On bodies, as well as on other geometric objects, you can perform operations - a set of actions on one or more original bodies, which leads to the birth of a new body. One of the main operations for two bodies is Boolean operations.

Boolean operations are called the operations of union, intersection and subtraction of bodies, since they perform the same operations on the internal volumes of bodies (over sets of points in space located inside the bodies). The Boolean operation of combining bodies will be denoted by the formula , where are the original bodies, S is the resulting body. The Boolean operation of intersection of bodies will be denoted by the formula. The Boolean operation of subtracting bodies will be denoted by the formula. In the order of the operand bodies, we will call them the first body and the second body.

Rice. 6.6.1. Two source bodies

Rice. 6.6.2. Merging bodies

The result of the operation of combining two bodies is a body that contains points belonging to the internal volume of either the first or second body. The result of the operation of intersection of two bodies is a body that contains points belonging to the internal volume of both the first and second bodies.

The result of the operation of subtracting two bodies is a body that contains points that belong to the internal volume of the first body, but do not belong to the internal volume of the second body.

In Fig. 6.6.1 shows the initial bodies for the Boolean operation. In Fig. 6.6.2 shows the result of the operation of combining bodies, in Fig. 6.6.3 shows the result of the intersection operation, in Fig. 6.6.4 and 6.6.5 show the results of the operation of subtracting bodies.

The operation of subtracting bodies can be reduced to the operation of intersecting bodies; To do this, you need to turn the second body inside out and find points of its volume that simultaneously belong to the volume of the first body. A body S turned inside out will be denoted by

Rice. 6.6.3. Crossing bodies

Rice. 6.6.4. Body difference

Rice. 6.6.5. Body difference

When turning the body inside out internal sides the faces become the outer sides, and the outer ones become the inner ones and the directions of the cycles change to the opposite, as a result of which the internal volume of the body becomes that part of the space that was previously located outside the body. Mathematically, the operation of subtraction is reduced to the operation of intersection of bodies. When designing, the designer uses the operations of union and subtraction (they can be called differently, for example, welding and drilling operations), and the mathematical apparatus performs the operations of union and intersection, respectively. All Boolean operations contain much in common and are performed using a single algorithm.

Merging bodies

Let's consider the Boolean operation of combining bodies. The essence of the operation can be briefly described as follows: you need to find the intersection lines of the faces of the bodies, remove that part of the first body that got inside the second body and that part of the second body that got inside the first body, and build a new body from the rest. We will conditionally divide the operation into three stages. At the first stage, we will construct lines of intersection of the surfaces of the faces and, on their basis, new edges. The constructed new edges will be called intersection edges, and the edges of the bodies will be called old edges. At the second stage, we will determine the intersection points of the new edges with the old edges and at these points we will cut the old edges into several edges. At the third stage of the operation, we will rebuild the cycles of the intersecting faces. After this, we add to the intersecting faces of the bodies the faces that are topologically connected by them. Let's look at the stages of body building in more detail.

We begin the first stage of the operation of combining bodies by constructing the lines of intersection of each face of the first body with each face of the second body, if any. To do this, we use an algorithm for intersecting face surfaces.

Let the faces of the first body be described by surfaces

and the faces of the second body are described by surfaces

Based on the intersection lines of the faces of the first and second bodies

(6.6.3)

Let's construct intersection edges. In this case, we give the intersection edges the direction of the vector product of the normal of the face of the first body with the normal of the face of the second body: The direction of the edge is determined by the sign that the direction of the derivative of the intersection line coincides with the required direction of the edge. For the positive direction of the face normal we take the direction outward of the body. The normal of a face can coincide with the normal of its surface or have the opposite direction, depending on the sign of their coincidence. In Fig. 6.6.6 shows the directions of the edges of intersection of the face of the first body with two faces of the second body.

The intersection edges must be constructed so that they lie completely inside the cycles of the faces of the original bodies. Intersection edges can only approach the edges of a face at their ends. At points A, B and C (Fig. 6.6.6), the old edges of the faces must each be divided into two edges, since only part of the original face will be included in the resulting body.

