Thinking in Fields - A Primer for Implicit Modeling

A detailed description of how fields are used in Implicit Modeling within Inspire, with videos and illustrative examples that will help you think intuitively in terms of field-driven design.

Most Computer-Aided Design (CAD) represents a 3D object by describing the surfaces that enclose it (i.e., a "boundary representation (BRep)"). This includes parametric surfaces (e.g., NURBS), tessellated surface meshes (e.g., planar triangles), and subdivision surfaces. Implicit modeling is not a BRep; it is a volumetric representation. It represents geometry as a field, which encodes information about every position in space that is external to an object, and every position in space that is internal to an object. The surface itself is implied as the boundary between the external and internal regions in space.

What do we mean when we say "field"?

The term "field" means different things to different people and different disciplines. For clarity, within the context of Implicit Modeling, the term field takes the following definition: at every coordinate in a space, usually within some practical bounds, the field returns a signed scalar value (e.g., 1.0, -3.1, 0.0, etc.). We use "field" as a shortening of the more complete term, "scalar field." The information encoded by the field is of the form {x-coordinate, y-coordinate, z-coordinate, signed scalar value}.

Implicit Modeling makes use of some important conventions within the context of geometry. Firstly, negative scalar values are assumed to be internal to the modeled object, and positive scalar values are external. The surface of the object, itself, appears at every location in space that returns a scalar value of zero. As such, the signed scalar values in the field give an immediate inside/outside check for any coordinate in space.

The following paragraphs describe the different types of fields that are useful in Implicit Modeling and that fall within the scope of this definition.

Signed-Distance Fields (SDFs)

SDFs are a special case in terms of scalar fields. The scalar value reported at each coordinate encodes the shortest distance to the surface of an object. As we move away from the surface in an outward direction, the positive scalar values become larger. Similarly, as we move away from the surface in an inward direction, effectively traveling deeper into the object, the negative scalar values become larger in magnitude. This concept is represented for the region external to the model in the image below. The modeled object is shown in solid blue, and the onion-peel layers represent isosurfaces of constant distance to the object's surface.

The image below is another way of visualizing a field. This image shows a section view of a field that represents a cylinder aligned to the green axis (the cylinder itself has been hidden). Both the internal and external regions of the SDF are visible, with the interior region shaded in blue and white bands, and the external region shaded in orange and white bands. The bands indicate how the scalar values change in space. For an SDF, these bands will be evenly spaced and not distorted, as shown below.

SDFs are particularly useful as they give instant inside/outside checks, and they tell you how close a point is to the surface of an object. This arrangement is optimal for creating accurate offset surfaces for a given geometry, and they also enable controlled blending between surfaces, filleting operations, and much more.

Unsigned Distance Fields (UDFs)

UDFs share all of the same properties as SDFs with one clear exception: all scalar values in the field have the same sign (+/-), irrespective of whether they are internal or external to the object being modeled. Some geometries produce UDFs by construction, such as lines, curves, and points. This is because there is no well-defined notion of inside and outside for these objects. However, it can sometimes be useful to construct a UDF in place of an SDF. This is achieved by taking the absolute value of the SDF, producing all positive distances.

Distance-Like Scalar Fields

Some fields behave similarly to SDFs (or UDFs), but their scalars don’t encode the exact distance to the surface. Often these fields have scalar values that are zero at the surface and increase in magnitude as a you get further from the surface (inwardly or outwardly). However, the rate at which the scalars change may be different than that of the true SDF. You can think about this in terms of how the scalar values change per unit of length of movement away from the surface. In an SDF, the scalars change in a ratio of 1:1 when comparing the scalar value with the distance moved away from the surface. Conversely, the scalars in a distance-like scalar field might vary by a ratio of 2:1, thereby not encoding the distance to the surface.

As a illustrative example, consider the following image, which shows a section view of the field of a Gyroid lattice. Despite some behind-the-scenes rescaling of the field to improve the distance-like properties, careful inspection shows that the bands (e.g., blue and white) are not evenly spaced, and they become distorted further away from the surface. This is because the underlying equation for the Gyroid lattice produces a distance-like scalar field and not an SDF.

Please note that many of the implicit modeling functions work equally well on distance-like scalar fields. However, care should be taken when blending, filleting, and offsetting these fields, as they may not create the precise thickness or radius that you require.

