are converted to threshold values normalized to a range of values used in a continuous density 3D input object. The density threshold matrix may be the same size as the density rank matrix (i.e., (Χ,Υ,Ζ)).

[0031] The 3D input object may be described as any object that is to be printed and includes a three-dimensional shape. The 3D input object may include discrete- space values that describe the density at each point of the object, from empty space, represented by zero, to solid, represented by a maximum value.

[0032] The bistate output object may be described as an object that includes discrete-space where each point is represented by one of two states: zero for empty space, or one for print-a-voxel.

[0033] In order to generate the bistate output object, a rendering system may be used. An example of a rendering system may include a 3D printer, and other such devices. For the rendering system, each element of the 3D input object, input(x, y, z), may be compared against a corresponding threshold value in the density threshold matrix, threshold (x, y, z), to print the bistate output object.

[0034] In order to generate the density rank matrix, elements of the density rank matrix may be generated form a skeleton line set using a line dilation technique as disclosed herein. For the line dilation technique, for each unassigned point of the density rank matrix, a minimum distance from the point to a skeleton line may be assigned. Each of the assigned points of the density rank matrix may be rank ordered by distance to generate the density rank matrix.

[0035] The skeleton line set may be described as a set of lines segments in a repeating rectangular volume that describe a specified lattice. An example of a lattice may include a tetrahedral-octahedral lattice. The tetrahedral-octahedral lattice may be described as a lattice that includes tetrahedral shapes in the interior thereof. For the apparatus and method disclosed herein, the tetrahedral- octahedral lattice may be rendered by reducing the lattice to a rectangular period.

[0036] The apparatus and method disclosed herein may provide a relatively high resolution structure definition. For example, the aspect of ranking with respect of each of the elements of the density rank matrix may provide a relatively high resolution structure definition.

[0037] Figure 1 illustrates an architecture of an apparatus for density rank matrix generation for three-dimensional printing (hereinafter also referred to as "apparatus 100"), according to an example of the present disclosure. Figure 2 illustrates an environment 200 for the apparatus 100, according to an example of the present disclosure.

[0038] Referring to Figure 1 , the apparatus 100 is depicted as including a density rank matrix generation module 102 to receive a skeleton line specification 104, and a density rank matrix size specification 106 (e.g., X). The operations disclosed herein with respect to the density rank matrix generation module 102 may be performed by a processor (e.g., the processor 2002 of Figure 20). The skeleton line specification 104 and the density rank matrix size specification 106 may be stored in and retrieved from a storage 120. As disclosed herein, the skeleton line specification 104 and the density rank matrix size specification 106 may be analyzed to generate a density rank matrix 108. The density rank matrix 108 may include rank values that range from 1 to n, where n represents a total number of elements of the density rank matrix 108. For example, for an X by Y by Z density rank matrix 108, the total number of elements n may include X times Y times Z elements.

[0039] A density rank matrix normalization module 110 may convert the rank values in the density rank matrix 108 to threshold values, where the threshold values are normalized to a range of values used in a 3D input object 112 to generate a density threshold matrix 114. The operations disclosed herein with respect to the density rank matrix normalization module 110 may be performed by a processor (e.g., the processor 2002 of Figure 20). The values used in the 3D input object 112 may be stored in and retrieved from the storage 120. The density threshold matrix may be the same size as the density rank matrix (i.e., (X, Y, Z)). The range of values used in the 3D input object 112 may represent the

specification of the 3D input object 112. For example, if the 3D input object 112 is specified with 8 bit elements, the range of values would be from 0-255. The density rank matrix normalization module 110 may normalize the rank values of the density rank matrix 108 by scaling the rank values to match an input amplitude range of input values of the 3D input object 112. For example, if the input values of the 3D input object 112 are 8 bits, then the input amplitude range may be between 0-255, and the rank values of the density rank matrix 108 may be normalized between 0-255 to generate the density threshold matrix 114.

