Page 1 of 17 A moment based metric for 2D and 3D packing John K. Dickinson and George K. Knopf Department of Mechanical and Materials Engineering, Faculty of Engineering Science, University of Western Ontario, London, Ontario, N6A 5B9, Canada Abstract The most common metric used today to evaluate the effectiveness of a packing technique is the percentage of space used. An inherent limitation of this metric is its inability to differentiate between two different packing arrangements of the same set of objects. This paper proposes an alternative metric for both the 2D and 3D cases, called the point moment metric, and expands on the theory behind the proposed metric. The metric is based on evaluating the compactness of the remaining free space in a packing arrangement. This measure is the ratio of a defined moment calculated for the current free space and the initial free packing space. The developed metric can be extended to N- dimensional packing problems where N = {1, 2, ...}. The arbitrary 3D shape packing problem is used as an illustration of an application of the metric. The point moment metric has not been developed to replace the measurement of the percentage of volume used but rather to complement it by allowing efficient comparison of apparently equivalent packing arrangements using the same volume of space. It is not suited for comparing two packing arrangements occupying different volumes. Keywords: Multi-dimensional packing; optimization; packing metrics. 1. Introduction Rapid prototyping, also known as solid freeform fabrication, is a recent addition to modern manufacturing techniques. What distinguishes rapid prototyping from the more conventional manufacturing processes is that arbitrarily shaped parts are formed directly from the geometric models generated by a CAD (computer aided design) system. Layered manufacturing, the most common approach to rapid prototyping builds a physical prototype of an object by forming or stacking layers, or slices, of material on previous layers until the object’s form is completed. Although several commercial layered manufacturing methods exist, they are all limited to working within a fixed volume, called the “build volume”. In general, a large portion of the build volume remains unused when creating only one object. As well, the time required to produce each layer is fairly constant and thus independent of a layer’s surface area or boundary details. Thus, the overall efficiency of the manufacturing process can be greatly improved by packing several objects in the same build volume. Two-dimensional (2D) and three-dimensional (3D) packing problems have been investigated for over twenty years and are reviewed by Dowsland [4], Dudzinski and Walukiewicz [6], Dowsland and Dowsland [5], Dyckhoff [7], and Li [10]. Research interest in spatial packing has not been limited to the field of operational research. Packing problems
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Page 1 of 17
A moment based metric for 2D and 3D packing
John K. Dickinson and George K. KnopfDepartment of Mechanical and Materials Engineering, Faculty of Engineering Science,
University of Western Ontario, London, Ontario, N6A 5B9, Canada
Abstract
The most common metric used today to evaluate the effectiveness of a packing technique is the percentage of space
used. An inherent limitation of this metric is its inabilit y to differentiate between two different packing arrangements
of the same set of objects. This paper proposes an alternative metric for both the 2D and 3D cases, called the point
moment metric, and expands on the theory behind the proposed metric. The metric is based on evaluating the
compactness of the remaining free space in a packing arrangement. This measure is the ratio of a defined moment
calculated for the current free space and the initial free packing space. The developed metric can be extended to N-
dimensional packing problems where N = { 1, 2, ...} . The arbitrary 3D shape packing problem is used as an ill ustration
of an application of the metric. The point moment metric has not been developed to replace the measurement of the
percentage of volume used but rather to complement it by allowing eff icient comparison of apparently equivalent
packing arrangements using the same volume of space. It is not suited for comparing two packing arrangements
Rapid prototyping, also known as solid freeform fabrication, is a recent addition to modern manufacturing
techniques. What distinguishes rapid prototyping from the more conventional manufacturing processes is that arbitrarily
shaped parts are formed directly from the geometric models generated by a CAD (computer aided design) system.
Layered manufacturing, the most common approach to rapid prototyping builds a physical prototype of an object by
forming or stacking layers, or slices, of material on previous layers until the object’s form is completed. Although
several commercial layered manufacturing methods exist, they are all li mited to working within a fixed volume, called
the “build volume”. In general, a large portion of the build volume remains unused when creating only one object. As
well , the time required to produce each layer is fairly constant and thus independent of a layer’s surface area or boundary
details. Thus, the overall eff iciency of the manufacturing process can be greatly improved by packing several objects
in the same build volume.