At the second stage, we cut the old edges of the body, to which the intersection edges fit. Cutting an old rib is done by cutting the curve on which the rib is based. From one curve we obtain two curves, which together replace the original edge curve. One of these curves will remain the geometric carrier of the cut edge, and on the basis of the second we will construct a new edge, which will receive all the necessary information from the original edge. As was said, each edge is built on the basis of the surface intersection curve (6.1.1). A surface intersection curve consists of two surfaces and two corresponding two-dimensional curves. It is precisely two-dimensional curves on two different surfaces that are subject to dissection. Both before and after dissection, these curves must have the same parameter definition areas and correspondence of points for all parameter values.

The intersection points of the new edge with the old edge of the face are sought as the intersection points of two-dimensional curves defined on the parameter plane common to them. From each edge in formula (4.8.6) of the points of intersection of lines, one two-dimensional curve is included in the intersection line.

The parameters of the intersection points of the edges and the coordinates of the curves themselves, which are surface parameters, must be determined with a given accuracy.

If the edge being cut is based on an intersection curve defined by individual points (two-dimensional lines are broken lines and exactly coincide in space only at characteristic points), then before cutting such a curve, additional points must be inserted into both lines, corresponding to the intersection points of three surfaces - two surfaces , lying on both sides of the cut edge, and the surface of the face of another original body. For example, if the line of intersection of surfaces with surface is cut, then additional two-dimensional characteristic points corresponding to the intersection of surfaces must be inserted into the lines. The problem of intersection of three surfaces is reduced to solving a system of six scalar equations (4.12.5) with respect to six parameters. The initial approximation of the solution is known quite accurately.

Since each edge of the original bodies is included in the cycles of two adjacent faces, then after cutting the edges of the original bodies it is necessary to adjust these cycles taking into account the cut edges.

After the first two stages, we received a set of intersection edges, oriented in a certain way, joining each other and with the edges of the original bodies only at the vertices. Next, you need to rebuild the cycles of the intersecting edges.

The third stage completes the Boolean operation. In order for each of those shown in Fig. 6.6.6 faces are cut into parts, you need to rebuild its cycles and, in accordance with the cycles, change the contours that describe the area for determining the surface parameters of the face. In Fig. 6.6.7 shows two intersecting faces (thin lines with arrows show the direction of cycles of the faces of the original bodies) and the intersection edge. In Fig. Figure 6.6.8 shows those parts of the faces that will be included in the union of bodies.

Rice. 6.6.7. Initial body faces

Rice. 6.6.8. Trimmed Edges

The arrows show the direction of the reconstructed cycles of the body faces. Each loop is a list of edges in the order they appear and a list of orientation flags for those edges in the loop.

It can be seen from the figure that with the accepted orientation of the intersection edges (), they will enter the cycles of the faces of the first body with a negative flag, and they will enter the cycles of the faces of the second body with a positive flag.

The old edges of the original bodies, which will be preserved in the resulting body, will enter the rebuilt cycles, retaining their flags. When rebuilding cycles of intersecting faces, we will use the following algorithm.

Let's consider one of the two intersecting faces belonging to the first body. We take any edge of intersection of the face under consideration and start compiling a list of cycle edges from it. The intersection edge must enter the face cycle of the first body with a negative flag, therefore the cycle will have the opposite direction to the first edge. To continue the cycle, among the intersection edges and among the old edges of the face, we look for all the edges that connect with a given edge at its initial vertex. Among the found edges, we choose the one that lies to the left of the others (turns to the left at a larger angle compared to other found edges, if viewed along the cycle with outside edges). We put the selected edge in the cycle list. If the selected edge is old, then it retains its flag in the loop. If the selected edge is an intersection edge, it will receive a negative flag. Intersection edges have priority over old edges to be selected. That is, if there are two matching edges to the left of the others, one of which is the old one and the other is the intersection edge, then the intersection edge must be selected to continue the cycle.

We will continue the process of rebuilding the face cycle until the cycle is closed. This completes the construction of the next cycle. If not all intersection edges of the face under consideration were used when constructing the cycle, then we begin to build another cycle of the face from any of the remaining intersection edges. We rebuild the cycles until we use all the intersection edges. In this way we will build in general case several new cycles of the considered face of the first body.

