The field of the invention is stereolithography, and more particularly improved stereolithography methods and apparatus for manufacturing parts or objects more rapidly, reliably, accurately, and economically.
It is common practice in the production of plastic parts or objects to first design the part and then produce a prototype of the part. This requires considerable time, effort, and expense. For example, tooling or molds may be required, even to produce just the prototype. The design is then reviewed and often times the laborious and expensive process is again and again repeated until the design has been optimized. After design optimization, the next step is production. Most production plastic parts are injection molded. Since the design time and tooling costs are very high, plastic parts are often only practical in high volume production. While other processes are available for the production of plastic parts, including direct machine work, vacuum-forming and direct forming, such methods are typically only cost effective for short run production, and the parts produced are usually inferior in quality to molded parts. Techniques have been developed in the past for making three-dimensional objects within a fluid medium. These techniques involve selectively curing the medium, e.g., a resin, with a beam of radiation. U.S. Pat. Nos. 4,041,476; 4,078,229; 4,238,840 and 4,288,861 describe some of these techniques. All of these techniques or systems rely on the buildup of synergistic energization, or curing energy, at selected points deep within the volume of the fluid medium, to the exclusion of all other points in the fluid volume. These systems however, encounter a number of problems with resolution and exposure control. The loss of radiation intensity and image forming resolution of the focused spots of the beam, as they are directed deeper into the fluid medium, create complex beam control situations. Absorption, diffusion, dispersion and diffraction all contribute to the difficulties of working deep within the fluid medium on an economical and reliable basis.
In recent years, xe2x80x9cstereolithographyxe2x80x9d systems, such as those described in U.S. Pat. No. 4,575,330, which patent is incorporated herein by reference as if set forth herein in full, have come into use. Stereolithography is a method for automatically building simple or complex parts (e.g., plastic parts) by successively xe2x80x9cprintingxe2x80x9d cross-sections or layers of a solidified fluid-like building material on top of each other, with all of the layers joined together to form a whole part. The building material may be, for example, a photopolymer which is solidifiable upon exposure to UV radiation or the like. Powder material, which forms a solidified mass when sintered by conducted or radiated heat from a heated element or source of IR radiation or the like, and powders which are solidifiable by the addition of a reactive chemical such as a binder, may also be utilized. This method of fabrication is extremely powerful for quickly reducing design ideas to physical form and for making prototypes.
One type of useful fluid medium, photocurable polymers (photopolymers) change from a liquid to solid when exposed to light. Their photospeed under ultraviolet light (UV) is fast enough to make them practical building materials. The material that is not polymerized when a part is made is still usable and remains in the vat as successive parts are made. In one embodiment an ultraviolet laser generates a small intense spot of UV. This spot is moved across the liquid surface with a galvanometer mirror X-Y scanner. The scanner is driven by computer generated vectors. After each successive surface is exposed by the laser, an elevator lowers the object further into the vat and allows another layer of fresh liquid to cover the surface of the object for formation of the next layer. Precise complex patterns can be rapidly produced with this technique.
The laser, scanner, photopolymer vat and elevator, along with a controlling computer and possibly a separate computer for creating appropriate cross-section data from initially supplied three-dimensional object data, combine together to form a stereolithography apparatus, referred to as an xe2x80x9cSLA.xe2x80x9d An SLA is programmed to automatically make a part by drawing its various cross-sections, one layer at a time, and building the part up layer-by-layer.
Stereolithography does not use tooling, molds, dies, etc. Since it depends on using a computer to generate cross-sectional layers or patterns, an SLA can be readily linked (i.e., a data link) to computer aided design as a computer aided manufacturing (CAD/CAM) apparatus.