Two-dimensional (2D) and three-dimensional (3D) packing problems have been investigated for over twenty years
and are reviewed by Dowsland [4], Dudzinski and Walukiewicz [6], Dowsland and Dowsland [5], Dyckhoff [7], and
Li [10]. Research interest in spatial packing has not been limited to the field of operational research. Packing problems
Page 2 of 17
Figure 1: Example of packing problems in 1D, 2D and 3D environments. In each case, of thetwo packing arrangements ill ustrated, the second arrangement is “ intuitively” better since theremaining free space or material is more useful.
are viewed as multidimensional extensions of the bin packing and backpack/knapsack problems found in the fields of
computer science and physics [6, 9, 8]. Packing problems which are evaluated solely on the eff icient use of the available
space, such as the stock cutting problem [7], can be considered “pure” packing problems. The objective is to minimize
the amount of material wasted when cutting parts from fixed stock pieces such as sheet metal, tubing or cloth.
When pure packing problems are evaluated based solely on the percentage of space used it is not possible to
differentiate between different packing arrangements of the same objects as depicted in Figure 1. In each example, the
second packing or cutting arrangement is “ intuitively” more eff icient than the first. The reason for this is that the second
arrangement allows a greater variety of object shapes and sizes to be added at a later time to the packing arrangements.
This ill ustrates why it makes more sense to arrange objects so that the remaining space or material is as compact as
possible.
In the 1D case, simply pushing each segment to be cut end to end as shown in the second arrangement leaves a single
piece of material behind instead of several small unusable bits. For the 1D case the packing arrangement which
maximizes the usefulness and compactness of the remaining material is easily found. Human beings attempt to do the
same for 2D and 3D situations. When a human being packs the trunk of a car they typically pack one object at a time,
placing each object in the container as “best as possible” before adding the next. To perform packing in a similar
fashion, a computer requires some way of picking the “best” position for an object from numerous potential positions
in the container. A measure of compactness of the remaining free space in the container could be used to differentiate
between potential object positions.
This paper briefly reviews the notions of compactness and continuity and then presents the point moment as a
measure of compactness for N-dimensional (ND) space. The paper goes on to develop a metric for comparing two
different packing arrangements of the same objects based on this moment. The mathematical properties that make the
Page 3 of 17
Figure 2: White region is an example of a continuous but notvery compact region of space. Subregions a and b are connectedby a narrow neck.
point moment and derivative metric useful for packing algorithms are then discussed. An algorithm is then presented
to calculate the metric and its computational costs are shown to be small. Finally, the metric is illustrated in use in the
arbitrary 3D shape packing problem.
2. Concepts of continuity and compactness
If it were possible to know ahead of time what particular shape would need to be packed at the last moment, then
the best packing would leave a perfectly shaped space for the object while filling in the rest of the container. This
information is not usually available as in the case when packing for effective use of build volumes in rapid prototyping.
Rapid prototyping bureaus typically take orders for prototypes from a large variety of industries including the
automotive, toy, medical and packaging industries. No study has yet been done to identify any trends in shapes or
dimensions of parts produced by an average rapid prototyping bureau. Similarly, no statistical information is available
on the sorts of shapes used in industries that employ die stamps to cut parts from sheet metal, or from the average
industry involved in cutting cloth materials or plastics sheets into parts. While lacking this a priori knowledge, the
usability of the remaining free space in a packing can only be measured by assessing its continuity and compactness.
Continuity can be defined as the property of being uninterrupted or unbroken. In more rigorous mathematical terms
a region of space, R, is continuous if for every two points within the region, there exists a path entirely in the region R
that connects the two points. Without continuity, the remaining free space maybe subdivided into small regions, each
being too small to contain a part on its own. Unfortunately continuity, in a purely mathematical sense, is not a very
useful criterion for real world packing applications since having a very narrow connection between two subsections of
a region (e.g. subregions a and b in Figure 2) is in general, no better than having two separate regions. Checking for
continuity is also computationally intensive as the only way to know if two subregions of space are continuously
Page 4 of 17
connected is by making a path through free space between them. Checks would have to be made to ensure each segment
of any generated path does not pass through any packed objects.
The term compact is defined as “having parts or units closely packed or joined” and “occupying a small volume
by reason of efficient use of space” [11]. Keeping the remaining free space compact helps ensure that the free space
is not a long set of connected but slender spatial regions within which no reasonable object could possibly fit (see Figure
2). An added benefit is that as a region becomes more compact it is less likely to be discontinuous. Hence, regions of
high compactness provide a reasonable assurance of continuity as well. A proposed measure of the compactness of a
region of ND space is discussed below.