It is convenient to search for joining edges and determine the angle of rotation at the junction point using the two-dimensional curves of the edges of the face. In this case, you can even work with edges that are contracted in space to a point (for example, an edge at the vertex of a cone or the pole of a sphere). In Fig. 6.6.9 shows that several edges join the new edge BA at point A. In this case, to continue the cycle started from the intersection edge BA A, edge AC should be selected.

We sort the newly constructed cycles into groups, each of which consists of an outer cycle and its internal cycles. The face in question may have one or more old cycles left intact. We will call untouched the old cycles of a face, not a single edge of which was included in the reconstructed cycles. Among the old untouched cycles, we will select those that need to be included in the description of the reconstructed face. They are the old inner cycles of the face lying inside the new outer cycles. We also need to determine whether the old outer edge loop will need to be included in the result if it remains intact. This must be done if for some new inner loops a new outer loop is not found and they lie inside the old outer loop. Sorting cycles is conveniently done using two-dimensional contours corresponding to each cycle.

If, as a result of sorting the outer cycles, more than one was obtained, this means that several faces were formed from the original face as a result of the operation. In Fig. 6.6.10 a - 6.6.13 a show options for the initial faces of the first body.

In Fig. 6.6.10 b-6.6.13 b. faces with added intersection edges are shown (intersection edges are highlighted). In Fig. 6.6.10 in - 6.6.13 in show the faces that resulted from the operation.

Rice. 6.6.10. Original face (a), face c. by adding intersection edges (b), the result of operation (b)

Rice. 6.6.11. The original face (a), the face with the addition of intersection edges (b), the result of the operation (c)

Rice. 6.6.12. The original face (a), the face with the addition of intersection edges (b), the result of the operation (c)

Rice. 6.6.13. The original face (a), the face with the addition of intersection edges (b), the result of the operation (c)

In Fig. 6.6.10, a face with two cycles was cut to produce one face with one cycle. In Fig. 6.6.11 a face with two cycles gave two faces. In the example shown in Fig. 6.6.12, it was necessary to use the old outer and inner loops. In Fig. 6.6.13 two faces were obtained from one face, and for one of them it was necessary to use the old outer loop.

The described reconstruction of cycles is performed for each intersecting face of the first body.

We will deal with the intersecting faces of the second body in the same way as we did with the faces of the first body, but with one small difference. The intersection edges must enter the rebuilt cycles of the faces of the second body with a positive flag (they entered the rebuilt cycles of the faces of the first body with a negative flag). This is the difference between rebuilding the cycles of the faces of the second body. All other actions on the faces of the first and second merged bodies are the same.

We have rearranged the intersecting edges of the merged bodies. We will include all of them in the shell of the resulting body. To obtain the resulting body, all that remains is to add to these faces in the body shell the faces that did not intersect in the operation, which are topologically related to the intersecting faces. To do this, we will take sequentially the edges included in the shell of the new body and include adjacent faces of the edges into the shell (if they are not there). Continuing these steps for the edges of all added faces, we obtain the shell of the resulting body.

Crossing bodies

Briefly, the essence of the Boolean operation of intersection of bodies can be described as follows: you need to find the lines of intersection of the bodies, remove that part of the first body that did not get inside the second, and that part of the second body that did not get inside the first, and build a new body from the rest.

This operation has much in common with the operation of combining bodies. Let's return to Fig. 6.6.6. It shows intersecting faces: one face of the first body and two faces of the second body. The faces cut each other, so that only parts of these faces will be included in the resulting shell.

The intersection of bodies will include part of the face of the first body, lying inside the second body, and part of the face of the second body, lying inside the first body (Fig. 6.6.14) (the union of bodies includes part of the face of the first body, lying outside the second body, and part of the face of the second body lying outside the first body).

Rice. 6.6.14. Source faces

Rice. 6.6.15. Rebuilt body intersection faces

This is the main difference between the operations of union and intersection of bodies.

We will divide the operation into three stages. The first and second stages of the operation of intersecting bodies completely coincide with the corresponding stages of the operation of combining bodies.

