Subject: Re: [Boost-bugs] [Boost C++ Libraries] #9672: PDF and CDF of a laplace distribution throwing domain_error
From: Boost C++ Libraries (noreply_at_[hidden])
Date: 2017-12-05 19:06:03
#9672: PDF and CDF of a laplace distribution throwing domain_error
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Reporter: HS <tan@…> | Owner: Paul A. Bristow
Type: Feature Requests | Status: closed
Milestone: Boost 1.56.0 | Component: math
Version: Boost Development Trunk | Severity: Showstopper
Resolution: fixed | Keywords: laplace
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Comment (by anonymous):
here we consider as say prof dr mircea orasanu as prof horia orasanu
with School children meet the number line in the early grades. By high
school algebra and geometry, the real number line has become a central
concept. But really, what is the real number line? Is it a figment of our
imagination? How do we define it as something more concrete?
A child’s intuition of the real number line as a straight line in a plane
or in space is derived from experience with straight line segments in real
life, as the edge of a ruler, the border of a page of paper, the lines on
graph paper, the edges of tables, or the lines where the walls meet the
ceiling. But what if the line is extended into space, say to Jupiter, or
beyond? What happens as the line approaches the outer reaches of space?
Even the concept of space itself is based on a precise notion for number
line.
And what are the individual real numbers? The child’s intuitive model for
a real number corresponds to a dot made with pencil on paper. But each dot
really corresponds to a multitude of points, a mound of graphite. Does the
heap of graphite represent something other than vacuum? What really are
“pi†and “the square root of 2�
An intuitively appealing construction of the rational numbers is based
upon Euclidean geometry. It runs as follows. One starts with a straight
line, one marks a point and labels it 0, and one marks a different point
and labels it 1. Then one constructs the other integers by marking off
steps of equal length, and one constructs the rational numbers by dividing
the segments between integers into equal parts. In this model, the real
number line, stripped of its arithmetic, is taken as a primitive concept
and subjected to the axioms of Euclidean geometry (say Hilbert’s axioms,
which are studied in a course on the foundations of geometry; Euclid
himself simply proceeded with blind faith that the constructions he
performed did not stumble into any holes). And how do we know there is a
model of Euclidean geometry? The canonical model for Euclidean geometry is
the Cartesian plane consisting of ordered pairs of real numbers, and the
verification of the axioms of Euclidean geometry depends on the properties
of the real number line. If we follow this route to construct the real
numbers from a Euclidean straight line, we find we have traveled in a
logical circle.
The circular reasoning that appears in some high school algebra textbooks
is not so subtle. In one of them, the rational numbers are defined as
quotients of integers, the irrational numbers are defined as the real
numbers that are not rational, and then the real numbers are defined as
the aggregate of the rational and the irrational numbers.
The book Mathematics for High School Teachers, by Usiskin, Stanley, et
al., treats the real numbers in Chapters 2 and 6. In Chapter 2, reference
is made to various methods of constructing the real numbers from the
rational numbers, without attempting to give a precise definition of the
real numbers. Then the authors take a straight line, mark off 0 and 1,
represent the rational numbers on the line, and go on to explore in some
detail the decimal representation of real numbers. They return in Chapter
6 to the field axioms, and they establish the uniqueness of a complete
ordered field. The question of existence is never completely nailed down.
Yet they come close, when they say: “In school algebra, real numbers are
commonly described as numbers that can be represented by finite or
infinite decimals.â€
EXERCISE: Suppose a persistent high school student asks you to explain
exactly what real numbers are. What explanation would you give the
student?
The goal of these notes is to bring you to a point where you can give the
student a satisfactory answer to this question. Your answer might be
brief, but you should feel confident that you can supply as much detail as
the student might insist upon. In particular, you should understand in
what sense the real numbers “are†the set of decimals.
THE REAL NUMBER LINE
Rather than specify concretely what a real number is, we will describe the
real number line by listing its properties. This is done by defining an
axiom system. The primitive concepts in the axiom system are points (real
numbers), the operations of addition and multiplication, and an order
relation. The list of axioms is quite long, but with one exception they
are not difficult to understand. They are familiar properties of the
rational numbers. The one exception is the “completeness axiom,†which
says that there are no “holes†in the real number line. We refer to any
model for the axiom system as “the real number line†or “the field of real
numbers.†In other words, the real number line is a set with arithmetic
and ordering that satisfies the “real number axioms.â€
There are two important facts that justify our use of the expression “the
real number line.†First, there is a model for the axiom system. Second,
any two models for the axiom system are isomorphic, that is, they can be
put in a one-to-one correspondence so that the arithmetic and the ordering
correspond. In other words, the real number line exists, and it is unique.