3. The point moment metric (� �
)
In 1D space, a continuous line segment is the most compact region possible. In 2D space, the disc is the most
compact region and in 3D space, the sphere is the most compact region. In higher dimensions, the most compact regions
are described as hyper-spheres. Correspondingly, a measure of the “spherity” of the remaining free space can provide
a good assessment of the compactness of the free space. Moments of areas, also known as area moments of inertia, are
a common concept in engineering and can be used to develop the theory supporting the proposed metric for measuring
the compactness of a region of space. Since moments of areas are defined for 2D spaces, the theory for the metric
proposed for 2D problems will be developed first and then extended an arbitrary number of dimensions.
3.1 Development of the 2D point moment metric (� �
)
Moments of areas (area moments of inertia) are part of the theoretical foundation of mechanics of materials and
are used to determine deflections under load [1]. For an area, A, defined in the x-y plane, as shown in Figure 3, the
second moment of an area with respect to the x axis (Ix) is defined to be;
(1)I d AAx y= ∫ 2
and with respect to the y axis (Iy);
. (2)I dAAy x= ∫ 2
The polar moment of inertia around the origin, i.e. with respect to both axes at once, in given by;
(3)J d A I Io A= = +∫ ρ 2
x y
where is the radius.ρ = +x y2 2
To facilitate extending the metric to multiple dimensions the polar moment can be rewritten in a vector notation.
Let x be a vector describing a point in the N-dimensional space in question. In this case, x is a 2-dimensional vector x
= [x, y]. The length of x is represented by � . The distance squared from any point to the origin can be expressed as the
dot product of the vector representing the point; i.e. � 2 = x � � x . Thus, the polar moment of inertia around the origin
can be written as
Page 5 of 17
Figure 3: An illustration showing each term found inthe formulas for moments of areas or area moments ofinertia.
. (4)J dAo A= ⋅∫ ( )x x
The location of the region under consideration relative to the origin will have a great impact on the value of the
moment of inertia around the origin (Jo). To eliminate this effect it is possible to take the moment, Jc , about the
area’s centroid xc . Note that xc is currently a 2-dimensional vector but will later be extended to an N-dimensional
vector. The moment about xc is given by
(5)J dAc c cA= − ⋅ −∫ ( ) ( )x x x x
Given regions of the same area but of different shapes, the polar moments of inertia around the centroids of the
various regions (Jc) is smallest for the region that most closely resembles a disc. In fact, Jc, will monotonically decrease
as a region with a fixed area becomes more compact. This will be discussed in Section 4. Consequently, Jc, can be used
to evaluate and compare the compactness of several possible regions of the remaining free space. Since Jc
monotonically decreases as the region gets more compact it is useful for optimization routines that seek to rearrange
packing layouts such that the remaining free space is in the most compact form possible. However, Jc has units of length
raised to the power of four (e.g. m4, in4) and its range of values could be from thousandths of a unit to thousands of
units. Working with a ratio of Jc for the remaining free space divided by Jc for the initial empty packing volume
provides a simple solution to these dimensional difficulties.
Now define the point moment metric (� �
) for the 2D case as being;
(6)� �
= J Jcfree a rea
co r ig a rea_ _
where is the centroid polar moment of inertia for the remaining free area available for packing and isJ cfree a rea_ J c
o rig a rea_
Page 6 of 17
the centroid polar moment of inertia for the initial empty packing space.
The point moment metric � �
has several useful properties. First, when the packing space is completely unused � �
= 1. Second, when the entire packing area is occupied � � = 0. Third, the metric is dimensionless and, therefore,
independent of the length of the unit used to describe the packing arrangement. Fourth and most importantly, � � will
monotonically decrease as the remaining free space becomes more compact just as does the measure Jc. Section 4
discusses the properties of the metric in more detail.
3.2 Development of the general ND point moment metric (� � �
The 2D metric was developed based on area moments of inertia that were determined around an axis perpendicular
to the plane of the region and passing through the centroid of the area. This moment, Jc, cannot be defined for regions
which are not areas such as regions that occur in 1D, 3D and ND spaces (not including 2D spaces). However, the
integral evaluated in the 2D case to get the moment is basically the integral of the distance from every point in the area
to the centroid. Define the point moment � � as a similar integral for ND spaces instead of area moment of inertia.