Binary and Piecewise Constant Fields

It is possible to create binary fields, which, by convention, are zero everywhere outside and on the surface and one everywhere inside of the surface of an object. These fields can be very useful for masking off regions in space to limit the scope of a downstream operation. For example, you may wish to apply geometry edits strictly within certain regions without applying them elsewhere. If speaking in loose terms, it is like saying “perform an operation in all of these positions, but do not perform the operation anywhere else.”

Similar to binary fields, it can be useful to create constant fields. Typically, these fields have one user-defined value inside and on the object, and either the same or a different user-specified values outside of the object. For example, you could create a field that has scalar values of 3 inside and on the surface of the object, and a value of 5 outside the object. These fields can be useful for setting (e.g., thicknesses of lattice elements in different regions).

Interpolating Scalar Fields

When using Inverse Distance Weighting, the scalar value at a position that is not specified in the point cloud data is computed as the weighted average of scalar values of points in the point cloud. The weighting with respect to each point in the point cloud is a function of the distance to each point in the cloud. Scalar values at a point specified in the point cloud data will be exactly equal to the supplied scalar value. Different exponent values can be used to control the fall-off for the distance weighting. Two example fields are shown below for a simple point cloud. Using a lower exponent (two in this case), results in each point's contribution traveling further through the field, decaying gradually, but being weaker in the vicinity of its source point. In this case, the field strength from this source point is inversely proportional to the square of the distance to the point. Increasing the exponent to 4 (second image) results in a much stronger influence in the vicinity of the source point, but a rapid decay as you move further away from this source. In this case, the strength of the contribution is inversely proportional to the distance raised to the power four.

Nearest Neighbor interpolation is simpler to visualize. Each position in the field adopts the scalar value of the nearest point in the point cloud. This technique forces abrupt changes in the resulting Interpolating Scalar field as nearby positions in space might "snap" to different scalar values. This effect is shown for the same point cloud configuration in the image below. You will note that there is no banding in the color scheme as there is no smooth variation in the field.

Using Fields to Create Shapes

As has been shown already, fields can be used to create geometry by extracting the isosurface that passes through all positions in space where the field is equal to zero. We will now give a simple example to explain how this works.

In 3D Cartesian coordinates, a popular equation for a sphere is

${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2=r^2$ ,

where $\left(x_{0,}{y}_{0,}{z}_{0,}\right)$ are the center coordinates of the sphere, and $r$ is the radius. However, to represent the sphere as a SDF, we present the equation of the sphere in the following format:

$||p-p_0||-r=0$

In this representation, $p_0$ is a 1x3 vector, identifying the centre position of the sphere. Similarly, $p$ represents the x-, y- and z-coordinate of a query point in space. As before, $r$ is the radius of the sphere. The double vertical lines, used as parentheses, denote the Euclidean 2-norm. In this case, their usage implies that we are calculating the length of the vector between the query point and the centre of the sphere. Piecing all this together, we subtract the radius of the sphere from the distance between the query point and the centre of the sphere. If the distance between the query point and center of the sphere is less than the radius, this equation produces a negative number, placing this query point inside the sphere. You can satisfy yourself that the scalar value returned by this equation will always return the signed distance from a query point to the surface of the sphere. Below is an sectioned view of the field created using this equation.

Using Fields to Edit Properties of Existing Shapes (i.e., Field-Driven Design)

We will now show that fields can be used to locally vary certain parameters or sizings for existing geometry. The image below shows a section view of a Strut Lattice, where the green face is the intersection between the lattice and the section plane. As you can see, the thickness of the struts reduces linearly from the top blue plane to the bottom blue plane. This is a simple example of field-driven design. The field was created based on the distance to the z=0 plane, where the range of distances for this model extend from -50mm to +50mm in the z-direction (the two blue planes). We use the 'Rescale' check-box option to map these distances into a suitable thickness range of 1mm to 4mm. To be clear, at -50mm to the reference plane, the thickness will be 1mm, and at +50mm from the reference plane, the thickness will be 4mm.

This principle can be applied throughout the Implicit Modeling work-flow in two ways. The first method is to define a field, usually based on some reference geometry, prior to its usage in the construction history. Worded differently, you construct the field before constructing the geometry that uses the field to control one of its parameters. To do this, you should click the Field icon in the main Implicit Modeling ribbon.

The second method is to create the field within the context of the geometry that will ultimately make use of that field. Continuing with the strut lattice example, above, both the strut diameter and the unit cell sizing can be controlled using field-driven design. This is made clear by the small field icon button highlighted in red, below. When using this method, you should set all of the parameters that are not field driven (such as the unit cell type), and then create the field, in situ, using whichever icon is relevant. In the example above, a field was created by clicking the field icon next to Strut Diameter and then defining the field using the global z-plane (z=0).