We may perform arithmetic operations on the set with confidence, without
pausing to consider where the set comes from or where it is going. (The
K-12 student is generally happy to perform arithmetic operations on real
numbers, oblivious of the defining properties of the real numbers,
confident that there is such an entity, and not the least concerned about
whether such an entity is unique.)
So what are the real number axioms? The axioms come in three batches
corresponding to arithmetic, ordering, and completeness. The axioms taken
together assert that the real numbers form a “complete ordered field.â€
The construction of the real numbers is usually carried out in a
foundational upper division course in analysis (Math 131A at UCLA). The
arithmetic axioms, in various combinations, are studied in more detail in
upper division algebra courses (Math 110AB and Math 117 at UCLA). The
arithmetic axioms assert that the real numbers form a field. The
completeness axiom in the form of the Least Upper Bound Axiom is usually
introduced in the first calculus course. Completeness is treated in more
detail in the foundational analysis course or in a more advanced topology
course (Math 121 at UCLA), in the context of metric spaces. The ordering
and completeness axioms also appear in some form in Hilbert’s axiom system
for Euclidean geometry, which is treated in a course on the foundations of
geometry (Math 123 at UCLA).
THE ARITHMETIC AXIOMS
The axioms for arithmetic assert that there are two operations, addition
and multiplication, and these operations satisfy certain rules.
There are four axioms for addition.
1. The associative law, (x + y) + z = x + (y + z), tells us we can
perform the operation of addition in any order. Thus the expression x + y
+ z has an unambiguous meaning.
2. The commutative law, x + y = y + x, allows us to switch the
orders of the addends.
3. There exists an additive identity, denoted by 0, that satisfies
0 + x = x = x + 0 for all x.
4. Each x has an additive inverse, denoted by -x, such that x +
-x = 0 = -x + x.
We define the operation of subtraction to be addition of the additive
inverse, so that
x minus y, written x – y, is defined to be x + -y. The usual rules for
subtraction hold. They are not axioms, but are consequences of the axioms
for addition. Subtraction is completely subservient to addition, in the
sense that any statement about subtraction can be restated as a statement
about addition and additive inverses.
There are four axioms for multiplication, and they are virtually the same
as the axioms for addition.
5. The associative law, (xy)z = x(yz), tells us we can perform the
operation of multiplication in any order. Thus the expression xyz has an
unambiguous meaning.
6. The commutative law, xy = yx, allows us to switch the orders of
the factors.
7. There exists a multiplicative identity, denoted by 1, that is
different from 0 and that satisfies 1x = x = x1 for all x.
8. Each x other than 0 has a multiplicative inverse, denoted by
1/x, such that x(1/x) = 1 = (1/x)x.
We define the operation of division to be multiplication by the
multiplicative inverse, so that x divided by y, written x/y, is
defined to be x(1/y). Note that division by y is defined only for those
y’s that have a multiplicative inverse. Division by 0 is not defined.
The usual rules for division hold. They are not axioms, but are
consequences of the axioms for multiplication. Division is completely
subservient to multiplication, in the sense that any statement about
division can be restated as a statement about multiplication and
multiplicative inverses.
Finally there is an axiom that guarantees that addition and multiplication
are compatible.
9. The distributive law, x(y + z) = xy + xz, relates the
operations of addition and multiplication.
A set with two operations, addition and multiplication, that satisfies
these axioms is called a field. Examples of fields abound. The rational
numbers form a field. So do the real numbers, and so do the complex
numbers.
EXERCISE: Deduce from the field axioms that 0 times anything is 0, so
that 0 cannot have a multiplicative inverse.
EXERCISE: Deduce from the field axioms that (-1)(-1) = 1.
EXERCISE: Suppose a high school student asks you why we cannot divide by
zero. What explanation would you give to the student?
There are some “funny†fields that do not look at all like the rational
numbers. One example of a “funny†field is a field consisting of just two
elements, which must be the additive and the multiplicative identities. In
this field we define addition by 1+1=0=0+0 and 1+0=0+1=1, and we define
multiplication so that 0 times anything is 0, and 1 times 1 is 1.