Calling the integral the point moment captures the nature of the moment as being around the centroid point no matter
how many dimensions the region has. The integral is no longer being taken over an area A but rather over an ND region
R. Define x as being the ND vector describing any point in the ND space and xc as being the ND vector defining the
region R’s centroid. Thus the point moment � � is defined as:
(7)� �
= − • −∫ ( ) ( )x x x xc cRd R
where R is the ND region in question, whether it is a length, area, volume, or hyper-volume. Good estimates of point
moments can be readily calculated for 3D regions from voxel or faceted B-Rep descriptions (Section 4).
Note that � � is equivalent to Jc for the 2D case. Now define the point moment metric � �
as:
(8)� � � � � �
= free space orig space_ _
where � � free_space is the moment for the remaining free ND space available for packing and � � orig_space is the initial
empty ND packing space. This definition is viable for ND spaces and keeps all the properties mentioned earlier for
the 2D case.
4. Point moment and metric properties
Several properties have been built into the point moment � � and point moment metric � �
by design. This section
expands on each of these properties and discusses their importance for use in packing algorithms. The theorems and
proofs referred to here appear in Appendix A.
Page 7 of 17
Though the following discussion applies to the metric in all dimensional cases, unless explicitly stated, it is easier
to refer to everything in common 3D terms. “Region” will be used to refer to a sub-set of the ND-space and is thus a
term specifying the geometry or the entity of the sub-set of space. “Volume” will be used to refer to the size of the ND-
space a region occupies (e.g. length for 1D spaces, area for 2D spaces, volume for 3D spaces or ND-hyper volume for
ND spaces) and is thus a scalar measure of the size of the region.
The first significant property is that the metric � � has a limited range, starting at 1 when the packing space is entirely
empty (free) and ending at 0 when no free space remains. This is the simple result of defining the metric as the ratio
of two point moments � � . The denominator of the ratio, the point moment of initially free space, remains as a positive
constant while the numerator, the point moment of the remaining free space, starts as equal to the denominator and drops
towards zero. The second significant property is that the metric is continuous as shown in Theorem 1 (Appendix A).
These properties are important since many common optimization routines perform best on functions with defined value
ranges or rely on the gradients of continuous functions to converge.
Note that � � will not have a unique value for every possible packing configuration. � � decreases as the free packing
space is used up and also as the remaining free space becomes more compact. Thus two different packing arrangements
of the same set of objects can have the same � � value (e.g. symmetric arrangements), as well as two arrangements of
different sets of objects. In fact, it is possible that the � � value for a packing that uses more volume but leaves the free
space more scattered can be larger than for a packing occupying less volume but leaving the remaining space more
compact. What can be shown is that the � � value will monotonically decrease as the existing free space is re-arranged
to be more compact (see Theorem 3, Appendix A) and this is the third significant property. Though not a necessary
for good performance of some optimization algorithms, monotonic reduction of the function is still highly desirable.
The fourth significant property is that the point moment of objects modeled in modern CAD packages can be easily
estimated from information supplied by the packages. Modern CAD packages can estimate or accurately calculate
standard mass moments of inertia (see reference [12] for a definition) for any solid modeled in them. Theorem 4
(Appendix A) shows that the point moment in 3D cases is simply half the sum of the three principle moments of inertia.
The fifth significant property is that point moment � � can be calculated for a region R made up of the union of non-
overlapping smaller regions {R1, R2, ... Rm}. This is because the defining integral can be broken into integrals over the
smaller regions as shown in Theorem 2. The point moment becomes the sum of the moments of each component region,
Ri, about the centroid for the combined region R. In other words, knowing the moment � � i , the location of the centroid
xi , and the ND volume Vi for each of the m component regions making up the combined region, the combined region’s
centroid x , volume V and point moment � � can be found by following Algorithm 1 below:
Page 8 of 17
Algorithm 1:
1) find the combined volume of the region: ,V V i
i
m
==
∑1
2) determine the new center of volume: ,V V i ii
m
x x==∑
1
3) calculate the magnitude of the distance (di) from the new center of volume to each region’s center of volume,
4) and sum the contribution of each region Ri to the point moment: .( )33 33a ll i i ii
m
V d= +=
∑ 2
1
This approach is convenient when the moments are known for sub-regions but not for the combined region as is
commonly the case when a larger region is made up of a combination of more primitive regions such as blocks and
spheres, or rectangles and triangles. The same methodology can be used to recalculate the point moment after removing
a portion of the region by using the algorithm above with negative volumes and negative point moments for any regions
being removed. The most important implication of this is that the point moment for the free space in a packing can be
found knowing only the point moments for each object packed, their centroids and volumes as well as the same
information for the initial packing space. Instead of having to re-evaluate the defining integral, updating the point
moment value becomes a simple case of a few arithmetic operations.
