## Natural Arch Dimensions

### Introduction

It is desirable to determine the dimensions of natural arches for
several reasons. For example, dimension measurements are useful for
comparison, description, classification, and analysis.

Through comparison we answer questions such as which arch is longest,
thinnest, tallest, etc. These questions are asked for a given region,
geology, arch type, etc. Description is useful for selecting arches
to visit or study, e.g., visiting only those arches larger than a given
size, or studying a sample falling within some size range. Classification
of arches into different size groups is a useful way to accomplish this.
Also, it may be possible to base a morphological classification scheme
on dimension measurements. (Note that the standard taxonomy
presented on this site is a genetic scheme rather than morphological.)
Finally, dimension measurements can be merged with other observable
arch attributes to perform various analyses. For example, analysis of
natural arch formation modes and evolutionary lifecycles require such
data.

For dimension measurements to be useful in satisfying these purposes,
they must be repeatable for any given arch and comparable for different
arches. By repeatable, we mean that two different investigators will
measure the same quantities when documenting an arch and, therefore,
obtain similar or identical results. By comparable, we mean that any
size comparison of two different arches will be based on measurements
of the same spatial attributes.

As an example of this, examine the two arches shown in Figure 1 below.

*Figure 1a*

*Figure 1b*

Different shapes or orientations can result in different
interpretations of dimensions. For the two arches in Figure 1 above,
L1 equals L2, while S1 is much greater than S2. Which of the two arches
is "longer"?

If one investigator equates the length of an arch to a
measurement of L while another equates it to a measurement of S, no
matter how accurately they measure they will report different "lengths."
Similarly, one investigator will say the two arches are about the same
length while the other will say one arch is much longer than the other.
The problem illustrated by this example becomes much more severe when
dealing with the full range of morphological complexity exhibited by
natural arches.

The solution to this problem is a set of universally accepted
definitions for natural arch dimensions. Unfortunately, most of the
dimension definitions found elsewhere, including those in the references,
are vague at best, containing ambiguous terms and phrases. These ambiguities
make repeatability and comparability difficult, if not impossible.

The set of dimension definitions below have been rigorously
crafted to avoid ambiguity. They are applicable to all natural arch
types. This standard set of dimensions, if carefully measured and recorded
by all field observers, will enable repeatability and comparability
and will thus permit the goals of comparison, description, classification,
and analysis to be achieved.

### Caveats

The definitions presented here are intended for use in
describing and measuring any type of natural arch with a single opening.
While these definitions can be successfully applied to many arches with
multiple openings, in some such cases their use is problematic.

All of the dimensions defined here are linear extents
of some part or component of an arch. Areas and volumes of various parts
of arches might be useful to describe the spatial characteristics of
arches, but are typically impossible to measure or even estimate in
the field. For this reason, they are only discussed when doing so helps
define a linear dimension of interest. While linear dimensions are also
often difficult to measure, they are much more obtainable than areas
or volumes.

This site does not discuss the several techniques by which
arch dimensions can be measured. It tries to define which spatial extents
should be measured, and from what point to what point the measurement
should be made, but does not discuss how the measurement is actually
taken, e.g. with a steel tape, or transit, or by visual comparison.
Obviously, the accuracy of measurements obtained is dependent upon that
choice.

### Preliminaries

Before examining the definitions for natural arch dimensions,
the reader should become familiar with two topics. First, the reader
should review the meaning of some mathematical terms whose use is necessary
to ensure the rigor of the dimension definitions. These terms are reviewed
at Basic Terminology. Second, the reader
should review the standard definitions for the basic components or parts
of natural arches. These definitions are presented at Natural
Arch Components.

### Morphology Independent Dimensions

Some dimensions are applicable to natural arches of any
shape, while others only have meaning if the arch's shape is of a certain
type. For example, the "vertical thickness of the lintel" has little
meaning for a pillar arch. Thus, there
are dimensions that do not depend on the morphology of the arch, and
there are dimensions that are morphology dependent. This section defines
some morphology independent dimensions.

Perhaps the most commonly used dimension existing in the
literature today is a natural arch's span. In reference
1, Vreeland defines the span of a natural arch to be "the maximum
horizontal length of the opening between supports, measured where the
full opening is at its minimum," and "the horizontal length of the unsupported
rock." There are two ambiguities in these definitions. It is not clear
what is meant by the phrase "where the full opening is at its minimum,"
nor is it obvious in all cases which parts of the rock are supported
and which are unsupported.