EXERCISE: Let p be a prime number, and let Zp be the set of congruence
classes of integers mod p. The addition and multiplication in Zp is
defined to be the usual addition and multiplication mod p. Show that
every m in Zp other than 0 has a multiplicative inverse. Remark: Zp
is a field with p elements.
A lot of effort in school mathematics goes into defining and interpreting
subtraction and division. From a purely mathematical point of view, the
definitions are quite simple. “Subtraction of x†is defined to be
“addition of the additive inverse of x.†“Division by x†is defined to
be “multiplication by the multiplicative inverse of x.â€
THE ORDER AXIOMS
The order axioms assert that there is a relation “ < †defined between
certain elements, which satisfies the following rules.
1. The trichotomy law asserts that exactly one of the relations x<y,
y<x, or x=y holds between any two given x and y.
We write x <= y as shorthand for x < y or x = y. Also, we write y >
x to mean x < y, and we write y >= x to mean x <= y.
2. The law of transitivity asserts that if x<y and y<z, then x<z.
3. The law of compatibility with addition asserts that if x < y,
then x+z < y+z.
4. The law of compatibility with multiplication asserts that if x <
y and a > 0, then ax < ay.
A field with an ordering that satisfies these axioms is called an ordered
field.
EXERCISE: Show from the axioms that -1 < 0 and 0 < 1.
EXERCISE: Show from the axioms that x2 >= 0 for any x in an ordered
field. Deduce from this that the complex numbers cannot be ordered to
become an ordered field.
EXERCISE: Show from the axioms that in an ordered field, the elements 1,
1+1, 1+1+1, 1+1+1+1, … are distinct.
If the elements 1, 1+1, 1+1+1, 1+1+1+1, … of a field are distinct, we
say that the field has characteristic zero. If these elements are not
distinct, there is a first positive integer p such that 1+1+…+1 [p
summands] is 0. In this case, we say that the field has characteristic
p.
EXERCISE: Show that the characteristic of a field is either 0 or a
prime, that is, show that the number p above is a prime number.
There is some standard notation that is convenient. In any field, we write
1+1 = 2, 1+1+1 = 3, 1+1+1+1 = 4, and so on. As usual, -n denotes the
additive inverse of n. If the field has characteristic zero, we identify
these elements with the integers Z, and we regard Z as a subset of the
field. Under this identification, addition and multiplication in Z are
the same as addition and multiplication in the field. Further, the
subfield generated by 0 and 1 (the smallest subfield containing 0
and 1) is isomorphic to the field of rational numbers. In other words, we
can regard the rational numbers as being a subset of any field of
characteristic zero, and in particular of any ordered field.
EXERCISE: Define the absolute value function by |x| = x if x>= 0, and
|x| = -x if x < 0. Show from the axioms that |x+y| <= |x| + |y|.
Hint: Consider four cases.
There is another important property of the ordering of the real numbers
that cannot be derived directly from the other order axioms.
Archimedean Order Axiom: If a > 0 and b > 0, there is an integer m>0
such that ma > b.
If the ordered field satisfies the Archimedean order axiom, we call it an
Archimedean ordered field. By taking a=1 in the Archimedean ordering
axiom we see that each b > 0 in the field is bounded above by some
positive integer m. Let n be the first integer such that b < n+1. Then
n >= 0, and n <= b < n+1. The integer n is the leading entry in the
decimal expansion of b. We return to decimal expansions later.
EXERCISE: In an ordered field, let (a,b) denote the open interval from
a to b, that is, the set of x in the field satisfying a < x < b. Define
[a,b), [a,b], and (a,b] similarly. Show that an Archimedean ordered field
is the union over integers n of the semi-open intervals [n,n+1). Show
that these semi-open intervals are pairwise disjoint.
The following exercise will be used later, in the discussion of the
uniqueness of the real numbers.
EXERCISE: In an Archimedean ordered field, show that if x > 0, there is a
positive integer n such that x > 1/10n.
THE COMPLETENESS AXIOM
The completeness axiom for the real numbers is the tersest, yet the most
difficult to understand. To state it, we need some preliminary
definitions. Let S be a subset of the ordered field. We say that b is
an upper bound for S if x <= b for all elements x of S. We say
that b is a least upper bound for S if b is an upper bound for S,
and
b <= c for any other upper bound c for S.
