The dimension definitions for natural arches presented in these pages
make extensive use of a few basic mathematical terms, especially certain
concepts from the discipline of topology.
Doing so avoids ambiguity. Since these mathematical terms have explicit
and rigorous meanings, they make the definitions concise, albeit somewhat
dense for many non-technical readers. Any attempt to maintain rigor
without their use would require excessively lengthy and clumsy definitions.
Therefore, it is necessary to define the basic set of mathematical terms
that are routinely used throughout these definitions. Readers who wish
to measure natural arches in accordance with the set of standard dimensions
presented here must have an appreciation of these concepts.
Surfaces, Points, Curves, and Chords
As a start, some readers may wish to review the following basic mathematical
In order to define some additional, more complex terms, we will be working
with points and curves drawn on two-dimensional surfaces that are curved
in three-dimensional space.
It is also important for the reader to understand the difference between
curve and an open curve. Similarly for surfaces, a closed surface
bounds a volume while an open surface does not. Thus, a balloon and
an inner tube are both closed since they bound the volume of air inside
them. On the other hand, a sheet of paper bounds no volume. It does
not make sense to talk about the volume of air "inside" the sheet of
paper. It is an open surface.
Imagine a large sheet of paper which is flexed in arbitrary ways. This
represents an arbitrary open surface. Many of the curves we will draw
will be confined to this surface, i.e. confined to the paper. Since
the paper is bent, these curves are also bent in arbitrary ways in three-dimensional
space even though they must remain on the paper. Thus, they are space
curves, but for simplicity we will just call them curves.
We will frequently draw line
segments to connect two arbitrary points on the surface as a measure
of the minimum distance between the two points in three dimensional
space. Thus, these line segments are not confined to the surface. Instead,
they connect points on the surface across a bend or fold of the surface.
These connecting line segments are called chords.
Figure 2 below illustrates these terms.
Start with the open surface A. Curve c is a space curve that is confined
to surface A. Points X and Y are on surface A, and point X is also on
curve c. Chord k connects points X and Y, and is not confined to surface
A. The length of chord k is the minimum distance between points X and
Y in three dimensions.
We will also make use of projections.
Typically we will project a chord onto a plane,
thus creating a new straight line segment with a different length and
a different orientation from the original chord. Although the analogy
is over simplified, imagine a straw suspended over a horizontal desktop
with a bright light above it. The straw may be at any arbitrary orientation.
It represents the original chord. The shadow of the straw on the desktop
is the projection of the chord onto the horizontal plane.
Another type of projection we will use is the projection of a chord
onto the vertical. In this case the resulting projection will be a straight
line segment parallel to the vertical having its highest point at the
same elevation as the highest point of the chord and its lowest point
at the same elevation as the lowest point of the chord.
Note that in both types of projection the length of the resulting line
segment is usually shorter, possibly equal to, but never longer, than
the original chord we decided to project. Also in both types of projection
the orientation of the resulting line segment is usually different from
the orientation of the original chord. Note that when projecting onto
the vertical, the resulting orientation is completely fixed. It must
be vertical no matter what the orientation of the original chord. When
projecting onto a plane such as the horizontal, the direction is constrained
but not fixed. It must be horizontal, but its azimuth matches the orientation
of the original chord.
Transformations, Loops, and Shrinking
We will also need to be able to move our points and curves around on
the surface, remembering to always keep them confined to the surface.
Moving a curve or point across the surface in such a way as to keep
it on the surface is called a transformation.
When we transform a curve along the surface, any chords and projections
associated with that curve move along with it, even though these are
not confined to the surface.
is a special curve that is drawn on the surface so that it connects
back with itself, i.e., it is closed. Loops divide the surface into
two areas, one inside the loop and one outside the loop. Curves that
are not closed are not loops. If it extends to the edges of an open
surface, an open curve might also divide the surface into two areas,
but these areas could not be characterized as being either inside or
outside the curve. Finally, a loop does not cross itself. For example,
a figure-8 shaped curve is not a single loop, but rather two loops which
happen to touch at one point.
As with any point or curve, we can move or transform loops drawn on
a surface. One special type of transformation for a loop is to make
it get smaller and smaller. This type of transformation is called shrinking.
Open curves cannot be shrunk. Closed curves, including loops, can be
What happens to a loop on our sheet of paper when we shrink it? For
a circular loop, the radius gets smaller and smaller until it goes to
zero. The loop has shrunk to a point. In this example, we kept the shape
of the loop the same, i.e., a circle, for simplicity sake. In general,
however, it is not necessary to preserve the shape of the original loop.
It is only necessary that the area interior to the loop decrease as
the loop shrinks. All loops drawn on a flexed sheet of paper can be
shrunk to a point, no matter how big or what shape they are originally.
This is not true for all surfaces, however. On certain types of surfaces,
some loops can only be shrunk so far. This important distinction is
fundamental to the system of arch dimension definitions presented here.
It is also key for defining what a natural arch is.
Holes and Orbits
In what follows, the term hole
is used as it is defined in the mathematical discipline of topology.