Many photopolymers have a xe2x80x9cminimum solidifiable thickness,xe2x80x9d i.e., a minimum thickness below which they cannot be sufficiently cured to form unsupported regions of transformed, cohesive material. For example, with presently preferred fluid photopolymers, if an attempt is made to try to form a feature of an object having a thickness less than the minimum solidifiable depth (MSD) or thickness, that feature will either simply fail to sufficiently solidify to become part of the object, or it will slump (i.e., fail to hold its shape) when the object or individual layer is moved relative to the vat of fluid photopolymer. The minimum solidifiable thickness of a building medium (e.g. photopolymer) is not only a characteristic of the building medium or material itself but it also depends on the synergistic stimulation source chosen (e.g. the solidifying radiation such as ultraviolet light) and the environmental conditions surrounding the material. For example, oxygen absorbed in a photopolymer can act as a reaction inhibitor. Therefore, as used herein, xe2x80x9cMSDxe2x80x9d refers to the minimum solidification depth obtainable with a given material/solidification environment combination. The minimum solidification depth can also be considered the depth resulting from the minimum exposure that is preferred for curing down-facing features of an object what ever the basis for this preferred minimum. It may be based on a desire to form a minimum solidified thickness of material from a single layer, which minimum thickness is selected for its ability to withstand curl distortion or to supply sufficient structural integrity. These definitions can apply to any fluid-like material whether liquid, powder, paste, emulsion, or the liquid. Furthermore, these definitions can also apply to building material that is applied in sheet form and then transformed.
Many liquid building materials also have a Minimum Recoating Depth, MRD, or thickness; i.e. a minimum coating thickness that can reliably be formed over previously solidified material. This minimum recoating depth, may derive from a dewetting phenomena that occurs between the liquid material and the previously solidified material. Alternatively, the MRD may simply be based on apparatus or process limitations regarding the timely formations of coatings; in other words, the minimum thickness may be set by a maximum acceptable recoating time or accuracy limitation. This alternative definition can be applied to both liquid and powder materials.
Stereolithography makes objects layer by layer. Since the MSD is the minimum solidification depth for forming unsupported regions of layers (i.e., down-facing features of the object), these regions must be given a cure depth of at least the MSD regardless of the thickness between individual layers or cross-sections from which the object is being formed. Therefore, due to the layer-by-layer formation process, even if the layers being used are thinner than the MSD, the accuracy of the stereolithographically reproduced object is limited by the MSD of the material being used.
Moreover, because of the layer-by-layer formation process of stereolithography, the MRD sets the minimum coating thicknesses that can be effectively utilized by standard stereolithographic techniques. This minimum coating thickness directly sets the vertical accuracy obtainable when using standard stereolithographic techniques.
In the remainder of the specification, the MSD and MRD may be expressed in either (1) particular units of length or (2) without units or followed by an xe2x80x9cLTxe2x80x9d. xe2x80x9cLTxe2x80x9d is an acronym for xe2x80x9clayer thicknessxe2x80x9d and in particular the thickness for layers having the desired resolution. If the MRD is expressed without units, or followed by an LT, it should be understood that it is expressed in terms of the number of layers, each possessing the desired resolution, that yield a thickness equal to the MRD.
Accordingly, it is an object of the invention to overcome the MSD limitation by providing a method and apparatus of practicing high resolution stereolithography when using a building material that is inherently incapable of making unsupported thicknesses of solidified material as thin as the desired level of accuracy when solidified by the chosen synergistic stimulation.
Additionally, it is accordingly an object of the invention to overcome the MRD limitation by providing a method and apparatus of practicing high resolution stereolithography when using a fluid-like building material that is inherently incapable of reliably forming coating over previously solidified material as thin as the level of accuracy desired when forming the object.
Another object of this invention is to provide a method and apparatus for enhancing object production by horizontally distinguishing regions of an object, to be formed, based on their distance from a given point, points, surface, or surfaces so as to form those regions with different building parameters.
Another object of this invention is to provide a method and apparatus for enhancing object production by vertically distinguishing regions of an object, to be formed, based on their distance from a given point, points, surface, or surfaces so as to form those regions with different building parameters.
A further object of the invention is to provide a method and apparatus for automatically performing xe2x80x9cZ-error correctionxe2x80x9d, i.e., correcting for the condition in which the relative distance between an up-facing and down-facing feature of an object produced through stereolithography is greater than a desired amount due to the MSD or MRD being greater than the desired LT of a layer situated between the up and down facing features.
Another object of the subject invention is to provide improved methods of representing an object which facilitate Z-error correction, the display of the object on a graphical display device which is not equipped to display a fine level of detail regarding the object, (e.g. insufficient processing power to display object and its movement in a timely manner) and various manipulations to the object representation such as scaling-up or scaling-down operations.
An additional object of the invention is to provide a method and apparatus for reducing rounding errors (e.g. errors associated with rounding of polygon vertices of NURB control points to slicing planes).
Further objects of the invention include utilization of the above objects alone or in combination with any two or more of the above objects into combination methods and combination apparatus to provide further enhancements to stereolithography. Other objects, useable alone or in combination, will be apparent to one of skill in the act from the teachings found herein.