EXERCISE: Show that a subset of an ordered field has at most one least
upper bound.
One version of the completeness axiom is the least upper bound axiom for a
fixed ordered field.
Least Upper Bound Axiom (LUB Axiom): If a nonempty subset of the ordered
field has an upper bound, then it has a least upper bound.
We say that a set is bounded above if it has an upper bound. The LUB axiom
can be restated simply: a nonempty set that is bounded above has a LUB.
An ordered field that satisfies the LUB axiom is called a complete ordered
field. Our goal is twofold. First, we aim to show that there exists a
complete ordered field. Second, we aim to show that any two complete
ordered fields are isomorphic. This complete ordered field, which is
essentially unique, is called the field of real numbers.
Before proceeding to the construction of the real numbers, we state a
theorem and give a formal proof to illustrate how the LUB axiom is used.
Theorem: A complete ordered field is Archimedean.
Proof: Fix a > 0. Let S be the set of multiples a, 2a, 3a, … of a. Let
c be an upper bound for S. Then (n+1)a <= c for all positive integers
n, so that na <= c – a for all positive integers n. Thus c – a is
also an upper bound for S, and further c – a < c. We conclude that S
does not have a least upper bound. By the LUB axiom, S is not bounded
above. Consequently for each b, there is some n such that b < na. Thus
the ordering is Archimedean.
EXERCISE: Write out a formal proof of the theorem starting with the
lines, “Suppose the field is not Archimedean. Then there are a > 0 and
b > 0 such that ma <= b for all positive integers m.â€
THREE MODELS FOR THE REAL NUMBER LINE
There are three methods that are often used to construct the real numbers.
Each method has its advantages and its disadvantages. Each method leads to
a model for the real numbers, that is, a set with addition,
multiplication, and ordering that satisfy the axioms for complete ordered
field. We shall refer to the three models respectively as the Weierstrass-
Stolz model (decimal expansions, the most intuitive model), the Dedekind
model (Dedekind cuts, the slickest model), and the Meray-Cantor model
(completion of a metric space, the most far-reaching model).
DECIMAL EXPANSIONS
It was Otto Stolz (1886) who pointed out that decimal expansions can be
used to define the real numbers. In the Weierstrass-Stolz model, we define
the real numbers to be the set of all decimal expansions a = a0.a1a2a3…,
where a0 is an integer (positive or negative), and a1, a2, a3, … are
integers between 0 and 9, except that we declare a decimal expansion that
terminates in all nines to be the same real number as the (terminating)
decimal expansion obtained by incrementing the last non-nine term by 1 and
replacing the subsequent 9’s by 0’s. Thus for instance we regard
3.2599999… and 3.2600000… as the same real number. (This unfortunate
complication is not an essential difficulty, but it does make the
verification of the arithmetic axioms into a tedious exercise.)
We think of the decimal expansion a0.a1a2a3… as representing the number
a0+(a1/10)+(a2/100)+(a3/1000)+… . For positive numbers this is the usual
decimal representation. For negative numbers, it’s not the usual decimal
representation, but it is the most convenient for establishing the
arithmetic axioms.
EXERCISE: What are the two possible interpretations of the decimal
-2.71828?
How would you respond to a student who asks about the ambiguity?
We define addition and multiplication of these decimals by following the
same procedures as we would for finite decimals, adding place by place and
carrying if necessary. Checking that this makes sense and that the axioms
for addition and multiplication hold is messy, but indeed the arithmetic
axioms are satisfied. Defining the order is quite easy, and it is a
straightforward task to establish the order axioms and the LUB axiom. The
Weierstrass-Stolz model is a complete ordered field. (We should be proud
of the model.)
EXERCISE: Define the order relation between two decimal expansions and
prove the trichotomy law.
Introducing arithmetic and ordering into the set of decimal expansions is
the most intuitive method for constructing the real numbers. It is the
model that appeals to school children. It is the model of most comfort to
teachers, who can explain with confidence to inquisitive students that
real numbers can be defined to be decimal expansions. The disadvantage of
the method is that checking the arithmetic axioms is a laborious task.