Some Common Misconceptions and Pitfalls when Working with Implicit Modeling

"I'm used to working with sketches"Traditional CAD work-flows tend to start with 2D sketches that get developed into 3D features using a handful of tools that loosely map onto manufacturing processes (extrude, revolve, etc.). Implicit Modeling is different in that is builds shapes up by defining, combining and blending between volumetric (3D) shape representations from the beginning. If you need to develop your model from sketches, you should follow a traditional CAD work-flow and then convert your model into an Implicit format when the time is right.

"Why can't I click on individual features?"Traditional CAD, driven by NURBS surfaces, explicitly defines the surfaces that envelop the modeled object. The trimming lines for each surface that form well-defined boundaries between surfaces and features are also available by construction. Implicit modeling has no such features and they cannot be readily selected, highlighted or edited. This is because the entire surface of the object is a single isosurface, implied as the boundary between the interior and exterior regions of the object's field. It is therefore difficult to select points on a surface, lines that separate features, and isolated faces. For the time being, this is the price that is paid for the additional flexibility in terms of creating complex, robust and field-driven geometry.

"Why isn't there a dimension tool?"As Implicit modeling is not sketch, feature and constraint-driven (like traditional CAD), a dimensioning tool is limited in its usefulness. Instead, geometry sizing and positions should be entered carefully at the point of creation, possibly controlled by a variable or field for added flexibility.

I want to convert/export my Implicit Modeling to Parametric CAD (NURBS)Almost every geometry representation can be converted into an Implicit format with little difficulty. This conversion process outputs a SDF for the object. However, Implicit geometry cannot always convert back to other formats as easily. Implicit geometry is easy to mesh, and can be outputted as .STL, .OBJ and .3MF formats. Users should be mindful that very complex implicit models may produce dense meshes with very high triangle counts. The route back to parametric CAD is via the PolyNURBS tool. This tool can be used to fit subdivision surfaces to the implicit model, making them editable CAD surfaces. Again care should be taken during these steps as you will be trading off compute time and accuracy during the fitting process. Parametric CAD is a poor choice for a very complex model as their will be too many surfaces to handle the model efficiently.

"Why do sharp edges look rounded?" Although many fields technically have infinite resolution, they ultimately end up being sampled using a 3D grid. The coarseness or fineness of this grid is controlled by the 'Visualization Quality'. Were the grid to be infinitely fine, sharp edges would be perfectly sharp. As the grid becomes coarser to match the computing resources available, sharp edges become somewhat rounded in both rendering and any subsequent conversion to e.g. a triangle mesh. To illustrate this, the two images, below, show the same lattice structure viewed on Very Low and Very High visualization quality, respectively. To minimise this effect, users should finish their design in the highest visualisation quality that their computer can tolerate, possibly even using a custom quality.

A Not-So-Obvious Pitfall: Not All Implicit Modeling Operations Preserve Distances When Combining SDFs

A common operation in implicit modeling is to combine separate bodies to form one connected body. This is often referred to as a Boolean union operation, but implicit modeling uses the shorter term "Combine." One would be forgiven for thinking that combining two bodies, each represented by an SDF, would result in another SDF. However, let's take a look at what happens to the field in this case. The images below were created using the following steps (the first image shows a section view of the field, and the second image shows a section view of the 3D object):

1. Create two cuboid primitives that are offset from each other but still overlapping.
2. Combine the objects (Boolean union).
3. Create an inward shell of the combined object to create a hollow object of constant wall thickness.

Careful inspection of the above images will show that the wall thickness is not constant. Specifically, the region where the wall thickness causes the internal corners to touch represents a wall thickness that is larger than expected. The corrected solution is shown in the image below. Note that the wall thickness is now constant, with appropriately rounded corners, as requested.

To summarize, simply combining multiple SDFs does not necessarily result in a SDF after the operation. The same is true of other operations, such as smoothing, morphing, and offsetting operations. It is useful to keep this in mind when using implicit modeling in general.

Concluding Remarks

You have now had an introduction to what is going on behind the scenes in Inspire Implicit Modeling. Keeping this information in mind when working with the software will enable you to exploit the major benefits of Implicit Modeling, including field-driven design, whilst avoiding some of the common pitfalls. Specific information on how to use each of the tools within the Implicit Modeling ribbon are given throughout the rest of the help files.