[0040] A rendering module 116 may compare each element of the 3D input object 112, input(x, y, z), against a corresponding threshold value in the density threshold matrix 114, threshold(x, y, z), at each location as follows: if lnput(x, y, z) > Threshold (χ', y', z')

then Output(x, y, z) = 1 (i.e., printer voxel)

else Output(x, y, z) = 0 (i.e., empty space)

where x' = x mod X; y' = y mod Y; and z' = z mod Z. The "mod" may represent the modulo operation. The operations disclosed herein with respect to the rendering module 116 may be performed by a processor (e.g., the processor 2002 of Figure 20).

[0041] In some examples, the modules and other elements of the apparatus 100 may be machine readable instructions stored on a non-transitory computer readable medium. In this regard, the apparatus 100 may include or be a non- transitory computer readable medium. In some examples, the modules and other elements of the apparatus 100 may be hardware or a combination of machine readable instructions and hardware.

[0042] Referring to Figure 2, the output of the rendering module 116 (i.e., Output(x, y, z) = 1 (i.e., printer voxel), or Output(x, y, z) = 0 (i.e., empty space)) may be used by a rendering system 202 to generate a bistate output object 204.

[0043] Determination of the density rank matrix 108 is described in further detail with reference to Figures 1 , and 3-16.

[0044] For the density rank matrix 108, the skeleton line specification 104 may describe a set of lines segments in a repeating rectangular volume that describe a specified lattice. An example of a lattice may include a tetrahedral-octahedral lattice. Another example of a lattice may include a cube-based lattice. Any type of lattice structure may be used with the apparatus 100.

[0045] An example of a structural lattice includes one formed by segments connecting centers of packed equi-sized spheres. The spheres may be packed in two ways.

[0046] A first way to pack spheres includes triangular pyramid sphere packing as depicted in Figures 4A and 4B. Figure 4A represents a pattern formed by packing spheres with a base that forms a triangle. Figure 4B represents a model of the underlying lattice suggested by the arrangement of Figure 4A. The

arrangement of Figure 4B may represent a structure that connects the center of the spheres of Figure 4A. The triangular packing technique of Figure 4A may also be described as hexagonal packing.

[0047] A second way to pack spheres includes square pyramid sphere packing as depicted in Figures 5A and 5B. Figure 5A represents a pattern formed by packing spheres with a base that forms a square. Figure 5B represents a model of the underlying lattice suggested by the arrangement of Figure 5A. The

arrangement of Figure 5B may represent a structure that connects the center of the spheres of Figure 5A.

[0048] For the arrangements of Figures 4A and 4B, and Figures 5A and 5B, the line segments may all include the same length.

[0049] The packing techniques of Figures 4A and 4B, and Figures 5A and 5B may represent the same packing technique in tessellating space. In this regard, the packing techniques of Figures 4Aand 4B, and Figures 5Aand 5B may differ by an angle of rotation Θ around an axis defined by any horizontal segment in either the triangular or square pyramids.

[0050] In order to determine the properties of the tetrahedral-octahedral lattice of Figure 7, Figure 6A illustrates a portion of a triangle pyramid lattice with a horizontal base ABC, according to an example of the present disclosure. Referring to Figure 6A, the solid lines represent equal segments of length one. The dashed lines represent perpendicular segments. The angle ZDAE= Θ may represent the angular difference between a triangular pyramid and a square pyramid.

[0051] Further, Figure 6B illustrates an equilateral triangle ABC with

perpendicular bisectors, according to an example of the present disclosure.