The third stage of the operation of intersection of bodies is performed similarly to the third stage of the operation of combining bodies, but has one difference in the flag with which intersection edges enter the cycles. The intersection edges are included in the rebuilt cycles of the faces of the first body with a positive flag, and in the rebuilt cycles of the faces of the second body - with a negative flag (in the union of bodies, the flags of the intersection edges in the cycles have the opposite meaning). All other actions on the faces of both bodies are the same in both operations.

Body difference

Briefly, the essence of the Boolean operation of subtracting bodies can be described as follows: you need to find the intersection lines of the bodies, remove that part of the first body that fell inside the second, and that part of the second body that did not fall inside the first, and construct a new body from the rest.

The Boolean operation of subtracting bodies is reduced to the Boolean operation of intersecting the body being reduced and the body being subtracted turned inside out. We will obtain a body turned inside out from the original body by reorienting the directions of the normals of the faces and the directions of the cycles of the faces. Reorientation of the direction of the face normal is performed by changing the sign of coincidence of the surface normal and the normal of its face. Reorientation of the direction of the face cycle is carried out by rebuilding the list of edges (changing to reverse order sequence of edges in the list) and replacing edge flags in lists with opposite ones. For an inverted body, the internal volume is the part of space located outside its shell. Therefore, at the intersection of a reduced body and a subtrahend body turned inside out, the resulting shell will contain that part of the volume of the reduced body that lies outside the subtrahend body.

Intersecting ribs.

The most labor-intensive and requiring a certain precision in the process of performing Boolean operations is the construction of intersection edges. The intersection edges must not have parts protruding beyond the boundaries of the face; they must necessarily join either with each other or with the old edges of the faces. If the surface intersection operation is performed correctly, these conditions are met. Intersection edges must not intersect each other beyond their extreme points.

In most cases, new edges do not intersect each other, but in some special cases this is possible. In Fig. 6.6.16 shows two cylindrical bodies of the same diameter, the axes of which intersect - a cross of cylinders. With a Boolean union of these bodies, a situation is possible when only two closed intersection edges are constructed. Such edges have two intersection points: A and B, at least one of which will not coincide with the vertices of the edges, but will lie somewhere on the edge. The intersection points of the ribs lie at the points of contact of the cylinders. These edges must be cut at points A and B, and the orientation of their parts must be specified, since when passing the point of contact of the surfaces in this case, the vector product of the normals to them, along which the new edges are oriented, changes its direction to the opposite. The intersection of new edges can be detected by the intersection of curves on the surfaces that make up the intersection line.

Rice. 6.6.16. At points A and B, the normals of the faces coincide

Matching edges.

The described algorithm for performing Boolean operations works correctly if the intersection edges do not coincide with the edges of the original bodies. Otherwise, it needs clarification. Let's look at examples.

In Boolean operations, a case is possible when any face of one body intersects another body along its edge. It is also possible that when constructing the edges of the intersection of the faces of one body with the faces of another body, we get two new edges that coincide in space with each other and with the edge of one of the bodies. All edges are different, since different faces join them. In Fig. 6.6.17 shows two bodies, when performing a Boolean operation on which, the intersection edges will coincide with the edges of the smaller body. Following the general algorithm, in this case we will get eight intersection edges, half of which should be omitted (or should not be constructed).

Rule for intersection edges.

If there are intersection edges that coincide with the edges of the faces of the original bodies, and in some other cases, we will perform the following check. Let us construct two vectors in the plane of each of the two intersecting faces, orthogonal to the intersection edge. In Fig. 6.6.18 these are vectors for a face with a normal and vectors for a face with a normal. Vectors rotate to the left of the intersection edge in the corresponding planes, and vectors rotate to the right of the intersection edge in the corresponding planes.

Rice. 6.6.17. Coincidence of intersection edges with edges of a smaller body

Rice. 6.6.18. Intersection of faces

Rice. 6.6.19. Intersecting faces along an edge

If the face does not extend beyond the intersection edge (the intersection edge partially or completely coincides with the edge of the body), then the corresponding vector is set equal to zero (Fig. 6.6.19). Using vectors and normals we can determine whether a given intersection edge will be used in the operation or should be omitted. Let us pay attention to the following circumstance.