Additional details about stereolithography are available in the following U.S. Patents and Patent Applications, all of which are hereby fully incorporated by reference herein as though set forth in full:
U.S. Pat. No. 5,321,622, referenced above, is particularly relevant to the instant invention. It describes the use of Boolean operations in determining which portions of each layer continue from the previous layer through the present layer and through the next successive layer and which portions are up-facing or down-facing or both. Therefore, this referenced patent describes methods and apparatus for comparing initial data associated with each layer, and comparing such data between layers to form resulting data that will be used in the process of physically reproducing the object. Additionally, this referenced patent describes the use of such operations to yield appropriately sized objects (e.g. undersized or oversized). Utility of the concepts of this referenced patent, to the instant invention, will become apparent by following the teaching of the instant invention as described hereinafter.
This invention allows the formation of higher resolution objects than traditionally thought possible when using a given building material (e.g. a given liquid photopolymer or powdered material).
The invention allows the use of materials, which have not been considered capable of producing high resolution objects by stereolithographic methods, to create many of these high resolution objects through improved stereolithographic techniques. In terms of photopolymers, these heretofore non-high resolution photopolymers typically have absorption and solidification properties which make them incapable of forming cohesive solid plastic of thickness less than some amount (e.g. 1 mm). In the normal practice of stereolithography, using one of these materials, all vertical features of an object occur at positions that are nominally integral multiples of a layer thickness which is greater than or equal to the MSD. Alternatively, if a finer layer thickness is used, all down-facing features are supplied with an amount of exposure that results in a net depth of cure greater than the layer thickness which results in relative displacement between up-facing and down-facing features which is greater than a desired amount, as well as other associated errors in object configuration.
The techniques described herein involve delaying the curing of at least portions of some cross-sections at least until recoating of at least one additional cross-section has occurred, after which sufficient exposure is applied to achieve the desired depth of cure. In other words, the techniques described herein provide an improved stereolithographic method for forming a three-dimensional object wherein data descriptive of at least portions of two cross-sections is modified by shifting said data from a first cross-section to a second cross-section which is located at least one layer thickness from said first cross-section and by differencing the shifted data from any other data on the second cross-section and on any intermediate cross-sections between said first and second cross-sections and wherein said modified data is used in forming said three-dimensional object.
In the practice of the present invention, the smallest single solid vertical feature is still equivalent to the MSD. However, vertical features of the object are no longer necessarily reproduced in steps that are integral multiples of a layer thickness that is greater than or equal to the MSD or wherein exposures are blindly applied that result in cure depth that cause relative displacement of up and down-facing features of the object. Using the techniques as taught herein, vertical features can be formed from smaller steps (layer thicknesses) than the MSD while simultaneously ensuring that relative displacement of up and down-facing features does not occur (as long as the feature""s minimum thickness is greater than the MSD).
The typical practice of stereolithography involves the transformation (curing), to a depth substantially equal to or greater than the layer thickness, of all areas of each cross-section prior to coating the partially formed object with a layer of untransformed or unsolidified material in preparation for formation of a next layer of the object. This typical practice may or may not involve the utilization of somewhat different depths of cure, wherein the depth of cure depends on whether the area being cured is used for adhesion to the previously formed layer or is being cured as a down-facing feature of the object. In the practice of the present invention deviations are made from the typical approach, wherein these deviations involve leaving untransformed material on at least one portion of one cross-section, at least until after the cross-section has been coated over with untransformed material in preparation for formation of an additional layer of the object, and wherein the portion(s) will be solidified by transformation of material after the formation of the coating.
Layer-to-layer comparisons are made to determine the depth to which the material can be solidified to ensure adequate adhesion to previously formed layers, and to ensure adequate strength (modulus of solidifiable material) while simultaneously ensuring that material is not solidified to a depth that causes penetration into a region that should remain unsolidified. Solidification depth is achieved by appropriate specification and control of synergistic stimulation which is used to exposure the surface of the material.
These comparisons form the basis of selective curing which enable individual portions of each cross-section to be transformed in association with the most appropriate layer and vertical level during object formation. By the selective curing aspect of the invention, a balance is maintained between necessary structural integrity, desired resolution, and the resulting accuracy.