Referring to Figure 6B, the solid lines represent equal segments of length one. The dashed lines represent perpendicular segments. The length of segment may be determined as follows: The angle Θ may be determined as follows:

[0052] With respect to the pyramid structures of Figures 4A and 4B, and Figures 5A and 5B, a repeating rectangular period may be determined in the square pyramid packing structure. A model of the repeating rectangular period is shown in Figure 7. The model of Figure 7 may represent a tetrahedral-octahedral lattice. Further, the skeleton line set corresponding to the repeating rectangular period is depicted in Figure 8. With respect to Figures 7 and 8, as the repeating rectangular period will tessellate three-dimensional space, lines that in part of the repeating rectangular period and lines that are part of neighboring periods may be identified. For example, as shown in Figure 8, the dashed lines may represent lines that are part of neighboring periods (i.e., lines that are not in the fundamental period). The top, back, and adjacent right side planes for the skeleton line set may include unique data, whereas the front, adjacent left side, and bottom planes of the skeleton line set do not include unique data. The repeating rectangular period may include a dimensional ratio

[0053] Compared to the representation of Figure 7 that represents a

tetrahedral-octahedral lattice, for the example of the cube-based lattice, the lines that in part of the repeating rectangular period may include the lines that form the outer edges of the cube-based lattice.

[0054] Referring to Figures 1 , 3, and 9, in order to generate the density rank matrix 108, the density rank matrix generation module 102 may receive as inputs the skeleton line specification 104 of a lattice as disclosed herein with reference to Figures 4A-7, and the density rank matrix size specification 106 (e.g., X). The density rank matrix generation module 102 may generate the density rank matrix 108 by beginning with the line-based skeleton (as specified by the skeleton line specification 104) as disclosed herein with reference to Figures 4A-7, and generating the density rank matrix 108 based on the distance of each element of the density rank matrix 108 from the line-based skeleton. The density rank matrix generation module 102 may take all of the elements of the density rank matrix 108 in the three-dimensional period (the rectangular period), and rank order the elements from 1 to n, where n represents the total number of elements in the density rank matrix 108.

[0055] The skeleton line specification 104 may include the size of the period and a unit skeleton line set. The size of the period may be expressed relative the x dimension of the lattice, or . For the example of the tetrahedral-octahedral lattice, . For the example of the cube-based lattice, s _y = 1 = s _z = 1. The unit skeleton line set may be expressed as a list of lines defined by two end points inside a unit volume. The unit skeleton line set for the tetrahedral- octahedral lattice may include a list of fourteen pairs of points as shown in Figure 10. For example, the fourteen pairs of points may be used to describe the bottom square, the top square, the diagonals, and the middle cross of the tetrahedral- octahedral lattice.

[0056] Referring to Figure 9, at block 900, the size of the density rank matrix 108 may be determined, and storage for the density rank matrix 108 may be allocated. The size of the density rank matrix 108 may be determined by a variable, the size in X. The other two dimensions Y and Z may be determined by a printer's voxel aspect ratio, and period size Sy and s _z . The size of the density rank matrix 108 may represent the size of the period of the structure that will be repeated.

[0057] The voxel aspect ratio is a property of the target rendering system 202. For example, for the rendering system 202, a model of a 3D printer may include a x and y resolution of 1200 dpi, and a z resolution of 254 dpi, which makes the 3D printer's voxels longer in the z direction by 1200/254 than they are in x or y directions. The voxel aspect ratio may be expressed relative to the size in x as v _y and v _z . For this example of the 3D printer, v _y = 1 and v _z = 1200/254 * 4.724. [0058] The density rank matrix 108 dimensions may be determined as follows:

For a specified size of X = 300, the other dimensions after rounding may be Y = 300 and Z = 90.

[0059] Thus, the density rank matrix 108 is a function of the skeleton line specification 104, the voxel aspect ratio, and the density rank matrix size specification 106. Once the density rank matrix 108 is determined, the density rank matrix 108 may be utilized to determine the density threshold matrix 114, which is utilized for any 3D input object 112 to generate the bistate output object 204.

[0060] At block 902, the "scale lines" process may multiply the triplets defining the endpoints of the unit skeleton set by X, Y, and Z. The resulting scaled skeleton line set is shown in Figure 11. In this regard, the unit skeleton set of Figure 10 for the tetrahedral-octahedral lattice may be scaled to include scaled values for the bottom square, the top square, the diagonals, and the middle cross of the tetrahedral-octahedral lattice.