In the operation of uniting bodies, we can rebuild the face of the first body if it has a continuation to the right of the intersection edge outside the second body, and we can rebuild the face of the second body if it has a continuation to the left of the intersection edge outside the first body.

Thus, in the Boolean operation of uniting bodies, the conditions for the intersection edge must be met:

Otherwise, the intersection edge in question should not be constructed in the Boolean operation of uniting bodies (if it is constructed, it should be omitted).

In the Boolean operation of intersection of bodies, we can rebuild the face of the first body if it has a continuation to the left of the intersection edge inside the second body, and we can rebuild the face of the second body if it has a continuation to the right of the intersection edge inside the first body. Thus, in the Boolean operation of intersection of bodies, the conditions for the intersection edge must be met:

Otherwise, the intersection edge in question in the Boolean operation of intersection of bodies should not be constructed (if it is constructed, it should be omitted).

As already mentioned, the intersection edges have a priority right over the old edges of the body to be included in the rebuilt cycle.

That point belongs to the internal space of the body. Otherwise, no.

Overlapping edges.

In Boolean operations, one can quite often encounter cases of partial coincidence of some faces of two original bodies. In Fig. Figure 6.6.20 shows two bodies, some of whose faces partially overlap each other.

Rice. 6.6.20. Coinciding faces of bodies

Rice. 6.6.21. Merging bodies

In this case, not all edges of the intersection of faces will be included in the Boolean result; some edges should be omitted (or not built). The result of combining bodies is shown in Fig. 6.6.21.

A similar situation arises when performing a Boolean operation on two coaxial cylinders of the same radius and in many other cases. Faces can overlap completely, partially, or just along one line. The normals of partially overlapping faces must be in the same direction at common points. Overlapping faces are those that can be transferred to one common surface. There may be more than two such faces. If the body operands have overlapping faces, all edges of the intersection of these faces must be checked to see if they are present in the Boolean result. In the operation of merging bodies, we only need those edges of intersection of overlapping faces whose adjacent faces have a continuation outside one of the bodies. In the operation of intersecting bodies, we only need those edges of intersection of overlapping faces whose adjacent faces extend into one of the bodies.

Cycles of overlapping faces of bodies must be rebuilt using a slightly different algorithm than for other faces. First of all, all edges of these faces must be brought to one common support (to the surface of one of the overlapping faces). In the operation of merging bodies, old edges of overlapping faces of the first body that lie inside the second body, and old edges of overlapping faces of the second body that lie inside the first body, will not be included in the resulting body. In the operation of cutting bodies, the old edges of the overlapping faces of the first body, lying outside the second body, and the old edges of the overlapping faces of the second body, lying outside the first body, will also not be included in the resulting body.

Let's start constructing a certain cycle of overlapping faces with an edge that should definitely be included in the result. To continue the cycle, among the intersection edges and among the old remaining edges of the face, we find all the edges that connect to a given edge at its corresponding vertex. Among the found edges, we choose the one that lies to the right of the others (it turns to the right at a larger angle compared to the other found edges, if you look along the cycle from the outside of the face). We put the selected edge in the cycle list. The difference in the algorithm is that as we continue the loop, we select the rightmost (rather than the leftmost) edge.

Bodies with several shells.

If any of the operand bodies has voids and, accordingly, is described by several shells, then the internal (external) shells untouched by the operation must be checked to see if the Boolean result is included in the shell. In general, the resulting body can also have several shells.

Body construction tree.

Boolean operations on bodies show the need to use topological concepts to construct bodies.

The bodies considered in the previous paragraphs will be called simple, in contrast to the bodies obtained as a result of operations, which we will call complex. Data structure simple bodies contains a minimum of information from which all faces of the body can be constructed. In the data structure of the body obtained as a result of a Boolean operation, we put the data structures of the original bodies and the type of the Boolean operation. Body shells can always be constructed using this information. Thus, the data structure of a body obtained as a result of repeated execution of Boolean operations will contain a construction tree. The resulting body is at the root of the tree, and its branches begin at simple bodies.