When using a photopolymer material, the minimum solidification depth (MSD) is related to the wavelength of radiation used. The MSD is typically directly related to the penetration depth of the material. Use of various penetration depths in the stereolithography process is described in U.S. Pat. No. 5,182,056. The methods of this referenced application can be combined with the teachings of the present invention.
Many objects that cannot be built accurately with standard stereolithography while using one of these materials of relatively high MSD, can be built accurately with the techniques of this invention. However, even with the techniques of the present invention some objects i.e., those having solid vertical features thinner than the MSD may suffer from accuracy problems. However, these can be handled in various ways as described herein.
The present method leads to more accurate creation of objects than is possible by use of typical stereolithographic techniques for a given building material with a given MSD and it also provides a more rigid xe2x80x9cgreenxe2x80x9d part or object. Reduction in distortion may also be achieved due to increased green strength along with staggered solidification of the material. Not all material to be solidified in a given area of a cross-section is necessarily solidified on that cross-section. It may be solidified through and simultaneously with a higher cross-section or layer, i.e., with the solidifying radiation penetrating downward through higher layers into the appropriate region.
As noted above, a portion of a layer forming a down-facing feature should only be cured to a depth of one layer thickness. However, in actual practice, down-facing regions are typically given a cure depth significantly greater than one layer thickness. This excess cure depth results in a distortion of the vertical dimensions of the object. The instant invention may be used to correct this over-curing problem by delaying the exposure of a down-facing region of the object for one or more layers until the object thickness is at least as great as the minimum cure-depth that will be obtained when exposing the down-facing region. In this embodiment, the MSD is defined as the minimum solidification depth that will be used in forming down-facing regions. This embodiment becomes especially attractive when the desired resolution becomes extremely small. For example, as resolution demands become higher, layer thicknesses become smaller. Typical layer thicknesses are on the order of 4, 5, 6 or 10 mils, but are steadily being pushed to 2 mils, 1 mil or even less, whereas with current materials the exposures typically preferred result in cure-depths for down-facing features of typically between 8 and 16 mils. Furthermore, in this class of embodiments the cure depth applied to down-facing features may be based on the exposure applied in a single layer or alternatively it may be based on the exposure applied to two or more consecutive layers wherein the depth of cure, as measured from the top of the first layer, is increased by print through of exposure from one or more higher layers.
Another feature of the present invention is the use of cross-sectional slices thinner than the MSD. These thin cross-sections in combination with the present solidifying or curing techniques will yield higher resolution parts than those obtainable using cross-sectional slices equal to the MSD. The minimum feature thickness will still be the MSD. Any errors due to this minimum feature thickness will present a problem in only a small percentage of the objects that can be built using stereolithography.
The invention also contemplates a method for making the surface of an object built with a particular layer thickness (for the bulk of the object) appear as if it were constructed from finer layers. In addition, the instant method relates to not only making the surface appear more continuous (i.e., finer layers) but also building the bulk of the object with thick layers at the same time while maintaining the overall accuracy associated with finer layers. This method is based on creating cross-sections having a vertical spacing equal to the desired resolution and comparing these cross-sections two or more at a time. This may be used to determine which portions of a cross-section require building at fine layer increments and cure depths (e.g. 5 mils or less); which portions can be built using greater cure depths; and which portions can be skipped altogether for building at even coarser layer increments (e.g. 10, 15, or 20 mils or more). Many of these methods require the use of a material that has the capability of being solidified to unsupported thicknesses at least as thin as the fine layer increments (e.g. 5 mils or less). Several embodiments to these novel methods are described herein.
A further embodiment of the instant invention can be used to form higher resolution objects than normally considered possible when using a building material that will not reliably form coatings thinner than a given amount. The minimum thickness for reliably forming a coating is referred to as the MRD. For example, one may wish to form an object with a layer thickness, or resolution, of 5 mils or less when using a material that will not reliably form coatings thinner than 10 mils or more. These recoating problems can be based on two phenomena: (1) excess time involved in forming coatings of the desired thinness when a thicker coating can be formed in a more timely manner; and/or (2) dewetting of all or portions of the previously formed layer of the object when one attempts to form a coating which is too thin. This dewetting phenomena is based on an incompatibility between the solidified building material and the liquid material. It has been found that the seriousness of this dewetting phenomena is greatly influenced by the exposure style which is used to solidify the layer which is being coating over. For a more detailed discussion of exposure styles, reference is made to concurrently filed Lyon and Lyon Docket No. 212/055, which is a continuation-in-part of U.S. patent application Ser. No. 08/233,026, filed Apr. 25, 1994. This concurrently filed application is incorporated by reference herein as if set forth in full. In this embodiment, layer comparisons are utilized to derive exposure data for each layer wherein the net thickness of unsolidified material for each region, to be exposed, is greater than or equal to the minimum coating thickness that can be reliably formed. It is preferred that all portions of each layer which are to be cured in association with a given cross-section should be exposed only when the coating thickness is greater than or equal to the minimum coating thickness. However, it is envisioned that in certain situations it may not be critical to ensure proper coating depth prior to exposure for some regions. In these circumstances only critical portions of layers may be involved in the potentially staggered curing required by this embodiment.