[0061] Referring again to Figure 9, for each unassigned element, or unassigned point of the density rank matrix 108 at block 904, at block 906, the distance to each line in the scaled skeleton line set may be determined. An example of one such a point is shown in Figure 12 at 1200, where Figure 12 illustrates the closest line to the point at 1200 being line 1202. The point at 1200 may be replicated, for example, above, below, left, right, front, and back, periodically in three-dimensions in accordance with the skeleton line set.

[0062] A technique of determining the closest distance from a point to a line defined by two other points is illustrated in Figure 13. Referring to Figure 13, at 1300, in order to determine the distance d from point to the line defined by the two points and in three dimensional space, at shown at 1302, vectors a and b

may be specified as follows:

The magnitudes of the vectors and may be specified as follows:

The geometric relationship involving the magnitude of the vector cross product may be specified as follows: area defined by the parallelogram (illustrated at 1302)

Thus, the distance d may be specified as follows:

Since , a, and are constants for the independent variable d

may be specified as follows:

Referring to Figures 9 and 11, for each of the scaled skeleton lines in Figure 11, at block 908, the line data generator may store a vector and the scalar a to make subsequent distance calculations efficiently. For each point in the density

rank matrix 108, referring again to block 906, Equation (1) may be executed to determine the distance to each line.

[0063] At block 910, the minimum distance in element may be determined and stored. That is, for each unassigned element p for the density rank matrix 108 at block 904, at block 910, a minimum distance associated with the unassigned element may be assigned to the element .

[0064] At block 912, the processing at blocks 904-912 may be repeated until all elements of the density rank matrix 108 are assigned as associated minimum distance.

[0065] At block 914, the elements of the density rank matrix 108 may be sorted. For example, the elements of the density rank matrix 108 may be sorted from smallest to largest distance value.

[0066] At block 916, the element values may be replaced with the element ranks. For example, a unique rank value may be assigned from 0 to (X-1)(Y-1)(Z- 1 ). The smallest ranks may begin along the skeleton line, then gradually dilate until all elements in the rectangular period are used. In this manner, the density rank matrix 108 may tessellate all of three-dimensional space.

[0067] Referring to Figures 14A-1 D, examples of application of the rendering module 116 to generate the bistate output object 204 for fixed density inputs for four example values are illustrated. For Figures 14A-14D, the size of the bistate output objects may be 2x2x2 periods, which is used to illustrate the nature and periodicity of the dilated structures. For the case of an input density of 99% for figure 14D, the output may be visualized by the remaining empty space or holes.

[0068] As an example of a variable density input object, Figure 15 illustrates a sphere with a 10% density in the center at location 1500, gradually increasing to 100%, or solid, density at the outer surface at location 1502. A radial cross section of part of the input object is depicted in Figure 15, with bands representing changes in density. One octant of the rendered bistate output object 204 is shown Figure 16.

[0069] With respect to material blending, the bistate output object 204 may be printed with a single material, or used as a placeholder to be filled by other mixtures of materials. Rendering this substrate to include material blending may provide for establishment of a relatively strong connected structure.

[0070] Figures 17-19 respectively illustrate flowcharts of methods 1700, 1800, and 1900 for density rank matrix generation for three-dimensional printing, corresponding to the example of the apparatus 100 whose construction is described in detail above. The methods 1700, 1800, and 1900 may be

implemented on the apparatus 100 with reference to Figures 1-16 by way of example and not limitation. The methods 1700, 1800, and 1900 may be practiced in other apparatus. The steps disclosed herein with respect to the methods 1700, 1800, and 1900 may be performed by a processor (e.g., the processor 2002 of Figure 20).

[0071] Referring to Figures 1-17, and particularly Figure 17, for the method 1700, at block 1702, the method may include determining, based on a skeleton line specification 104 of a lattice structure and a density rank matrix size specification 106, dimensions of the density rank matrix 108 (e.g., see also block 900 of Figure 9).

[0072] At block 1704, the method 1700 may include determining, based on the dimensions of the density rank matrix 108, a number of elements of the density rank matrix 108. For example, for an X by Y by Z density rank matrix 108, the total number of elements n may include X times Y times Z elements.