An example of a tree is shown in Fig. 6.6.22. The nodes of the tree contain bodies. The tree has several tiers. Operations between bodies are indicated by appropriate signs. Operations are performed between bodies of the same tier.

Rice. 6.6.22. Body construction tree

The body data structure in the form of a construction tree can be used for all bodies - both for bodies obtained as a result of some operation, and for simple bodies.

Operations > Boolean

A Boolean operation is designed to create a new body based on two or more existing bodies. As a result of the operation, a new body is created, which is a combination of the original bodies.

Basic concepts. Operation options

Types of Boolean Operation

There are three types of Boolean operations:

● Addition . The result of the operation is a body that combines all the parts of the bodies involved in the operation.

● Subtraction . The result of the operation is a body obtained by subtracting one body from another.

● Intersection . The result of the operation is a body obtained by the intersection of the bodies involved in the operation and consisting of common parts of these bodies.

Operands of a Boolean operation

The bodies involved in a Boolean operation are called " Operands " Boolean operation. They are divided into first and second operands. The bodies on which transformations are performed are called "First operands" Boolean operation. The bodies with which the transformation is performed are called "Second operands". Depending on which operand the bodies were chosen as, we get different results Boolean operation.

A Boolean operation can also be performed between a set of bodies, some of which can be selected as the first operands, and other bodies as the second. In the process of performing such an operation, all first operands are combined into one body and all second operands into another body. The selected Boolean operation is then performed between these bodies. The result is one multicomponent body.

Boolean operations results

Both solid and sheet bodies can be selected as operands. The ability to select different combinations of operands and set parameters allows you to generate different results of Boolean operations.

It should be noted that in some situations, when combining certain types of operands in Boolean operations, it is impossible to obtain a result.

Below we will consider the main combinations of operands and the results of the operation.

As a result of Boolean operations, in some cases, both a solid and a sheet body can be formed. The user can select the type of the resulting body using the "Result of the operation".

This option is advisory in nature and allows you to select, other things being equal, one or another type of the resulting body. If it is impossible to form the selected body type as a result of the operation, then the value of this parameter will be ignored.

Interpretation of a sheet body

A sheet body used as an operand in a Boolean operation can be defined in a command as either a sheet body itself or a half-space. Depending on the interpretation and which operand the body is chosen as, different results will be obtained.

If the operand - a sheet body was defined as a half-space, then the normal vectors to the surface of the sheet body will be displayed in the scene. The direction of these vectors indicates which half-space is selected and will affect the outcome of the operation.

Operation "Addition"

Results of the Boolean operation "Addition" when combined various types operands:

Solid + Solid

If the first and second operands are solids, then as a result of the operation one body will be formed, combining all parts of the bodies involved in the operation.

Sheet body + Solid body

As a result of the operation, a sheet body will be formed, which represents the “punching” of the sheet body by a part of the solid body. Depending on the selected half-space, a “punching” of either the lower or upper part is formed.

It should be noted that the sheet body must completely intersect the solid body.

Solid + Sheet body

If a rigid body was selected as the first operand and a sheet body was selected as the second, then the default result of the Add operation is to create a rigid body. The faces of the sheet body will form additional volume for the solid body.

The sheet body in this case must be defined in the command as a half-space.

It should be noted that to form this kind of Boolean operation, it is necessary that the sheet body, when added to a solid body, closes the volume. Otherwise, it is impossible to obtain the result of the operation.

When selecting a sheet body, you should pay attention to the direction of the normal vectors. With a certain direction of the normals, the resultant body may be incorrect. In this case, you need to select a different half-space in the "Interpretation of the operand".

Leaf body + Leaf body

As a result of the addition of two sheet bodies, a sheet body is obtained. To successfully create an operation, it is necessary that the faces of the sheet bodies either have a common edge or coincide in a certain area. This feature is similar to the "Stitch" operation. The difference is that the Boolean operation " Addition "does not remove gaps between bodies, but" Stitching " is trying to create a new continuous surface.

