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.
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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.
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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.
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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.
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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.
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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?