Chapter 6
6.1 Basic Theorems of Hyperbolic Geometry
We still have our 6 axioms of Neutral Geometry and now for our seventh, we will adopt the Hyperbolic Parallel Postulate:
For every line l and for every point P that does not lie on l, there are at least two lines m and n such that P lies on both m and n and both lines are parallel to l.
This is a negation of the Euclidean Parallel Postulate.
Theorem 6.1.1 For every triangle , .
Corollary 6.1.2 For every triangle , 0°< .
Theorem 6.1.3 For every convex quadrilateral, , .
Corollary 6.1.4 The summit angles in a Saccheri quadrilateral are acute.
Corollary 6.1.5 The fourth angle in a Lambert quadrilateral is acute.
Theorem 6.1.6 There does not exist a rectangle.
Since the Saccheri quadrilateral summit angles are acute, it is not a rectangle!
Theorem 6.1.7 In a Lambert quadrilateral, the length of a side between two right angles is strictly less than the length of the opposite side.
Corollary 6.1.9 In a Saccheri quadrilateral, the length of the altitude is less than the length of a side.
The altitude of a Saccheri Quadrilateral is the segment joining the midpoint of the summit to the midpoint of the base. It is, by an earlier proof, perpendicular to each. The length of the altitude is called the height.
Corollary 6.1.8 In a Saccheri quadrilateral, the length of the summit is greater than the length of the base.
Theorem 6.1.11 AAA
If is similar to , then is congruent to .
Strategy: We will show one pair of sides is congruent and have the triangles
congruent by ASA. We’ve got the angles all congruent by hypothesis,
we will assume no side pairs are congruent and show a contradiction.
Proof: Let and be 2 similar triangles. This gives us all 3 cooresponding angle measures are congruent.
Suppose . Now there are 3 side pairs so 2 pairs have one side longer or 2 pairs have one side shorter than
in . We will assume that . If this is not the case, a similar proof will work with the 2 pairs that are longer.
Chose points on so that a smaller internal triangle congruent to is created….
Now by SAS and quadrilateral C’D’B’C is convex (Thm. 4.6.7).
Further . Now, note that the angles at vertices C’ and B’ are linear. Thus the convex quadrilateral is a rectangle with the angle sum = 180°.
This contradicts that we are in Hyperbolic geometry! (see Thm 6.1.6). So at least one of the inequalities on the sides. So our original triangles are congruent. QED
Thus there is no such concept of similarity in Hyperbolic geometry. All triangles are congruent.
Similarly, all Saccheri Quadrilaterals are congruent.
Theorem 6.1.12 If and are two Saccheri Quadrilaterals such that
, then .
Read and understand the proof, please.
6.1.A The Poincaré Disc is a 2D model of Hyperbolic Geometry.
The points are those for which , known as the Unit Disc (as opposed to the Unit Circle).
Lines are arcs of orthogonal circles. These are circles that intersect the Unit Circle with tangents at the point of intersection that are at 90° to the circle.
6.2 Common Perpendiculars
We assume that parallel lines are equidistant from one another…this is not the case in Hyperbolic Geometry
Theorem 6.2.1 If l is a line, P is an external point, and m is a line such that P lies on m,
then there exists at most one point Q such that , Q lies on m and
d(Q, l) = d(P, l).
Let’s look at what this is really saying!
This says there are at most 2 distinct points from one parallel line to another at a given distance between the parallel lines. Note, too, unstated – one of the angles of intersection must be 90° for each point pair.
Proof:
Let l and m be two lines. Suppose there are 3 points P, Q, and R on m such that . Let P’, Q’, and R’ denote the feet of the perpendiculars on l, respectively. As these distances are all non-zero, none of P, Q, or R lies on l. Thus at least 2 of the 3 points must lie on the same side of l, suppose these are P and Q. We then see that
quadrilateral is a Saccheri Quadrilateral. Thus l and m are parallel (Thm 4.8.10, part 4) and further all 3 points lie on the same side of l since they are collinear.
Assume that P*Q*R (if not, then rename the points). Now both and are Saccheri Quadrilaterals. From this we see that are acute. This contradicts the fact that they are supplements! So our supposition that all three distances are the same is not true.
Definition:
Lines l and m admit a common perpendicular if there exists a line n such that n is perpendicular to both l and m.
Theorem 6.2.3 If l and m are parallel lines and there exist two points on m that are equidistant from l, then l and m admit a common perpendicular.
Theorem 6.2.4 If lines l and m admit a common perpendicular, then that common perpendicular is unique.
Theorem 6.2.5 Let l and m be parallel lines cut by a transversal t. Alternate interior angles formed by l and m with transversal t are congruent if and only if l and m admit a common perpendicular and t passes through the midpoint of the common perpendicular.
Let’s sketch this!
Some familiar items from Chapters 3 and 4:
Vertical angles are still congruent:
The Exterior Angle Theorem in Hyperbolic Geometry:
6.3 The Angle of Parallelism
Construction: Let l be a line and let P be an external point. Drop a perpendicular from P to l and call the foot of the perpendicular A. Let be a point on l. For each real number r with
there exists a point on the same side of as B such that .
Define a set K:
K is called the INTERSECTING SET for P and .
Note that K is a subset of [0, 90).
Theorem 6.3.1 Let K be the intersecting set for P and . If , then
1. for every 0 < s < r.
2. there exists a such that t > r.
K is an open interval of the form where is called the critical number for P and .
Any ray from P that makes an angle less thanintersects l. And any ray that makes an angle more than doesn’t intersect l on the side. This is why is critical.
Definition: Angle of parallelism
FM is the HLine just past the one that would define .