Operation "Subtraction"

Results of the Boolean operation "Subtraction" when combining different types of operands:

Solid - Solid

The result of performing the Boolean operation “Subtraction” of one solid body from another will be a body that rearranges the first operand, from which the volume of the second operand is removed.

When creating a Boolean operation " Subtraction "A situation where the resulting body will have areas with walls of zero thickness is not allowed. In this case, the topology of the model is violated.

Sheet body - Solid body

When subtracting a rigid body from a sheet body, a hole will be created in the sheet body in the shape of the second operand - the solid body.

As a result of subtracting a sheet body from a solid, a part of the solid body remains, cut off by the sheet body. In this case, the sheet body must be defined as a half-space. Depending on the selected half-space, either the top or bottom part solid body. This type of Boolean operation is analogous to the "Cut" operation.

Operation Crossing

Results of the Boolean operation "Intersection" when combining different types of operands:

Solid - Solid

The result of the intersection of solid bodies is a body that represents common part all operands.

Sheet body (half space) - Sheet body (half space)

If the sheet bodies coincide in a certain area, then as a result of performing the Boolean operation "Intersection" their common part remains.

Sheet body - Solid body

If a sheet body was selected as the first operand, and a solid one as the second, then as a result of performing the Boolean operation "Intersection" a sheet body will be formed, which is the common part of the two operands.

Rigid Body - Sheet Body (Half Space)

As a result of performing the Boolean operation "Intersection" of a sheet body, interpreted as a half-space, and a solid body, a part of the solid body will remain, cut off by the sheet body. The result of this operation is similar to the result of the Boolean operation “Subtract” a sheet body (half-space) from a solid body. The difference is that as a result of performing the “Intersection” operation, another part of the solid body remains. In this case, the order in which the operands are selected does not affect the result.

Boolean operations between sheet bodies (half-spaces)

One way to use Boolean operations between sheet bodies is to create corners. To do this, the sheet bodies in the team must intersect and be defined as half-spaces. Depending on the direction of the normals to the surfaces of the sheet bodies and the type of Boolean operation used, different resulting sheet bodies are obtained. In this case, the directions of the normals on the resulting body are preserved.

In the case where one sheet body does not completely intersect another, the Boolean operation will not work. To correct the situation, the parameters of the Boolean operation provide the ability to extend the edges of the first or second operand to the nearest edge or boundary of the body. In this case, the line of intersection of these two bodies will be extended tangentially to the selected edge or boundary of one of the operands.

Global and local Boolean operations

If sheet or rigid bodies are selected as operands, then these types of Boolean operations will be called " Global ". When creating such a Boolean operation, the system itself determines the faces of the bodies participating in the operation and their intersections. If the Boolean operation involves geometrically complex bodies or uses enough large number operands, then searching for all intersections can greatly slow down the execution of the command.

In addition to the "Global" Boolean operation in T-FLEX there is the concept of " Local "Boolean operation. Unlike the "Global" Boolean operation, where only bodies are selected as operands, in the "Local" Boolean operation, in addition to bodies, the faces of the operands are selected. You can select the faces of the operands in the operation parameters on the "Faces" tab. In some cases, use The “Local” Boolean operation provides a significant gain in model regeneration time compared to the “Global” Boolean operation, because as a result of the operation, intersections of only the specified faces will be searched. For example, if you need to make a hole in a fairly complex body and it is known that. If the hole intersects a small number of faces, it makes sense to specify these faces in the operation parameters.

It should be noted that you need to carefully select the faces for the "Local" Boolean operation. In the case of incorrectly selected faces, an incorrect body may be formed.

Selecting individual operand faces in " Local "Boolean operation with parameter enabled"" allows you to form such bodies as shown in the figure below. In this case, the intersection lines of the faces of the operands will first be determined, and then closed loops will be formed along these intersection lines and the boundaries of the faces of the first operands to form a Boolean operation.

In some cases it may be convenient to use the option "Use adjacent faces" for the first or second operands. It is enough to select one face of the operand so that the system itself defines the neighboring faces in such a way that the lines of intersection of the faces of the operands form closed cycles. In this case, when changing the number of faces, the component closed loop, if the operands are modified, the result of the Boolean operation will remain correct.