This ambiguity can be removed by recasting Vreeland's
definition using the terms described in Basic Terminology. The new,
rigorous definition for the span of a natural arch is *the min/max
of the lengths of the horizontal projections of the chords of all opening
orbits*. In other words, consider all the opening orbits on the arch's
surface. On each one, draw all possible chords and project them onto
the horizontal. For each opening orbit, select the longest of these
projections. This is just the maximum horizontal extent of the orbit.
From this set of longest projections, one for each opening orbit, select
the shortest. This is just the shortest maximum horizontal extent of
any opening orbit. The length of the selected projection is the span.
It corresponds to S in Figure 1.

Note that the span was defined as a scalar value, i.e.,
a length. It is useful also to determine the orientation of the projection
that was selected by the min/max operator. Obviously, this projection
is confined to the horizontal, but its azimuth is arbitrary. Therefore,
another important attribute of the arch which should be measured is
the orientation, or azimuthal angle, of the projection whose length
is the span.

It is possible that more than one projection will produce
the same min/max length we call the span. These projections might be
associated with completely different orbits and hence have different
azimuths. For example, this would be true for a hypothetical arch shaped
like a pipe with a bend in the middle. Of course, natural arches seldom
exhibit this degree of symmetry. In practice the projection which produces
the span will be unique and the azimuth will also be unique.

Another dimension which is useful to cast in the min/max
notation is the length of light opening defined by Stevens and McCarrick
in reference 2. The new, rigorous definition
is the *min/max of the lengths of the chords of all opening orbits*.
This corresponds to L in Figure 1.
As with the span, it is useful also to determine the azimuthal angle
of the chord which produces the length of light opening. Furthermore,
since this chord is not confined to the horizontal, the angle the chord
makes with the vertical should also be determined.

Analogs to the span and length of light opening exist
for the lintel. These dimensions use lintel orbits rather than opening
orbits. Thus, *the min/max of the lengths of the horizontal projections
of the chords of all lintel orbits* is a recast, rigorous definition
for both Vreeland's width (reference 1) and Stevens/McCarrick's horizontal
thickness (reference 2).

The other analog, the *min/max of the lengths of the
chords of all lintel orbits*, does not correspond to any currently
published dimension, but seems to be a useful one, especially for vertically
oriented lintels. We call this new dimension the lintel breadth.

At this point, we have defined four primary dimensions:
span, length of light opening, width, and lintel breadth. We now define
dimensions that are orthogonal (perpendicular) to the orientations of
these primary dimensions. Two of these orthogonal dimensions apply to
any arch and are defined next. Others apply only to certain well-defined
morphological classes of arches and are therefore defined in the next
section.

The Stevens/McCarrick dimension called depth of light
opening (reference 2) can be defined as the *min/max of the lengths
of the opening orbit chords orthogonal to the length of light opening
chord*. Similarly, we define the lintel depth to be *the min/max
of the lengths of the lintel orbit chords orthogonal to the lintel breadth
chord*. These two definitions complete our set of morphology independent
arch dimensions.

### Morphology Dependent Dimensions

The dependence on arch morphology for our dimension definitions
is minimized by dividing all arches into classes based upon three characteristics
that are easily determined for any arch shape. Arches that do not have
any vertical opening orbits must be treated as special cases. This is
also true for arches that don't have any vertical lintel orbits. Finally,
arches that have horizontal opening orbits should be treated differently
from arches that do not.

For arches with vertical opening orbits, Vreeland's height
(reference 1) is redefined as the *min/max of the lengths of the vertical
projections of the chord of all vertical opening orbits*. For arches
without vertical opening orbits, Vreeland's height is redefined as *the
length of a vertical line segment drawn down to the ground from the
min/max of the elevations of all points on all opening orbits*.

For arches with vertical lintel orbits, both thickness
(reference 1) and vertical thickness (reference 2) are redefined as
*the min/max of the lengths of the vertical projections of the chords
of all vertical lintel orbits*. For arches without vertical lintel
orbits, we artificially create an analog to them by considering the
volume of rock above the opening to be detached and suspended in air.
This volume must have vertical orbits. These are analogous to vertical
lintel orbits, but aren't constrained to the arch surface. Rather they
are constrained to the surface of the volume of rock above the opening
considered in isolation. Using this construct, thickness and vertical
thickness are redefined as *the min/max of the lengths of the vertical
projections of the chords of all vertical orbits of the volume of rock
above the opening*.

For arches with horizontal opening orbits, Vreeland's
width of opening (reference 1) is redefined as the *min/max of the
lengths of the horizontal opening orbit chords orthogonal to the span
projection*. For arches without horizontal opening orbits, Vreeland's
width of opening is not applicable. In some of these cases, either the
separation dimension or the divergence dimension is of interest. These
two dimensions are defined next.