Suppose P, A, and B are as given in the definition of intersecting set. Suppose is the critical number for P and . Let D be a point on the same side of as B such that . The angle is called the angle of parallelism for P and .
As you would expect there is another set of rays on the other side of that are symmetric to the
ones shown. Think of as a kind of limit as the angle measure goes to 90°.
Theorem 6.3.5 The critical number depends only on d(P, l).
As the distance from P to l increases the critical number decreases.
Thus we view it as a function on the real numbers.
Theorem 6.3.7 is a nonincreasing function; that is a < b implies
.
Theorem 6.3.8 Every angle of parallelism is acute and
every critical number is less than 90.
This is every bit as much a distinguishing feature between Euclidean Geometry and Hyperbolic Geometry as angle defect. In Euclidean geometry the angle of parallelism is exactly 90°. In Hyperbolic geometry it is strictly less than 90° and varies with the location of P with respect to l.
In fact, the angle of parallelism is a measure of how much deviation from the Euclidean ideal a parallel line has. Just as defect measures how far from a Euclidean triangle a Hyperbolic triangle is.
6.4 Limiting Parallel Rays
Definition: Two rays and are called limiting parallel rays, denoted
If B and D are on the same side of and and every ray between and intersects .
The angle of parallelism provides the basic example of limiting parallel rays.
Theorem 6.4.2 If , then .
If two rays are limiting parallel rays, then they are parallel.
(asymptotically)
Theorem 6.4.3 Symmetry of Limiting Parallelism
If , then .
Theorem 6.4.4 Endpoint Independence
If is a ray, and P, Q, and D are points such that Q*P*D, then
if and only if .
The points need to be collinear.
Theorem 6.4.5 Existence and Uniqueness of Limiting Parallel Rays.
If is a ray and P is a point that does not lie on , then there exists a unique ray such that .
Theorem 6.4.7 Transitivity of Limiting Parallelism
If , , are three rays such that and , then either or and are equivalent rays.
Lemma 6.4.8 If , , are three rays such that and , then
either and are equivalent rays or .
Lemma 6.4.9 If , , are three rays such that and , then
there exists a line l such that l intersects each of , , and .
6.5 Asymptotic Triangles
Asymptotic triangles are also called Omega triangles.
It’s not really a triangle because at least one vertex is not really in our space.
Notation: check the book: an open triangle followed by 4 letters for 2 vertices and 2 points on the limiting rays.
The Exterior Angle Theorem for Asymptotic Triangles
Theorem 6.5.2 If DPAB is an asymptotic triangle and C*A*B, then .
Corollary 6.5.3 Angle Sum Theorem
If PAB is an asymptotic triangle, then .
We cannot measure an “angle” with a “vertex” that is not a point in our space!
Theorem 6.5.4 Side-Angle-Side Congruence Condition
Let EPAB and FQCD be two asymptotic triangles. If and then .
6.6 The Classification of Parallels
Theorem 6.6.2 Classification of Parallels
Let l and m be parallel lines.
Part 1 If l and m are asymptotically parallel, then l and m do not admit a common perpendicular.
Part 2 Either l and m admit a common perpendicular or they are asymptotically parallel.
Theorem 6.6.3 Suppose . Let P, Q, and R be points on m such that P*Q*R and let
A, B, and C be the feet of the perpendiculars from P, Q, and R to l.
1. If, then
2. If , then
1.
2.
Theorem 6.6.
If l and m are parallel lines that admit a common perpendicular, then for every positive number
there exists a point P on m such that . Furthermore P may be chosen to lie on either side of the common perpendicular.
What does this say about distances being bounded in the Poincare Disc?
Summarizing:
For every line l and point P not on that line, there are two lines through P asymptotically parallel to l. There is no common perpendicular between l and these two lines. Every line between these two particular parallels is also parallel to l, but will allow a common perpendicular. These are the divergently parallel lines. See illustration below.
6.7 Properties of the Critical Function
Construction: Let l be a line and let P be an external point. Drop a perpendicular from P to l and call the foot of the perpendicular A. Let be a point on l. For each real number r with
there exists a point on the same side of as B such that .
Define a set K:
K is called the INTERSECTING SET for P and .
Theorem 6.7.1 is a strictly decreasing function; that is a < b implies
.
This is a strictly Hyperbolic result.
K is nonincreasing in Neutral Geometry.
Theorem 6.7.2 If l and m are asymptotically parallel lines, then there exists a point T on m such that .
Theorem 6.7.3 .
Theorem 6.7.4 .
Theorem 6.7.5 K is onto; that is, for every number there exists an such that K(x) = y.
6.8 The defect of a triangle
Theorem 6.8.1 For every there exists an isosceles triangle such that
and .
Proof Let be fixed and given. Select a large enough so that
. Let be an isosceles right triangle with A being the right angle and both legs having measure a. Now intersects thus the measure of is less than the critical number (). The measure of is also less than the critical number because the triangle is isosceles. Place a copy of this triangle adjacent to the one we constructed, sharing side . Now we have another isosceles triangle with 2 angles measuring and one angle measuring so the angle sum is less than .
Theorem 6.8.2 For every there is a right triangle such that
and .
This means we may have triangles with a small defect.
Theorem 6.8.3 For every positive number there is a positive number such that if
is a triangle in which every side has length less than d, then
.
This means that when we have a triangle with a very short side, the defect will be small. It turns out that defect is proportional to the area of the triangle.
Theorem 6.8.4 For every pair of points A and B and for every positive number there is a number d > 0 such that if C is any point not on with AC < d, then
.
Theorem 6.8.5 Continuity of Defect
The function defined in the text is a continuous function
6.9 Is the Real World Hyperbolic?
Read this in preparation for your term paper.
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