[0073] At block 1706, the method 1700 may include sorting the elements based on an analysis of a property of each of the elements relative to the lattice structure (e.g., see also block 914 of Figure 9).

[0074] At block 1708, the method 1700 may include generating, based on the sorting of the elements, the density rank matrix 108 (e.g., see also discussion with respect to Figure 9).

[0075] The method 1700 may further include normalizing the density rank matrix 108 to generate a density threshold matrix 114, receiving specifications of a three-dimensional input object 112, and comparing each of the specifications of the three-dimensional input object 112 to a corresponding threshold value in the density threshold matrix 114 to determine whether to generate a voxel. For example, the rendering module 116 may compare each element of the 3D input object 112, input(x, y, z), against a corresponding threshold value in the density threshold matrix 114, threshold(x, y, z), at each location, to determine whether to turn on the voxel (i.e., to generate the voxel). [0076] For the method 1700, comparing each of the specifications of the three- dimensional input object 112 to the corresponding threshold value in the density threshold matrix 114 to determine whether to generate the voxel may further include applying a modulo operation to each threshold value in the density threshold matrix 114. Further, a determination may be made as to whether a specification of a three-dimensional input object is greater than a corresponding modulo based threshold value. The corresponding modulo based threshold value may represent a threshold value to which the modulo operation is applied. In response to a determination that the specification of the three-dimensional input object is greater than the corresponding modulo based threshold value, the method 1700 may include indicating generation of the voxel. In response to a

determination that the specification of the three-dimensional input object is less than or equal to the corresponding modulo based threshold value, the method 1700 may include indicating not to generate the voxel. For example, if lnput(x, y, z) > Threshold (χ', y', z'), then Output(x, y, z) = 1 (i.e., printer voxel), else Output(x, y, z) = 0 (i.e., empty space), where x' = x mod X; y' = y mod Y; and z' = z mod Z.

[0077] For the method 1700, receiving the skeleton line specification 104 of the lattice structure may further include receiving the skeleton line specification 104 of a tetrahedral-octahedral lattice structure (e.g., see also discussion with respect to Figures 7 and 8).

[0078] For the method 1700, the tetrahedral-octahedral lattice structure may include a repeating period including a dimensional ratio of 1 , 1 , and V2,

respectively, relative to first, second, and third axis of a line skeleton of the tetrahedral-octahedral lattice (e.g., see also discussion with respect to Figures 7 and 8).

[0079] For the method 1700, determining, based on the skeleton line

specification 104 and the density rank matrix size specification 106, dimensions of the density rank matrix 108 may further include determining a dimension of the density rank matrix 108 along a first axis, and determining the dimensions of the density rank matrix 108 along second and third axes as a function of a period size of the lattice structure and a voxel aspect ratio of a printer that is to generate an output object 204 based on the density rank matrix 108 (e.g., see also block 900 of Figure 9).

[0080] The method 1700 may further include determining, based on the skeleton line specification 104, a unit skeleton line set as a list of lines defined by two end points, and scaling the unit skeleton line set by multiplying values of the unit skeleton line set with corresponding dimensions of the density rank matrix 108 along the first, second, and third axes (e.g., see also discussion with respect to Figures 10 and 11).

[0081] For the method 1700, determining, based on the dimensions of the density rank matrix 108, the number of elements of the density rank matrix 108 may further include multiplying the dimensions of the density rank matrix 108 along first, second, and third axes to determine the number of elements of the density rank matrix 108.

[0082] For the method 1700, sorting the elements based on the evaluation of the property of each of the elements relative to the lattice structure may further include determining, for each of the elements, distances of each element from each specified line of a line skeleton of the lattice structure, determining, from the distances, a minimum distance for each of the elements, sorting the elements according to corresponding minimum distances defined by the minimum distance for each of the elements, and replacing each of the elements with a rank value that corresponds to an order of the sorting of the elements (e.g., see also blocks 904, 906, 910, 912, and 914 of Figure 9).