Some arches are adjacent to a cliff wall that provides
no support to any part of the lintel, but which noticeably occludes
the opening. Two obvious examples are Black
Arch and Surprise
Arch in Arches National Park. In such cases, the separation of the
arch from the occluding cliff wall is of interest. This separation is
defined as *the minimum distance found between any of the set of points
on all planar lintel orbits and any of the set of points on the cliff
wall surface*. Note that if this minimum distance is zero, then the
cliff wall is providing some support to the lintel and this dimension
is not applicable. In this case, either use width of opening if the
arch has horizontal opening orbits or divergence if it does not.

Some arches are adjacent to a cliff wall which provides
some support to the span and occludes the opening, but the arch still
has no horizontal opening orbits. An example of this is Mesa
Arch in Canyonlands National Park. In cases like this, the separation
dimension defined above will be zero. What remains of interest, however,
is the maximum divergence of the lintel from the cliff wall. For this
particular morphology, divergence is defined as *the maximum distance
found along a straight line segment orthogonal to the span projection
between any of the set of points on all planar lintel orbits and any
of the set of points on the cliff wall surface*. Note that the restriction
of measuring orthogonal to the span projection does not restrict the
divergence dimension to be horizontal. Although the divergence will
be horizontal for many arches, in some cases it will not be. The divergence
of Mesa Arch, for example, is not horizontal because the span extends
above the top of the adjacent cliff wall.

### Refinement of Semantics

Various authors use different words to label the natural
arch dimensions they have defined. Looking across the literature, the
reader will find many examples of the same word meaning different things
and the same dimension labeled with different words. For this reason,
we have chosen to modify some of the semantics used by these authors
and recast the definitions of the standard set of dimensions using a
standardized set of dimension labels. This standardized semantics for
dimensions is based on the following analysis.

A popular dictionary contains the following definitions:

*length* – the greatest of the two or three dimensions of anything
*span* – the distance between two ends or supports; as, the
span of an arch
*width* – distance from side to side
*breadth* – measure of anything from side to side; width
*height* – the distance from the bottom to the top
*thickness* – dimension from surface to opposite surface
*depth* – the distance from the top straight downward, from
the surface or edge inward, or from front to back

Several conclusions are drawn from these:

*Length* is a poor choice for labeling arch dimensions because
it implies the longest (greatest) dimension. The definitions that
currently include *length* in their name are not always the longest
dimension for all morphologies. Indeed, no dimension can be consistently
applied for all arch morphologies and always be the longest dimension.
*Span* is a good choice for labeling the dimension Vreeland
calls *span*. Indeed, *span* is used throughout the general
literature on arches in ways similar to the meaning Vreeland intends.
It seems unnecessary and even inappropriate to replace it.
*Width* is a good choice for labeling arch dimensions as long
as we have a common understanding of what "side to side" means for
arches. What seems natural is to use *width* to label horizontal
dimensions.
*Breadth*, like *width*, is a good choice. However, since
its dictionary definition is identical with *width*, is there
a need for it? Since we used *width* to label horizontal dimensions,
we use *breadth* to label dimensions that are "from side to side"
but are not necessarily horizontal.
*Height* is a good choice and seems to be used correctly and
consistently by most authors.
*Thickness* is a good choice. Although no orientation is implied
by the definition, both references 1 and 2 use it to label vertical
dimensions. Unfortunately, reference 2 also uses it for a horizontal
dimension. Since *thickness* is often paired with *width*
in common usage, we constrain its use to vertical dimensions orthogonal
to *width*.
*Depth* is a good choice. Since it is often paired with *breadth*,
we use it for dimensions orthogonal to those described using *breadth*.

All but two of the terms used to label dimensions in reference
1 and 2 are consistent with the above. To remove those exceptions, we
have renamed length of light opening from reference 2 to opening breadth.
Similarly, horizontal thickness from reference 2 has been changed to
width. These changes are reflected in the synopsis of dimension definitions
below.

### Dimension Definition Synopsis

The following are rigorous definitions for the standard
set of natural arch dimensions that should be measured and reported
when documenting a natural arch. These dimensions are divided into two
subsets. The first contains dimensions which are aligned with the vertical
as a reference direction. These are called aligned dimensions. The second
contains dimensions which are not tied to the vertical. These are called
unaligned dimensions. Both subsets are further divided into dimensions
pertaining to the opening and dimensions pertaining to the lintel.

**Aligned Dimensions for the Opening:**

**SPAN** - the min/max of the lengths of the horizontal projections
of the chords of all opening orbits.

**HEIGHT** - the min/max of the lengths of the vertical projections
of the chords of all vertical opening orbits; if there are no vertical
opening orbits, then the length of a vertical line segment drawn down
to the ground from the min/max of the elevations of all points on
all opening orbits.

**WIDTH OF OPENING** - the min/max of the lengths of the horizontal
opening orbit chords orthogonal to the span projection; if there are
no horizontal opening orbits, then this dimension is not applicable.