DEDEKIND CUTS
The subtlest method for constructing the real numbers is due to Richard
Dedekind (published in 1872). It is the model that appeared in the first
chapter of the first edition of Walter Rudin’s classic textbook,
Principles of Mathematical Analysis. Wading through Rudin’s construction
of real numbers by Dedekind cuts became trial by fire for many college
mathematics majors. The method is so slick that many mathematics majors
find it hard to digest; they regard Dedekind cuts as being rather unkind.
By the third edition, a kinder and gentler Rudin had relegated Dedekind
cuts to an appendix.
For this construction, one begins with the rational numbers. The idea is
that a real number x is the right endpoint of a unique semi-infinite open
interval (– ∞, x), and this interval is uniquely determined by the
rational numbers in the interval. With this idea to guide intuition, one
defines a Dedekind cut to be a set E of rational numbers such that (1)
if x is in E and y < x, then y is in E, (2) neither E nor its
complement is empty, and (3) E does not contain a largest number.
EXERCISE: What sets E of rational numbers satisfy (1) but not (2) or
(3)?
In this model, the real numbers are defined to be the set of Dedekind
cuts. Order is easy to define. We declare E < F if E is a subset of
F. Addition is also easy to define. The sum of the cuts E and F is the
set of all sums x + y, where x is in E and y is in F. The product
is a little more complicated to define. Once defined, it is
straightforward to verify the real number axioms. The main disadvantage of
this method is the level of sophistication required to organize and
execute these “straightforward†verifications.
In any event, the Dedekind cuts form a complete ordered field. The
additive identity in the Dedekind model is the open interval from minus
infinity to 0. The multiplicative identity is the open interval from
minus infinity to 1. More generally, each rational number r corresponds
to the cut (– ∞, r), and this correspondence allows us to identify the
rational numbers with a subfield of the Dedekind model for the real
numbers.
COMPLETION BY CAUCHY SEQUENCES
The most far-reaching method for constructing the real numbers is due
independently to Charles Meray (1869, 1872) and Georg Cantor (1872, 1883).
Again one begins with the rational numbers. One considers the set of all
sequences {xn} of rational numbers such that xn-xm tends to zero as n
and m tend to infinity. Such sequences are called Cauchy sequences. We
introduce an equivalence relation in the set of Cauchy sequences by
declaring two Cauchy sequences {xn} and {yn} to be equivalent if xn
– yn tends to zero as n tends to infinity. The real numbers are then
defined to be the set of equivalence classes of Cauchy sequences. Addition
and multiplication are easy to define. The sum of the equivalence classes
represented by two such sequences {xn} and {yn} is defined to be the
equivalence class of {xn + yn}, and similarly for the product. It is
straightforward to verify the axioms of an ordered field, and a little
more complicated to verify the completion axiom. The main disadvantage of
the method is the excess labor and the level of sophistication required
for working with equivalence classes rather than just sequences. The
advantage of the method is that it can be used in a fairly general context
to embed metric spaces in “complete†spaces. (A metric space can be
embedded as a dense subset of a complete metric space, which is
essentially unique.)
OTHER VERSIONS OF THE COMPLETENESS AXIOM
There are several other versions of the completeness axiom that are
introduced and used in the calculus course sequence and the basic analysis
course. In an ordered field, each of these is equivalent to the LUB axiom.
Every bounded increasing sequence converges.
A decreasing sequence of nonempty finite closed intervals has nonempty
intersection.
Every Cauchy sequence converges.
In the context of metric spaces, the latter version of the completeness
axiom becomes a definition. We say that a metric space is complete if
every Cauchy sequence converges.
EXERCISE: Formulate a definition of a convergent sequence in an ordered
field. Use your definition to show that in an ordered field with the LUB
axiom, every bounded increasing sequence converges.
UNIQUENESS OF THE FIELD OF REAL NUMBERS
The uniqueness (up to isomorphism) of the field of real numbers is
established in outline as follows. We start with a complete ordered field,
and we show how to assign to each x in the field a decimal expansion. The
first step is to choose an integer a0 such that
a0 <= x < a0+1.
There is then a unique integer a1, 0 <= a1 <= 9, such that
a0 + (a1/10) <= x < a0 + (a1/10) + 1/10.
We continue in this manner, selecting at the nth stage the unique integer
an such that
a0 + (a1/10) + … + (an/10n) <= x < a0 + (a1/10) + … + (an/10n) +
1/10n.