An additional embodiment involves the use of layer comparisons to define additional types of regions to be cured. These layer comparisons can be used to define a variety of regions. For example, these layer comparisons can be used to defined extra regions to be skinned or to indicate which portions of the object are located above down-facing features by a particular distance and/or located below up-facing features by a particular distance. More particularly, for example, skin regions can be defined which are one, two or more layers above a down-facing feature. It is to be understood that these extra skin regions can be in addition to any shifting of skins required by the utilization of an embodiment which compensates for MSD values greater than the layer thickness. The supplying of multiple skins to strengthen up-facing and down-facing regions has particular advantages when attempting to form drainable parts than can be used as patterns for investment casting. Using multiple skins on objects in this manner is discussed further in concurrently filed Lyon and Lyon Docket No. 212/055, and its parent application from which it is a continuation-in-part, U.S. patent application Ser. No. 08/233,026.
An additional aspect of this invention, which can be used independently or in combination with the other embodiments disclosed herein, involves the use of horizontal comparison techniques to further differentiate cross-sections into separate curable regions. These different regions may be obtained from the cross-section as a whole, from within a single boundary type on the cross-section, from a combination of boundary types on a given cross-section, or from intermediate or final boundaries involved in the layer comparison techniques described herein. The horizontal comparison technique may utilize an erosion or buildup technique to define the distance between points on a cross-section or even within the object as a whole. This erosion or buildup technique may involve the use of one or more positive cure width (i.e. line width) compensation type manipulations (i.e. reductions in area) or negative cure width compensations type manipulations (i.e. expansions in area). For example, based on these operations each portion of the cross-section can be designated as being a certain distance from the exterior surfaces of the cross-section. As an alternative to labeling each portion of a cross-section, boundaries can be defined that separate selected portions of the cross-section into designated regions. As a second example, portions of the cross-section can be designated as being located a certain distance from the deepest interior point of the cross-section. As an additional example, each portion can be labeled based on its distance from the centroid of the cross-section or an axis around which the object is be rotated. Cure parameters appropriate to each defined regions or location can be defined. These cure parameters may be related to scanning speeds, vector density, vector types, or the like. For example, when forming an object that is to be built with enclosed liquid building material that is to be drained from the interior walls of the object after formation, these techniques can be used to increase the spacing between hatch vectors as one moves deeper into the interior of the object. An object can thus be formed with its internal regions being separated into two or more regions where each region is given a different exposure style.
The techniques of the present method may also be implemented solely as a distortion reduction technique and may be implemented by manipulating vectors defining boundaries, a manipulations of pixels or voxels, by other data manipulation techniques, by any other technique that results in appropriate treatment of the cross-sections, or by any combination of these.
Typically, in stereolithography, objects are built on webs or some other form of supporting structure. With the present method, the selection and placement of support structures should be carefully considered. Because of the possibility of staggering the formation of various regions of an initial cross-section to different layers, support placement is critical. Supports should be designed and placed to catch the regions that will be locally cured in association with the lowest layers.
A method is also provided for automatically performing Z-error correction by manipulating a three-dimensional object representation. The term xe2x80x9cZ-errorxe2x80x9d refers to the error which typically occurs when the MRD or MSD is greater than the desired layer thickness of a layer situated between an up-facing and down-facing feature of an object. The problem typically manifests itself in the form of a relative displacement between the up-facing and down-facing features in the built part which is different from the desired amount.
One embodiment involves performing Z-error correction after or during the process of slicing the object representation into a plurality of layer representations. Through appropriate comparisons between the data representing multiple layers, the building of down-facing surfaces can be deferred, or alternatively, the building of up-facing surfaces can be advanced, such that the Z-error is corrected in the built part.
