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.