Another consequence of this is that it is easy to calculate the moment for 3D objects described using voxel
representation. Voxel representations of objects describe the object as the union of lots of little cubes arranged in a 3D
grid. As mentioned above, the point moment of a cube can be found by taking half the sum of its principle moments
of inertia as found in any dynamics text (e.g. reference [12]). Given that the point moments, volumes, and centroids
of each contributing cube are known the point moment for the object can quickly be determined.
Finally, it is also worth noting that the point moment � � is orientation and position independent because the distance
from any point in the region to its centroid remains the same no matter what orientation or position of the region is in.
5. Computational costs of object position optimization
When building a possible packing solution in a serial fashion, the computer places each object to be packed into the
available space one after the other. Each object should be placed in the best possible position before the next object
is added to the packing arrangement. As mentioned earlier, if no information is available about the sort of objects still
to be packed the “best” partial packing is assumed here to be the packing that leaves the remaining free space as compact
as possible. One measure of this compactness can be found using the point moment metric � � developed above.
Optimization routines can be used to minimize the metric as each object is added to the work space so that the remaining
Page 9 of 17
free space is rearranged to more and more resemble a disk, sphere or hyper-sphere, and thus be in as compact form as
possible. For the general packing problem, it is hoped that this provides a better opportunity for packing subsequent
arbitrarily shaped objects.
Algorithm 2 details a way to calculate an updated metric value after an object has been added to a packing
arrangement. It assumes that the centroid, ND volume and point moment � � are known for the object being added as
well as for the region of remaining free space. It is also assumed that the point moment is known for the completely
empty packing space before the packing procedure had commenced. The following is a list of steps and the
computational difficulty required in performing the steps. Subscripts k and k+1 refer to before and after the object has
been added to the packing space and N is the dimensionality of the packing problem.
Algorithm 2:
1) Compute reduced free space volume: V V Vk k ob j+ = −1 cost: 1 addition
2) Determine new center of volume: xx x
k+
k k o b j o b j
k+
V V
V11
=( - ) cost: 1 addition,
3N multiplications.
3) Find new point moment: � � � � � �k k k k k o b j o b j k o b jV V+ + += + − − + −
1 1
2
1
2
x x x xcost: 4N addition,
2N+3 multiplications
4) Calculate new metric: � �� �
� �kk
in itia l+
+=11 cost: 1 multiplication
Total Cost: 4N + 2 additions and 5N + 4 multiplications
If the packing problem is three dimensional, i.e. N = 3, then the total cost to calculate a new metric value is only 14
additions and 19 multiplications. This is a very quick set of calculations on today’s processors and could even be further
reduced for processors with built in vector calculating capabilities. This means that optimization routines can go through
many iterations in their search for the best solution without demanding too much CPU time to evaluate the efficiency
of a packing, which might otherwise make finding a good packing too expensive.
6. 3D serial packing example
To illustrate an application of the point moment metric an example was taken from tests of a 3D packing program.
The program was designed to sequentially pack arbitrary 3D shapes into a specified box volume. A simulated annealing
optimization method was used by the program to minimize the point moment metric � � to locate the best position for
each object. Additional constraints were used to stop objects from intersecting the container walls or overlapping other
objects. The example presented here was selected to demonstrate how the point moment metric changes as an object
Page 10 of 17
Figure 4: The knight chess piece near the center of thebox early in the packing program’s attempts to placethe part. At this stage the metric � � = 0.98656 .
Figure 5: The knight chess piece is in the sameposition as in the previous figure but with a differentorientation. The metric � � remains at 0.98656 .
being packed moves through the container. The example is of 6 chess pieces being packed into a box, 2 rooks, 2 knights
and 2 pawns. In this case the packing algorithm was unable to place the final pawn without it intersecting other parts.