**SEPARATION** - the minimum distance found between any of the
set of points on all planar lintel orbits and any of the set of points
on an adjacent cliff wall surface; only applicable if there are no
horizontal opening orbits.

**DIVERGENCE** - the maximum distance found along a straight
line segment orthogonal to the span projection between any of the
set of points on all planar lintel orbits and any of the set of points
on an adjacent cliff wall surface; only applicable if there are no
horizontal opening orbits and the separation is zero.

**Aligned Dimensions for the Lintel:**

**WIDTH** - the min/max of the lengths of the horizontal projections
of the chords of all lintel orbits.

**THICKNESS** - the min/max of the lengths of the vertical projections
of the chords of all vertical lintel orbits; if there are no vertical
lintel orbits, then the min/max of the lengths of the vertical projections
of the chords of all vertical orbits of the volume of rock above the
opening volume.

**Unaligned Dimensions for the Opening: **

**OPENING BREADTH** - the min/max of the lengths of the chords
of all opening orbits.

**OPENING DEPTH** - the min/max of the lengths of the opening
orbit chords that are orthogonal to the opening breadth chord.

**Unaligned Dimensions for the Lintel: **

**LINTEL BREADTH** - the min/max of the lengths of the chords
of all lintel orbits.

**LINTEL DEPTH** - the min/max of the lengths of the lintel orbit
chords that are orthogonal to the lintel breadth chord.

### Discussion

The dimensions defined above fall into groups. This is
no accident since they are all either directly based on, or natural
extensions of, the groups of dimensions found in references
1 and 2. Vreeland's span and height are an orthogonal pair aligned
with the vertical. As such, the pair attempts to characterize the arch's
defiance of gravity. This intent is reflected in Vreeland's use of the
phrase "the horizontal length of the unsupported rock" in his definition
of span. A set of dimensions aligned with the vertical provides valuable
information on both the appearance of the arch and its structural character.

When Vreeland's width of opening dimension is applicable,
it is orthogonal to both the span and the height, creating a three-dimensional
set. When width of opening is not applicable, one of three cases obtains,
depending on whether or not the arch is next to a cliff wall. For two
of these three cases, new dimensions have been defined to complement
span and height. For the last case, no third dimension is needed.

First, if there is an adjacent cliff wall that occludes
the arch opening but provides no support to the lintel, then the separation
dimension replaces width of opening as the third member of the trio.
Second, if the occluding cliff wall does provide support to the lintel,
then the divergence should be measured as the third dimension. The divergence
will be orthogonal to the span, but may or may not be orthogonal to
the height. Lastly, if there is no adjacent cliff wall, then no third
dimension is needed. For example, there is no reason to measure the
distance from Delicate
Arch to the nearest projecting rock mass, since this has no effect
on the structure or appearance of the arch.

The alternative opening dimensions used by Stevens and
McCarrick, labeled opening breadth and opening depth in the Synopsis
above, are not aligned with the vertical. Rather they are free to conform
to the orientation of the arch. They do form an orthogonal pair. These
dimensions provide valuable information on the overall size of the arch
opening regardless of its orientation. They also complement the Vreeland
opening dimensions. In fact, a comparison of span and height to opening
breadth and opening depth can provide an indication of the gross shape
and orientation of the opening.

Vreeland's width and thickness dimensions for the lintel
are also an orthogonal pair aligned with the vertical. The Stevens and
McCarrick lintel dimensions, horizontal and vertical thickness, are
identical to width and thickness respectively.

The newly defined lintel breadth and lintel depth are
an orthogonal pair that are not aligned with the vertical and are free
to conform to the orientation of the arch. They are an obvious extension
of the opening breadth and opening depth. As with those opening dimensions,
this unaligned pair nicely complements the vertically aligned pair to
grossly characterize the fragility, size, and orientation of the lintel.

In addition to the standard set defined above, there are
many other possible natural arch dimensions that could be defined. Two
that are found in references 1 and 2 respectively are total height and
extent, but the list is endless. Some can be defined using the same
rigorous methods employed for the standard set. For example, total height
could be defined as the difference between the maximum elevation of
any point in the lintel and the minimum elevation of any point in the
opening. Others probably are not definable using this method, for example
extent.

The dimensions included in the standard set were selected
for the following reasons. First and foremost, they are now rigorously
defined and hence can be applied repeatably. They are also generically
defined so that they can be applied to any arch and hence permit comparability
between different arches. Finally, they are based on, or are obvious
extensions of, already published dimensions that are in common use.
It is important to note that the standard set implies a minimum set
of dimensions to be measured to document an arch. Nothing prevents an
investigator from including measurements of other dimensions in an arch's
documentation. Should this be done, however, it is hoped that they will
be well defined.