Selective Boolean Operations

As a result of the intersection of the operand faces, common edges are formed. These edges divide the bodies of the operands into parts. Parts of operands can be internal (common parts of operands) and external. When creating a Boolean operation, it is possible to specify the external parts of the operands that will be left in the resulting body or, conversely, excluded from it.

Part of the operand is defined by topological elements: vertices, edges or faces. To select a part of the operand, just specify in the operation parameters on the " tab Operand parts " one of these elements. The figure below shows the possibility of using a selective Boolean operation "Addition", during which only the selected parts of the second operand remain in the resulting body.

In some situations, the selected topological element does not explicitly define the desired part. For example, as shown in the picture below, the selected edge belongs to several outer parts of the operand. In order to unambiguously determine the required part, you need to build a 3D node on this face and use the option " Point selection " select this 3D node. There is no need to select a face.

In cases where the operands have matching areas, the command provides the ability to additionally determine the topological correspondence of the areas of these operands. Based on this information, the team will choose required algorithm forming a Boolean operation.

This feature allows you to:

1. Speed ​​up the process of recalculating a Boolean operation by specifying a specific type of contact.

2. In some cases, without specifying a match, it is impossible to obtain the result of a Boolean operation. For example, when there is contact between spline and analytical surfaces (Boolean operation between the cylinder and the face of the spiral in the picture below).

The command provides four types of matches:

● Complete match– the boundaries of topological elements coincide with the specified accuracy.

● Turn on – the boundaries of the second element lie within the boundaries of the first element.

● Partial match– the boundaries of one element partially coincide with the boundaries of another element.

● Projection – coincidence of fingerprints of the boundaries of topological elements.

The table below provides a description of the match types for different operand topologies:

The match can be exact or approximate. Exact matching assumes that the operands have only the described contacts and there are no other matches or intersections. If you establish an approximate match, then in addition to the specified contacts, the system will look for other intersections of operands. This method takes longer to complete than the exact method, but in some cases it is the only way to obtain the result of a Boolean operation.

The figure below shows the assignment of a correspondence in the Boolean operation "Addition" between a hemisphere and a cone that have matching bases. If the match is set to coincide with the faces of the base and selected exact method, then in this case it will be impossible to obtain the result of a Boolean operation, because when the operands intersect, there is one more common edge that is not described in the correspondences. In this case, the only possible approximate method will be the implementation of which this intersection will be found.

Smoothing New Edges

Another feature of Boolean operations is smoothing new edges. By new edges we mean edges resulting from the intersection of operand faces. When the geometry of the operands changes, or the number of new edges decreases or increases, the system will automatically track this and generate smoothing. In the command, you can select the smoothing type: rounding or chamfer and set the smoothing radius.

Missing one of the operands

When creating a Boolean operation, you can set the parameter "It is possible that one of the original operations is missing". This technique is often used in parametric models, when for a certain design of the model it is necessary to exclude one of the elements. As shown in the figure below, the nut can have two designs: with and without a groove. In order to implement these designs in one model, you can suppress the second the operand will save all subsequent operations created on the basis of a Boolean operation (in the example, this is saving a cosmetic thread on the edge of a Boolean operation).

If this option is disabled, a Boolean operation creation error will be thrown when suppressing an operand.

Creating a Boolean Operation

To work in a team, the automenu and the properties window are used together. They work synchronously and complement each other. The state of the automenu depends on the stage of specifying the operation, on the purpose and type of selected objects.

To create an operation you need to do the following:

1. Select Boolean operands.

2. Select the type of Boolean operation.

3. Select parts of the operands to form a selective Boolean operation (optional).

4. Set element correspondence (optional).

5. Set additional parameters (if necessary).

6. Set anti-aliasing parameters (optional).

7. Confirm the creation of the operation.

Basic parameters of the operation

Operand selection

Selecting Operands for a Global Boolean Operation

To select the operands of a global Boolean operation, use the following option:

<1> Select Boolean Operands

In this case, the following options become available in the automenu:

Select first body

Select second body