The box container was designed to be just tall enough for the knight pieces so that the packing algorithm would place
them vertically to simplify visualization. This limits the possibility of tipping the chess pieces since any angle from
vertical would cause the base or top of the piece to touch or pass through the container walls.
Figures 4-7 show the first piece, the knight, being tested in different positions in the box and the corresponding point
moment metric values. The first figure in the series, Figure 4, shows the knight near the middle of the box where the
metric is almost at its maximum. Figure 5 shows the metric remains the same as the knight changes orientation without
changing position. This position was manually introduced into the packing sequence for illustration purposes of this
paper. It is highly unlikely that an optimization routine would repeat exactly the same position for a part while
attempting to generate a packing arrangement. The next figure in the series, Figure 6, shows the knight against the right
wall and approaching the back corner of the container as the metric’s value drops. The knight’s final resting position
is depicted in Figure 7. By placing the knight in this corner the simulated annealing algorithm has located the lowest
metric value that it could.
The last picture (Figure 8) shows the final packing arrangement generated for all the parts by the packing program.
Each successive figure shows how the metric decreases as the object is placed farther into a corner leaving the remaining
free space as usable as possible. It should be noted that the chess pieces being packed here are hollow and have interior
Page 11 of 17
Figure 6: The knight is now near the right wall of thecontainer and getting close to the back corner. Thevalue of the metric � � = is now down to 0.976313.
Figure 7: The knight is in its final position in thecorner and � � = 0.96455. Though difficult to see, thepiece just fits vertically in the container and thuscannot be tilted much off the vertical in the corner.
detail. This means that their point moments are smaller than if they were solid and thus their impact on the metric is
correspondingly smaller as well. Nevertheless, the simulated annealing algorithm is able to arrange the chess pieces
in the container.
More extensive testing of the implementation has been done [2, 3]. Results have shown that the serial packing
method based on the point moment metric generated solutions faster than previously published approaches.
7. Conclusions
It has been shown that the point moment metric � � is highly suitable for use with standard optimization algorithms.
This is due to the fact that it has a limited range of 1 to 0, and that it continuously decreases as the remaining free space
in a 1D, 2D, ... ND packing arrangements gets smaller and more compact. More specifically, as an arrangement of free
space becomes more compact the metric will monotonically decrease. When viewed from an implementational
perspective, it has also been shown that the point moment metric can be cheaply recalculated from the previous metric
value, the container’s initial point moment and the point moment, volume and centroid of the object being added to the
packing arrangement (Algorithm 2, Section 5). Furthermore, estimates or exact values for the point moment of 3D
objects modeled in CAD packages or stored in voxel representations can also be easily found.
Page 12 of 17
Figure 8: Final position of all the hollow chess pieces packed. Thepoint moment metric � � = 0.837802 . The tall pieces can’t be tiltedwithout their bases or tops intersecting the container.
This combination of features make the point moment metric an ideal tool for optimizing partial packing arrangements
found in serial approaches to packing problems.
8. Appendix A : theorems and proofs
This appendix includes the theorems referenced in Sections 4 and 5 and their proofs. These theorems provide the
foundation for the mathematical properties of the point moment and the point moment metric that make them useful for
evaluating partial packing arrangements. Theorem 1 shows that the metric is continuous and thus can be used with
standard continuous optimization routines. Theorem 2 shows that the point moment can be determined for a region
based on simple knowledge of its constituent sub-regions. Theorem 3 shows the metric monotonically decreases as a
region get more compact. Finally Theorem 4 demonstrates that the moment can be quickly determined for a region if
common moments used in dynamics are known for the same region.
Page 13 of 17
Figure 9: Region R with its centroid coincident with the origin O and vector xrepresenting the distance of the infinitesimally small portion of region R, dR,from the centroid and vector x + p representing the distance of dR from pointP.
Theorem 1: The point moment metric � � is a continuous function.
Proof: As the point moment metric in Equation 8 is by definition only � � divided by a constant, it will vary only as � �
varies. However, � � is continuous as it is an integral of a continuous function (i.e. the distance to the centroid) defined
for any region belonging to the ND space. Any slight change in the region of space considered will only slightly change
the integral’s result. Thus, as � � is continuous, so too must the point moment metric � � be. �
Theorem 2: The equivalent of the parallel axis theorem for area moments of inertia applies to the ND point moment
P. In other words, if the point moment were to be taken around any point P positioned in space at p instead of the
region’s centroid, then the resulting point moment Pp can be found using the formula where Pp is� � � �
p c V d= + 2
the point moment about the region’s centroid, d is the distance from the centroid of the region to the point P, and
V is the volume of the region R.