A second embodiment involves manipulation of the .STL representation of the object prior to or at the beginning the slicing process. In this embodiment, the vertices of down-facing triangles, including all or a portion of flat and down-facing and near-flat down-facing triangles, are moved upwards to correct for Z-error. The amount shifted is preferably the same for all down-facing triangles; however, the amount may vary with triangle slope. In particular, it may be advantageous to decrease shifting as triangles become steeper. Alternatively, the vertices of up-facing triangles, including all or a portion of flat-up-facing and near-flat up-facing and possibly also near-flat up-facing triangles, are moved downwards to correct for Z-error. In one version of this embodiment, the .STL triangles are manipulated before they are rounded to slicing planes. In a second version of this embodiment, the .STL triangles are manipulated after they have been rounded to slicing planes.
A third embodiment involves formation of a new object representation, known as the .CTL format, and manipulation of it to correct for Z-error. In the .CTL format, an object is represented by a first list of unique vertices of polygons such as triangles which substantially span a surface of the object, and a second list of polygonal representations defined in terms of unique identifiers of the vertices in the first list. With the .CTL format, Z-error correction is performed simply by manipulating selected vertices in the first list. Individual manipulation of each individual polygonal representation is not required.
A fourth embodiment involves formation of another new object representation known as the three-dimensional run-length encoded (3D-RLE) format. In the 3D-RLE format, the object is represented by a plurality of groupings of Z-values, one grouping for each cell of an XY grid. In effect, each Z-value represents intersection points between the object and a plurality of lines emanating from cells of the grid. Though it is preferred that the grid be formed from orthogonal components and that the lines project perpendicular to the grid, other relationship are possible. With the 3D-RLE format, Z-error correction is performed simply by manipulating the Z-components of selected intersection points.
A method of forming a .CTL representation of an object from a .STL representation is also provided. The method involves hashing all vertices from the .STL file into a hash table and removing redundant vertices, assigning all non-redundant vertices unique identifying indicia, and representing the polygons which span the surface of the object in terms of the identifying indicia.
A method of forming a 3D-RLE representation of the object from a first object representation is also provided, The method involves associating a planar grid of cells (e.g., coplaner with the XY plane) with the first object representation, overlaying the first object representation with a plurality of lines emanating perpendicularly (e.g. parallel to the Z-axis) from the cells in the grid, determining the Z-components of all intersections points formed by the intersections between each line and the first object representation, and associating the Z-components with the cells of the grid from which the lines emanated that were used to form the respective intersection points.
A method of displaying an object at a reduced level of detail is also provided. The method preferably involves forming a .CTL representation using an artificially high rounding error value, thereby causing many triangle vertices to collapse into one another, and many triangles to become degenerate. The method involves eliminating the degenerate triangles, leaving non-degenerate triangles which expand to cover the object at a reduced level of detail.
Methods of forming a scaled-up or scaled-down representation of the object are also disclosed. These methods preferably exploit the fact that, with a .CTL file, the object representation can be scaled up and down by manipulating selected ones (e.g. all) of the vertices in the list of vertices without requiring individual separate manipulation of each polygonal representation.
Methods of forming a shell instead of a solid object are disclosed. These methods preferably utilize a combination of two .CTL representations of the object which are scaled relative to each other with the normal orientations of the triangles reversed on the relatively scaled down representation.
Methods of forming an object utilizing different building parameters in two or more shell-like zones, and methods to obtain the data necessary for such building are disclosed. For example, the object may be divided into an exterior zone of specified thickness and an interior zone. These methods utilize a combination of three or more object representations scaled relative to each other, with the normal orientations of the triangles of some representations reversed, and with pairs of consecutive representations utilized to define distinct zones.
Methods of forming objects with reduced slice layer rounding error distortion (Slice Layer Rounding Error Minimizationxe2x80x94SLREM) and the data necessary for such formation are disclosed. These methods utilize slicing planes spaced at the desired vertical resolution, the rounding of the vertices or control points to these high resolution slicing planes, and additional data processing to yield cross-sectional data having a lower but presumably more buildable resolution. This additional data processing may be similar to the techniques described herein for handling MRDs and/or MSDs which are greater than the building resolution desired or alternatively it can be varied depending on the exact relationship between the MRD, MSD and SLREM factor.
Apparatus for implementing the above methods are also disclosed.