Thus each x determines the infinite decimal a0.a1a2a3…. We must show
that the correspondence between x and the decimal expansion is a one-to-
one correspondence that respects arithmetic and order, so that it is an
isomorphism of the complete ordered field and the Weierstrass-Stolz model
based on decimal expansions.
If y is different from x, say y > x, then there is n such that y –
x > 1/10n. (Recall the exercise based on the Archimedean ordering.) Then
x and y do not belong to the same interval of length 1/10n, so the
first n+1 entries in the decimal representation of y cannot be the same
as those of x, and the decimal representation of y is different from
that of x.
Next note that x = a0 + (a1/10) + … + (an/10n), which belongs to the
field, corresponds to the terminating decimal a0.a1a2a3…an000…. On the
other hand, the correspondence does not yield any decimal that terminates
in 9’s. Indeed if y corresponds to the decimal a0.a1a2a3…an999…, where
an < 9, and if x is the rational number with terminating decimal
a0.a1a2a3…(an+1), then x>y and x – y < 1/10m for all large m, so x =
y, contradicting the fact that the decimal corresponding to y terminates
with 9’s, not 0’s.
To show that the correspondence between x in the complete ordered field
and the decimals in the decimal model is one-to-one, it suffices now to
show that each decimal representation that does not terminate in 9’s
arises from some x in the field. This step depends crucially upon the
completeness axiom. Suppose a0.a1a2a3… is a decimal that does not
terminate in 9’s. The set S of elements in the field of the form
a0 + (a1/10) + … + (an/10n), for n >= 1, is bounded above by a0 + 1. By
the completeness axiom, the set S has a least upper bound, call it x.
One checks that the decimal corresponding to x is a0.a1a2a3…, as
required.
To complete the proof of the uniqueness, we must show that the
correspondence preserves the arithmetic and ordering. That the
correspondence respects the ordering follows directly from the definition.
It is straightforward but somewhat of a hassle to show that the
correspondence respects the arithmetic. That does it.
EXERCISE: Sketch an argument to show that an Archimedean ordered field is
isomorphic to a subfield of the real numbers.
EPILOG
We have defined “the real number line†to be something that satisfies the
real number axioms, that is, we have defined it to be a complete ordered
field. We have sketched the proof that there is a complete ordered field
and that it is unique (up to isomorphism). The idea of this approach is
quite simple in hindsight, yet it was quite difficult historically for
mathematicians to arrive at this point of view. This approach has the
effect of divorcing the concept of the real numbers from its geometric
origins. This may seem simple, but actually it was quite a difficult step
for mathematicians to take (and it is a step that we would not ask school
children to take). As mathematicians such as Weierstrass and Dedekind were
preparing their calculus lectures, they became ever more acutely aware,
over a period of years, that the concept of the real number line was not
on a firm footing. Though various ideas had been percolating for some
time, the critical year in the historical development of the real number
line was 1872, which saw the appearance of Dedekind’s monograph and papers
of Meray, Cantor, and Heine (a student of Weierstrass).
The degeometrization of the real numbers was not carried out without
skepticism. In his opus Mathematical Thought from Ancient to Modern Times,
mathematics historian Morris Kline quotes Hermann Hankel (a brilliant
mathematician, died in 1873 at age 34), who wrote in 1867:
Every attempt to treat the irrational numbers formally and without the
concept of [geometric] magnitude must lead to the most abstruse and
troublesome artificialities, which, even if they can be carried through
with complete rigor, as we have every right to doubt, do not have a higher
scientific value.
It is not clear that even Dedekind grasped the import of what he had done.
According to Kline again, when Heinrich Weber told Dedekind that he should
say that an irrational number is no more than the cut, Dedekind responded
(in a letter of 1888) that in fact the irrational number is not the cut
itself but something distinct, which corresponds to the cut and brings
about the cut.
We may compare the divorce of the construction of the real numbers from
geometry to the divorce of the foundations of geometry from its origins in
the Euclidean geometry of space. Those divorce proceedings lasted through
the nineteenth century and beyond with the development and discovery of
non-Euclidean geometries and various axiomatic approaches to geometry,
including finite geometries.2 Exponential, Logarithmic and Hyperbolic
Functions
Definition 3:
If , the logarithm to the base a of x is .
Definition 4:
The number e is defined by
.
Note:
.
