Spherical geometry.
Hyde, Hartley
Many teachers approach the teaching of spherical geometry by
considering the earth as a near spherical object and well-known
geographical locations as points on the sphere. If we were handed a
scale model of the earth, reduced to the size of a basketball, the
little bit of dampness would soon evaporate and we would be left holding
a very smooth ball. The minute variations corresponding to mountain
ranges and deep-sea trenches would be insignificant.
There are no lines. Well, not straight lines. I consider the term
'straight line' a tautology and have enjoyed the reaction of
art students who are quite at ease with 'curved lines'.
There are plenty of curves in spherical geometry. The 'lines
of latitude' are circles with their centre on the earth's axis
of rotation. Only one of these, the equator, is a 'great
circle' because its centre corresponds to the centre of the sphere.
All of the meridians, or 'lines of longitude', are great
semicircles. Aircraft routes approximate to great circles because an arc
of a great circle is the shortest distance between two points on the
sphere. Even so, many antipodean tourists are surprised to fly over
Russia and Scandinavia before landing at Heathrow. The great circle arcs
are usually called geodesics.
[FIGURE 1 OMITTED]
Students may enjoy listing some reasons why the US state of
Colorado, which has borders defined in terms of lines of latitude and
longitude, does not form a rectangle.
A geodesic can be drawn using Cabri-3D.
* Hide the contents of the opening screen except for the central
point; label it C.
* Construct a sphere with centre C through a new point in space
which is then labelled A.
* Locate a point B on the surface of the sphere.
* Draw the plane defined by A, B and C.
* Find the great circle which is the locus of points at the
intersection of the sphere and the plane.
* Place a point D on the arc between A and B (this point is only to
identify the minor arc).
* Draw the arc ADB which is the geodesic between A and B on the
surface of the sphere.
* Find some well contrasted colours and it will look much more
impressive than this example.
Having explained how each geodesic is constructed, let us do it the
easy way. Go to http://merganser.math.gvsu.edu/easel/ and download the
program Spherical Easel written by David Austin and William Dickinson.
At the time of writing it is still free. You probably already have a
Java applet installed, but if not, follow the instructions on the
download page.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Spherical Easel is used much like any other dynamic geometry
package. The buttons down the left-hand side allow you to rotate the
sphere, move an element, insert a new point, insert a new great circle
(called a 'line'), insert a new geodesic (called a
'segment'), or insert a new circle. The pull down menu items
enable a feast of construction and measurement options. All new objects
are automatically labelled. The labels can be edited or hidden. You can
also change the colour, size and visibility of each element.
Constructions can be stored as text files and the diagrams can be saved
in a postscript format that is recognised by Photoshop.
I checked the exterior angle of a cyclic quadrilateral and found
that it did not equal the interior opposite angle. What a
disappointment! The angle measurements appear in a small information box
that can be moved to a convenient blank space as shown above.
Having established that some familiar theorems do not have
equivalents on the surface of a sphere, I gave some thought to what
might work and what would probably not work.
The four familiar conditions of congruency survive and any
construction which depends only on congruency will likely succeed on the
surface of a sphere. If we add up the angles of a triangle, the total
relates directly to the size of the triangle. Therefore triangles which
are equiangular are also congruent: a fifth condition of congruency
(A.A.A).
The construction shown in Figure 4 illustrates that the ambiguous
case also exists on a sphere:
* Draw a geodesic AB.
* Add a point M and draw the great circle AM.
* Draw a perpendicular at M.
* Draw a circle centre B to pass through R.
* Find the intersections of the circle centre B, with the great
circle AM and label one C.
* Reflect AB, and the circle in the perpendicular.
* Label the reflected image with primed labels.
* Of the two intersecting points in the reflection, label the point
C' which is not a reflection of C.
* Draw the geodesics BC, AC, B'C' and A'C'.
* The model can be varied using the handle R.
[FIGURE 4 OMITTED]
The geodesics AB and A'B' are equal and the angles BAC
and B'A'C' are equal because of the reflection. The
geodesics BC and B'C' are equal because they are radii of
reflected circles. Thus we have two triangles with two sides equal, but
the angles are not included.
The construction of the in-centre (Figure 5) shows that the
antipode D of the in-centre E, is also an in-centre and as the vertices
of the triangle ABC are moved about the sphere, the point D can take the
place of E as the more obvious in-centre.
[FIGURE 5 OMITTED]
The closest we come to parallel lines are the lines of latitude.
They are better thought of as concentric circles about the poles. Any
two great circles will intersect in two points that are antipodes of
each other. There are no parallel lines, there is no exterior angle of a
triangle theorem, the angles of a triangle do not add up to 180[degrees]
and there are no parallelograms.
The angle subtended at the centre of a circle is seldom double the
angle subtended at the circumference. Angles at a circumference
subtended by the same chord are only equal in symmetrical figures and
the exterior angle of a cyclic quadrilateral does not 'behave'
either.
However, some of the circle theorems remain. The geodesic from the
centre of a circle to the mid-point of a chord is still perpendicular to
the chord because the proof depends only on congruent triangles. In the
limiting case, the tangent of a circle is still perpendicular to the
radius at the point of contact.
When we consider two tangents to a circle we still have a pair of
congruent triangles (R.H.S) and so the tangent segments are equal in
length, are equally inclined to the common geodesic and they support
equal angles at the centre of the circle.
The midpoint theorems involve parallel lines and similar triangles.
We would not therefore expect that the midpoint constructions would be
of any interest. However, if we draw a triangle ABC using three great
circles then we find that a congruent triangle XYZ forms on the opposite
side of the sphere. If we find the midpoints N and M of AB and AC and
draw a great circle through N and M we find that it intersects the sides
XY and XZ at their midpoints Q and P.
Such outcomes relate to the symmetry of the sphere rather than
either congruency or similarity.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
If students have experience using Cabri or Geometer's
Sketchpad before they attempt to use Spherical Easel the software is
very easy to learn. The main difference is that we form a geodesic by
dragging one point to another rather than clicking on the end points.
Given that we often introduce spherical geometry by considering the
geography of the Earth, I would like to see an option that places a thin
outline of the continent shapes onto the surface of the sphere--like the
first diagram of this article.
For those of us who know some plane geometry, Spherical Easel is an
entertaining toy that can consume many hours. It forces us to think
carefully about the logical structures of the geometries of the plane as
well as the sphere. And for a new generation of students, spherical
geometry is an excellent way to compare and better understand one's
knowledge of plane geometry.
However, while there are so many non-mathematicians teaching
mathematics, any type of geometry is likely to fall off the end of
teaching programs. Most of the teachers who studied Euclidean geometry
at school have already retired. If a teacher's main subject is not
mathematics, catching up geometry becomes a low priority. Those who do
make a brave attempt to follow the textbook are unlikely to venture as
far as using geometry software. It is possible for students to reach
senior years having very little knowledge of geometry or experience of
any geometry software.
I wonder what would happen if a student encountered geometry for
the first time by considering only the geometry of a sphere?
Hartley Hyde
cactus.pages@internode.on.net
www.users.on.net/~hhyde/cactus
