Area
DABC is a Scalene triangle. Segments AX, BY, and CZ are the three altitudes of DABC. Notice that, as in Euclidean Geometry, the three altitudes intersect at a single point. Measuring this particular triangle gives the following data:
Now, if we calculate the area by A=1/2bh, we find that
1/2(AB)x(CZ) is NOT EQUAL to 1/2(BC)x(AX) nor equal to
1/2(AC)x(BY). In Hyperbolic Geometry, the equation A=1/2bh
gives three different answers depending on which side you use as the
base. Therefore, A=1/2bh, will not work as an area function in
Hyperbolic Geometry..
6.2 A=kd
In Euclidean Geometry, we define a square region that has edges of
length 1 unit to have an area of 1 square unit. In
Hyperbolic Geometry, rectangles (quadrilaterals with 4 right angles)
do not exist, and, therefore, squares (a special case of a rectangle
with four congruent edges) also do not exist. In Hyperbolic
Geometry, if a quadrilateral has 3 right angles, then the forth angle
must be acute (see figure 6.2a).
A regular quadrilateral is a quadrilateral that has four sides of equal length, and four angles of equal measure. A square it is a special case of regular quadrilateral where the four angles are right angles. In Euclidean Geometry, all regular quadrilaterals are squares. In Hyperbolic Geometry, regular quadrilaterals exist, but they all have four acute angles. Regular quadrilaterals in Hyperbolic Geometry cannot be used to form the basic unit of area the way squares do in Euclidean Geometry. One reason for this is that Hyperbolic, regular quadrilaterals do not fit together without leaving gaps. Figure 6.2b shows how nine, 1x1, Euclidean, regular quadrilaterals form a single 3x3, Euclidean, regular quadrilateral. Figure 6.2c shows five, 1x1, Hyperbolic, regular quadrilaterals - notice the gap in the upper right.
6.3 Defect of a
Triangle
As we saw earlier, in Hyperbolic Geometry, the sum of the three
angles of a triangle is always less than 180°. The
defect of a triangle is defined as 180° minus the sum of
the three angles of the triangle. When we construct a few
Hyperbolic triangles, and measure the defect of each, we find that
for small triangles the defect is small (the angle sum is almost
180°). In fact, as the perimeter of triangle approaches
zero, the angle sum approaches 180°. This is consistent
with the idea that a relatively small piece of Hyperbolic Space
looks, and behaves very much like Euclidean Space.
Contrariwise, we find that large triangles have a large
defect. As the length of the three sides of a
triangle get closer and closer to infinity, each of the angles gets
closer and closer to zero degrees - and, therefore, the defect gets
closer and closer to 180°.
Before proceeding, it would be a good idea to try constructing some example triangles in NonEuclid. Use the "Measure Triangle" command from the "Measurement" menu: this command will display the angle measure of each vertex, and the length of each side. It is important to get a "feel" for how it is that large triangles have large defects.
The larger the triangle, the larger the defect - but more than that: just like area, the defect is additive. The whole equals the sum of the parts. For example, in figure 6.3, the defect of DBAM is 77.4°, the defect of DCAM is 43.7°, and the defect of DABC is 121.1° (77.4 + 43.7 = 121.1). This works for all triangles in Hyperbolic Geometry - regardless of how you cut them into smaller triangles. Use NonEuclid to try a few examples.
6.4 Defect of a
Polygon
Any polygon can be cut up into a finite number of non-overlapping
triangles. Figure 6.4 shows two different ways that the same polygon
might be cut up.
The defect of a polygon is defined to be the sum of the
defects of a set of triangles that it can be cut up into. A
polygon can be cut up into triangular regions in infinitely many ways
(to save space, only two are shown for the polygon above).
However, the sum depends only on the polygon that we started with,
and is independent of the way in which we cut it up. Try
constructing and measuring a few examples.
6.5 Invariance of Defect with
Translation
We have already seen, that objects appear to shrink, and flatten as
they move from the center of the Boundary Circle toward the
edge. Whatever we use to measure area must remain invariant as
an object moves from one location to another. Recall that in
spite of the fact that objects appear to shrink and flatten, the
length of all sides, and the measure of all the angles remains
constant as an object moves. Therefore, the defect remains
constant as an object moves (since the defect is calculated by
measuring angles only).
6.6 Properties Necessary for an
Area Function
n summery, an area function must have the following properties
[Moise-74]:
It can be proved that every function that satisfies A-1 through A-4 has the form of a simple constant (k) times the defect (d) or A=kd . [Moise-74]
The choice of the constant, k, is important. It is what
links the units of length to the units of area.
6.7 Upper Bound to
Area
One interesting consequence of A=kd is that the maximum area of any
triangle is bounded. Since the defect of a triangle can never
be greater than 180°, the area can never be greater than
k(180). Is it also true that the area of a polygon is
bounded? It could be argued that the area of a polygon would
have to be bounded as follows: lets say that you
constructed a polygon a defect of 200°. Then construct a
triangle that completely includes the polygon. The defect of
the triangle equals the sum of the defects of each of its parts,
therefore the defect of the triangle must be greater than 200° -
but this is impossible. Therefore, it must be impossible for a
polygon to have a defect of 200°. This however, is not a
proof because it contains an assumption - see the exercises on
Polygons for more details.
NonEuclid
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