[0083] Referring to Figures 1 -16 and 18, for the method 1800, at block 1802, the method may include determining a dimension of the density rank matrix 108 along a first axis.

[0084] At block 1804, the method 1800 may include determining dimensions of the density rank matrix 108 along second and third axes as a function of a period size of a lattice structure that is to be used to generate an output object and a voxel aspect ratio of a printer that is to be used to generate the output object based on the density rank matrix 108 (e.g., see also block 900 of Figure 9).

[0085] At block 1806, the method 1800 may include determining, based on all of the dimensions of the density rank matrix 108, a number of elements of the density rank matrix 108.

[0086] At block 1808, the method 1800 may include sorting the elements based on an analysis of a property of each of the elements relative to the lattice structure (e.g., see also blocks 904, 906, 910, 912, and 914 of Figure 9).

[0087] At block 1810, the method 1800 may include generating, based on the sorting of the elements, the density rank matrix 108 (e.g., see also discussion with respect to Figure 9).

[0088] Referring to Figures 1 -16 and 19, for the method 1900, at block 1902, the method may include determining, based on a skeleton line specification 104 of a structure that is to be used to generate an output object 204 and a density rank matrix size specification 106, dimensions of the density rank matrix 108 (e.g., see also block 900 of Figure 9).

[0089] At block 1904, the method 1900 may include determining, based on the dimensions of the density rank matrix 108, a number of elements of the density rank matrix 108 by multiplying the dimensions of the density rank matrix along first, second, and third axes to determine the number of elements of the density rank matrix 108. For example, for an X by Y by Z density rank matrix 108, the total number of elements n may include X times Y times Z elements.

[0090] At block 1906, the method 1900 may include sorting the elements based on an analysis of a property of each of the elements relative to the structure (e.g., see also block 914 of Figure 9).

[0091] At block 1908, the method 1900 may include generating, based on the sorting of the elements, the density rank matrix 108 (e.g., see also discussion with respect to Figure 9).

[0092] Figure 20 shows a computer system 2000 that may be used with the examples described herein. The computer system 2000 may represent an operational platform that includes components that may be in a server or another computer system. The computer system 2000 may be used as a platform for the apparatus 100. The computer system 2000 may execute, by a processor (e.g., a single or multiple processors) or other hardware processing circuit, the methods, functions and other processes described herein. These methods, functions and other processes may be embodied as machine readable instructions stored on a computer readable medium, which may be non-transitory, such as hardware storage devices (e.g., RAM (random access memory), ROM (read only memory), EPROM (erasable, programmable ROM), EEPROM (electrically erasable, programmable ROM), hard drives, and flash memory).

[0093] The computer system 2000 may include the processor 2002 that may implement or execute machine readable instructions performing some or all of the methods, functions and other processes described herein. Commands and data from the processor 2002 may be communicated over a communication bus 2004. The computer system may also include the main memory 2006, such as a random access memory (RAM), where the machine readable instructions and data for the processor 2002 may reside during runtime, and a secondary data storage 2008, which may be non-volatile and stores machine readable instructions and data. The memory and data storage are examples of computer readable mediums. The memory 2006 may include a density rank matrix generation for 3D printing module 2020 including machine readable instructions residing in the memory 2006 during runtime and executed by the processor 2002. The density rank matrix generation for 3D printing module 2020 may include the modules of the apparatus 100 shown in Figure 1.

[0094] The computer system 2000 may include an I/O device 2010, such as a keyboard, a mouse, a display, etc. The computer system may include a network interface 2012 for connecting to a network. Other known electronic components may be added or substituted in the computer system.

[0095] What has been described and illustrated herein is an example along with some of its variations. The terms, descriptions and figures used herein are set forth by way of illustration and are not meant as limitations. Many variations are possible within the spirit and scope of the subject matter, which is intended to be defined by the following claims - and their equivalents - in which all terms are meant in their broadest reasonable sense unless otherwise indicated.