Proof: To simplify the algebra without loss of generality assume that the centroid of the region over which the point
moment should be calculated is at the origin (see Figure 9). The vector p points from the region’s centroid to the point
the moment should be taken around and thus d 2 = p � p. Note the vectors x and p can be expanded into component form
to be [x1, x2, x3, ... xN] and [p1, p2, p3, ... pN] respectfully.
Page 14 of 17
(9)( )
� �
� � � �
pRR
R RR
i iRi
N
RR
c c
d R d R
d R d R d R
d R d R d R
V V d
= + • + = • + • + •
= • + • + •
= • +
+ •
= + + = +
∫∫
∫ ∫∫
∫∑ ∫∫=
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
x p x p x x x p p p
x x x p p p
x x p p
p
2
2
2
0
1
2 2
p x
The first integral is just the definition of the point moment P of a region when the centroid of the region is centered
at the origin. Since the region’s centroid has been placed at the origin, the definition of the centroid requires that the
integral for each component xi , of the vector x. As the second integral breaks down into a sum of these( )x iR
d R∫ = 0
integrals times the constant components of the vector p the second term equals zero. The third integral is just the volume
magnitude of the region times d2. �
Theorem 3: The � � value monotonically decreases as a region of space, R, becomes more compact.
Proof: Assume that the region R occupying the ND volume VR is made up of two subregions occupying volumes V1 and
V2 with point moments � � 1 and � � 2. Thus VR = V1 + V2. Given that these regions have their centroids at locations x1 and
x2 respectively, the centroid of region R, xR, can be found using the equation:
(10)x x xRR R
V
V
V
V= +1
12
2
Let the distance between x1 and x2 be represented by d1 (see Figure 10). The outer circle drawn in the figure is
centered at x1 and is of radius d1. Since the combined region is made up of only two regions, it follows that the centroid
xc lies on the line between the component region’s centroids x1 and x2. In fact, the centroid xR is always a distance of
(V2 /VR )d1 from x1 and (V1 /VR )d1 from x2 along the line. Let d2 = (V1 /VR )d1 be the distance the combined region’s
centroid is from x2.
Now if region 2 is moved slightly closer to the centroid xc, the overall region R becomes more compact. Label
region 2' s new centroid x2*. Reviewing the definition of the metric for both positions of region 2, it can be seen that
the denominators match. Therefore to show that the point moment metric after region 2' s shift � �
* is smaller than the
original metric � �
only requires showing that � � * is smaller than � � .
Page 15 of 17
Figure 10: View of a plane passing through the centroid points x1 and x2
of the two component regions as well as through the centroid x*2 forRegion 2 after it is moved towards the centroid of the combined regionsxR as described in the proof for Theorem 2. The plane represents thewhole space for the 2D case and a slice of the space for the ND caseswhere N > 2.
(11)� � � �� �
� � � �� �= =
free space
orig space R
free space
orig space**_
_
_
_an d
Expanding both point moments gives:
(12a)� � � � � �
R R RV V= + − + + −1 1 1
2
2 2 2
2x x x x
(12b)33 33 33R R R* V V= + − + + −1 1 1
2
2 2 2
2x x x x* * *
and subtracting � � * from � � gives:
(13)( ) ( )� � � �R R R R R RV V− = − − − + − − −* * * *1 1
2
1
2
2 2
2
2
2x x x x x x x x
Page 16 of 17
Equation 13 will show that P* is smaller than P if it can be shown that the distance || x1 - xR || > || x1 - x*R || and
|| x2 - xR || > || x*2 - x*R ||. The smaller circle of radius d2 in Figure 10 encloses the entire region where region 2 can be
moved so that its centroid x*2 is closer to the initial centroid xR than x2 (remember that the plane view in Figure 10 was
picked to pass through points x1, x2 and x*2). Since, the smaller circle is completely contained in the big circle of radius
d1 it shows that no matter where region 2 is placed, it gets closer to x1. Thus the distance d3 between x1 and x*2 must
be less than d1. But x*R lies along the line between the two centroids x1 and x*2 and maintains a fixed proportional
distance from each. Therefore, if d3 < d1 then x*R must be closer to both x1 and x*2 than xR is to x1 and x2 so