Formulation

NOTE:
This page is posted to the web in order to share results with the swept volume community. The material below introduces the basic theory and demonstrates a simple example of sweeping using the Jacobian rank deficiency condition.

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Formulation

Before getting involved in the mathematical formulation, we shall first demonstrate the need for such mathematics through a simple example. Consider the planar sweep of a circle along an axis. The circle is defined by the equation

(1)

The axis along which the circle will be sweeping (or in this case can be called extruding) is also defined by an equation

(2)

where this axis shall start at the origin and extending to . Therefore, the parameter varies as as shown in Fig. 2.1.

Fig. 2.1

To understand the sweep concept and especially when dealing with multiple parameter sweeps, it is helpful to think of a point P on the swept entity (the circle) moving along the circle at all times. This point traces the circle for each round it makes. Sweeping of this circle while this point is moving along an axis yields a trace covering the swept area illustrated in Fig. 2.2(a). The swept area consists of all points touched (swept) by P.

Fig. 2.2a A geometric entity is represented by a point circulating along its trajectory

Fig. 2.2b Sweeping of a planar entity

Fig. 2.2c Resulting swept area

The task at hand here is to identify the boundary to this swept area. At the beginning of this sweep, the original circle is clearly identified part of which is the boundary as shown in Fig. 2.3. At the end of the sweep, i.e., at , a circle is also identified and a segment of which is also on the boundary.

Fig. 2.3 Boundaries to the swept area

In order to find these boundaries and without presenting any complicated proofs, we shall present a simple mathematical formulation for delineating the boundary.

The totality of the swept area of the sweep described above is characterized by

(3)

(4)

There are two types of borders to be identified.

Boundary borders: Those are due to sweep or entity limits. For this example, there is a boundary at and . Substituting these values into yields the first circle and the second circle both of which contribute to the boundary.
Jacobian borders: The other type of boundary is not as straightforward. This type can be thought of as resulting from the sweep of entities that are curved, where this curvature forms the outer fiber in the direction of the sweep. Figure 2.4 illustrates this concept. A curve shown in 2.4(a) has its maximum point in the direction of sweep at point P (although vague at this point, this concept will be formalized in the following section). The boundary borders are readily identified. The Jacobian border is identified by the sweep of the point P to the end position and is identified by the dotted line in Fig. 2.4(b). To find this maximum point P for any curve and to generalize it to any surface of any number of parameters, the Jacobian, hereon referred to as the Sweep Jacobian, is introduced.

Fig. 2.4 (a) A curve with its maximum at P (b) Sweeping and identifying borders

Returning to our simple example, we shall first compute the Jacobian of . The Jacobian is

(5)

Similar to finding the maximum point in a function by setting its derivative to zero, the determinant of the sweep Jacobian is set to zero such that

(6)

which identifies two solutions for and for . Substituting these solutions into yields the other two boundaries as

which are the two lines closing the boundary to the swept area shown dotted in Fig. 2.3. Now that all boundary curves have been determined, the swept area can be identified by a function given by Eq. () and its boundary curves as shown in Fig. 2.5.

Fig. 2.5 Removing internal branches

It is interesting to see if this concept works if the circle is to be swept along the y-axis.

For this case, the circle is still defined by Eq. (), and the y-axis (sweep axis) is given by

(7)

The sweep function is then

(8)

and the sweep Jacobian is

(9)

The determinant of the Jacobian set to zero is

(10)

Solving this equation yields two points on the circle at and as shown in Fig. 2.6.

Fig. 2.6 (a) Sweep along the x-axis (b) along the y-axis

The method for characterizing the swept volume and its boundary, illustrated above in rather general terms, will be presented in the next section in a rigorous mathematical formulation.

Formulation

Consider a geometric entity parametrized in terms of one or more variables as a vector given by , where , i.e., the geometric entity can be a curve, a surface, or an entity in a higher dimensional space. In order to generalize the formulation, we shall also consider a boundary imposed on in the form of constraints on the parameters characterized by inequality constraints in the form of

(11)

Consider a second geometric entity parametrized in terms of one or more variables as a vector , where , where also this entity can be a curve, a surface, or a volume. This entity also has a defined boundary characterized by

(12)

The accessible set generated by the sweep of on can be defined as (the term swept volume is reserved to indicate the actual volume of the set)

(13)

where q is the extended vector defined by

(14)

and is the rotation matrix defining the orientation of the swept entity. In fact, characterizes the set of all points on the surface and on the boundary of the swept volume. The aim of this section is to determine the boundary to this set.

Consider the derivative of the accessible set (Equation 3) with respect to time

(15)

where the dot denotes a derivative with respect to time and the matrix

(16)

is called the Jacobian of the sweep and has dimension . The left-hand side of Equation 5 is the velocity of a point moving in the swept volume having a parameter velocity vector .

Square Jacobian-No Limits

In the case of a square Jacobian, the determination of singular values is directly obtained. The determinant of a square Jacobian is equated to zero and solved for its roots.

Example of a Three-Parameter Square Jacobian

To demonstrate the formulation for sweeps that do not have limits in their definitions, consider the sweeping of the entity generated above (swept area) along an axis to give it an extrusion height.

Let the swept entity be given by Eq. () as

(17)

that is to be swept on the entity

(18)

with an orientation specified by the rotation matrix

(19)

where and . The function characterizing this sweep is given by

(20)

(21)

where . The sweep Jacobian is computed as

(22)

Since this is a square Jacobian, the determinant is evaluated and equated to zero as

(23)

Solving for the roots yields and . These are the singular sets due to the Jacobian. We shall denote these sets by such that and . Other singularities are due to the limits imposed on and such that , , , and .