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In plane geometry, lines have two important characteristics. A line represents the shortest path between two points, and the slope of such a line is constant. When describing lines on the surface of a spheroid, however, only one of these characteristics can be guaranteed at a time.

A *great circle* is the shortest path between
two points along the surface of a sphere. The precise definition of
a great circle is the intersection of the surface with a plane passing
through the center of the planet. Thus, great circles always bisect
the sphere. The equator and all meridians are great circles. All great
circles other than these do not have a constant azimuth, the spherical
analog of slope; they cross successive meridians at different angles.
That great circles are the shortest path between points is not always
apparent from maps, because very few map projections (the Gnomonic
is one of them) represent arbitrary great circles as straight lines.

Because they define paths that minimize distance between two
(or three) points, great circles are examples of *geodesics*.
In general, a geodesic is the straightest possible path constrained
to lie on a curved surface, independent of the choice of a coordinate
system. The term comes from the Greek *geo-*, earth,
plus *daiesthai*, to divide, which is also the
root word of *geodesy*, the science of describing
the size and shape of the Earth mathematically.

A *rhumb line* is a curve that crosses
each meridian at the same angle. This curve is also referred to as
a *loxodrome* (from the Greek *loxos*,
slanted, and *drome*, path). Although a great circle
is a shortest path, it is difficult to navigate because your bearing
(or *azimuth*) continuously changes as you proceed.
Following a rhumb line covers more distance than following a geodesic,
but it is easier to navigate.

All parallels, including the equator, are rhumb lines, since they cross all meridians at 90º. Additionally, all meridians are rhumb lines, in addition to being great circles. A rhumb line always spirals toward one of the poles, unless its azimuth is true east, west, north, or south, in which case the rhumb line closes on itself to form a parallel of latitude (small circle) or a pair of antipodal meridians.

The following figure depicts a great circle and one possible rhumb line connecting two distant locations. Descriptions and examples of how to calculate points along great circles and rhumb lines appear below.

In addition to rhumb lines and great circles, one other smooth
curve is significant in geography, the *small circle*.
Parallels of latitude are all small circles (which also happen to
be rhumb lines). The general definition of a small circle is the intersection
of a plane with the surface of a sphere. On ellipsoids, this only
yields true small circles when the defining plane is parallel to the
equator. Mapping Toolbox™ software extends this definition to include
planes passing through the center of the planet, so the set of all
small circles includes all great circles as limiting cases. This usage
is not universal.

Small circles are most easily defined by distance from a point. *All
points 45 nm (nautical miles) distant from (45*º*N,60*º*E)* would
be the description of one small circle. If degrees of arc length are
used as a distance measurement, then (on a sphere) a great circle
is the set of all points 90º distant from a particular *center* point.

For true small circles, the distance must be defined in a great
circle sense, the shortest distance between two points on the surface
of a sphere. However, Mapping Toolbox functions also can calculate *loxodromic
small circles*, for which distances are measured in a rhumb
line sense (along lines of constant azimuth). Do not confuse such
figures with true small circles.

You can calculate vector data for points along a small circle
in two ways. If you have a center point and a known radius, use `scircle1`;
if you have a center point and a single point along the circumference
of the small circle, use `scircle2`. For example,
to get data points describing the small circle at 10º distance
from (67ºN, 135ºW), use the following:

[latc,lonc] = scircle1(67,-135,10);

To get the small circle centered at the same point that passes
through the point (55ºN,135ºW), use `scircle2`:

[latc,lonc] = scircle2(67,-135,55,-135);

The `scircle1` function also allows you to
calculate points along a specific arc of the small circle. For example,
if you want to know the points 10º in distance and between 30º
and 120º in azimuth from (67ºN,135ºW), simply provide
arc limits:

[latc,lonc] = scircle1(67,-154,10,[30 120]);

When an entire small circle is calculated, the data is in polygon
format. For all calculated small circles, 100 points are returned
unless otherwise specified. You can calculate several small circles
at once by providing vector inputs. For more information, see the `scircle1` and `scircle2` function
reference pages.

**An Annotated Map Illustrating Small Circles. **The following Mapping Toolbox commands illustrate generating
small circles of the types described above, including the limiting
case of a large circle. To execute these commands, select them all
by dragging over the list in the Help browser, then click the right
mouse button and choose `Evaluate Selection`:

figure; axesm ortho; gridm on; framem on setm(gca,'Origin', [45 30 30], 'MLineLimit', [75 -75],... 'MLineException',[0 90 180 270]) A = [45 90]; B = [0 60]; C = [0 30]; sca = scircle1(A(1), A(2), 20); scb = scircle2(B(1), B(2), 0, 150); scc = scircle1('rh',C(1), C(2), 20); plotm(A(1), A(2),'ro','MarkerFaceColor','r') plotm(B(1), B(2),'bo','MarkerFaceColor','b') plotm(C(1), C(2),'mo','MarkerFaceColor','m') plotm(sca(:,1), sca(:,2),'r') plotm(scb(:,1), scb(:,2),'b--') plotm(scc(:,1), scc(:,2),'m') textm(50,0,'Normal Small Circle') textm(46,6,'(20\circ from point A)') textm(4.5,-10,'Loxodromic Small Circle') textm(4,-6,'(20\circ from point C') textm(-2,-4,'in rhumb line sense)') textm(40,-60,'Great Circle as Small Circle') textm(45,-50,'(90\circ from point B)')

The result is the following display.

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