The surface of an arch, i.e., the surface that bounds the rock frame
(and any fill material) from the air that surrounds it, has a topological
hole. If we draw loops in various places on the arch surface, it will
become apparent that some loops can be shrunk to a point while others
Figure 3 shows the four types of loops that occur on an open surface
with a hole, i.e., an arch surface:
Figure 3: A is an open surface with a hole. It is the surface
of a natural arch. Loop a is a point loop. It can be shrunk to a point.
Loop b is a boundary loop. It can't be shrunk to a point because it
surrounds the entire arch. Loop c can't be shrunk to a point because
it encircles the lintel. It is called a lintel orbit. Loop d can't
be shrunk to a point because it encircles the opening. It is called
an opening orbit.
Loops which can be shrunk to a point are called point loops, e.g.
loop a. Loops which cannot be shrunk to a point because they completely
surround the arch are called boundary loops, e.g. loop b. Remember in
shrinking the loop that all parts of it must remain on the surface at
Some loops cannot be shrunk to a point because they either encircle
the arch's lintel, e.g., loop c, or the arch's opening, e.g., loop d.
(See Natural Arch Components for definitions
of the opening and lintel.) Loops which encircle either the opening
or the lintel are called orbits. Orbits around the opening are called
opening orbits and orbits around the lintel are called lintel orbits.
Orbits, chords associated with orbits, and projections of those chords
are the basic building blocks used to define many of the standard set
of natural arch dimensions.
An arbitrary orbit, while confined to the arch surface, is still free
to wander anywhere on the surface, i.e., it can be transformed so that
it contains any point on the surface. Orbits that are also restricted
to a plane are a special subset of the general class of orbits. Such
orbits are defined by the intersection of a plane with the surface and
are called planar orbits. An example of a planar lintel orbit is depicted
in Figure 4, which shows the intersection of a plane with the arch's
lintel. One can also easily imagine a planar opening orbit defined by
the intersection of a plane with the arch's opening.
Figure 4: Plane P intersects the arch surface through the lintel,
thus defining a planar lintel orbit. If the plane happened to be vertical,
it would define a vertical lintel orbit. Planar opening orbits are
defined similarly, including vertical and horizontal opening orbits
when they exist.
Planar orbits defined by planes that are vertical or horizontal
form two other important subsets of orbits. These are called vertical
orbits and horizontal orbits respectively, where the word planar is
understood. Vertical lintel orbits, horizontal lintel orbits, vertical
opening orbits, and horizontal opening orbits are also key ingredients
of the definitions offered here.
One more key term remains to be defined in the set of
basic mathematical terms used in the definitions, the min/max operator.
This operator is somewhat complex and is best explained with an example.
Imagine a set of objects (for example apple trees in an
orchard) each of which has a set of other objects (for example apples)
associated with it. Furthermore, assume that each of these other objects
exhibits a characteristic that is measurable (for example the diameter
of an apple.) In this example of an apple orchard, we want to estimate
the productivity of a specific tree. One of the numbers that will help
estimate this is the diameter of the largest apple on the tree in question.
After the diameters of all the apples on the tree are measured, we determine
which diameter is the largest and then associate that value with the
tree. In short, let the maximum apple diameter produced by tree N be
designated as DN.
Now we want to do some analysis on the productivity of
the entire orchard. We want to quantify the minimum yield of any tree
in the orchard. A number that will help estimate this is the smallest
of all the maximum diameters, DN. We first determine DN for each tree
and then determine the smallest of these values. This is the minimum
DN for the orchard. The smallest of the largest.
The term min/max is a convenient notation that represents
the selection process described in the orchard example above. It selects
the minimum value from a set of maximum values, each of which was previously
selected from a set of some measurable quantity.
Now let's look at an example a little closer to our subject
than apples and orchards. Consider a chord drawn between two arbitrary
points on an orbit. The length of the chord measures the distance between
the two points. Next, consider all possible chords that could be drawn
between any two points on the orbit. There is obviously an infinite
number of chords, but don't be daunted. Of all those possible chords,
what chord is the longest, i.e., has the maximum length? Typically,
we don't have to draw all the chords to determine which is the longest.
In most cases it will be obvious. In a few cases, we might have to measure
two or three chords to see which is really the longest. In some very
special cases, e.g., a circular orbit, many chords will tie for being
the longest, but it will still be easy to determine that maximum chord
The maximum chord length we've just determined is a measure
of the maximum spatial extent of the orbit we started with. Now let's
examine a whole set of orbits which together form a surface. Consider
the entire set of maximum chord lengths for every orbit on this surface.
From this set of maximum chord lengths, select the shortest. We'll call
this value the min/max of chord lengths for all orbits of the surface.
This min/max of chord lengths, i.e., this shortest of the longest chords,
is a measure of the spatial extent of our surface where it is most constrained.
In fact, had we been dealing with opening orbits on an arch surface,
the min/max of chord lengths would be the distance L in Figure 1 below.
L is one of the standard dimensions of a natural arch.