Theorem 5:
is differentiable for and
.
[justifications of theorem 5:]
Theorem 6:
is differentiable for all x and
.
[justifications of theorem 6:]
Note:
.
Definition 5 (The hyperbolic cosine and sine functions):
The hyperbolic cosine function is defined as
,
while the hyperbolic sine function is defined as
.
Theorem 7:
.
Note:
.
Note:
since
Definition 6 (other hyperbolic functions):
The hyperbolic cosine function is defined as
Hyperbolic Functions ï€ ï€
Hyperbolic cosine of x:
(vi) Sector Area
Example For the cardioid , the total area enclosed by the curve will be
given by
Example For one of the leaves of the four leaved rose
(vii) Intersections of curves expressed in polar form.
Example Find the points of intersection of and . Find also the area
enclosed between the two graphs, outside the cardioid.
For intersections ïœ
ïœ
ïœ Points of intersection are in the form
and
The sketch below helps.
Shaded area = area of sector of circle
Hyperbolic sine of x:
Hyperbolic tangent:
Hyperbolic cotangent:
Hyperbolic secant:
Hyperbolic cosecant:
Identities
Derivatives
Integrals
Useful Identities
Derivatives of Inverse Logarithm Formulas for Evaluating
Hyperbolic Functions Inverse Hyperbolic Functions
Integrals of Inverse Hyperbolic Functions
Lesson Summary:
Students will measure distances and angles in Euclidean and Hyperbolic
space on intersecting line segments, circles and triangles to discover the
character of hyperbolic space. Students will use this knowledge to
construct a triangle and determine whether triangle in hyperbolic space
have circumcenters.
Key words:
H2, unit disc, d-line, d-segment, d-circle, boundary
Background Knowledge:
Students should be familiar with Cabri software. The hyperbolic menu
should be downloaded from the internet. The lab should be completed after
the students have studied the axioms and theorems in absolute and
Euclidean geometry. Specifically, students should be familiar with the
linear pair axiom, vertical pair theorem, properties of circles,
properties of triangles, isosceles triangle theorem and circumcenter of
triangles.
Learning Objectives
1. To become familiar with the concept of distance and angle measure
in hyperbolic space to test axioms and theorems which are true in
Euclidean space.
2. Understand the existence of perpendicular and parallel lines in
hyperbolic space.
3. Determine if isosceles triangle theorem holds in hyperbolic space.
4. Use understanding of the nature of hyperbolic space to determine
if hyperbolic triangles have circumcenters.
Materials
Cabri
Access to lab via the internet
Assessment
A lab report with answers to the questions and constructions illustrating
completion of the lab
Circmcenter of d-Triangles
Lab Goal: To determine if the hyperbolic triangles have circumcenters.
Activity:
Part I. Comparing Euclidean and Hyperbolic Space (open Hyperbol.men)
Euclidean Space (this section should confirm what you already know)
1. Create two intersecting lines AB, and CD. (Use line
tool)
Label the point of intersection P.
(intersection point tool )
2. Create segment AP, PB, CP, and PD.
(Use segment tool)
Hide the lines.
(Use hide/show tool)
3. Measure <APD and <APC. Use the calculator function to add the angles.
Record the value. What theorem or axiom have you just illustrated? Do you
predict that this will hold in hyperbolic geometry? Why?
(use the angle measurement tool)
4. Measure <CPB. Record the value. Compare to <APD. What theorem or
axiom have you just demonstrated? Do you predict that this will hold in
hyperbolic geometry? Why?
Hyperbolic Space – The hyperbolic menus appears on the last four buttons
on the toolbar. These will be referred to as:
12. Figure Menu
13. Construction Menu
14. Reflection Menu
15. Measurement Menu
5. Create your hyperbolic plane. (On the Measurement menu – Button 15,
create a unit disc by choosing a center point and a point on the x-axis
which will represent 1 unit. All constructions made here have the
properties of H2. Create a d-line (on the Figure Menu) by choosing two
points A and B and then choosing the unit circle. (Label A and B using the
Euclidean label tool.)
6. Create a d-segment AB on the d-line (Figure Menu) by choosing A and B
and then the unit circle. This segment is also called an “arcâ€. Create
d-line CD and d-segment CD such that AB and CD intersect. Label the
intersection P.
7. Measure the non-E distance of AB and CD (Using the Measurement menu,
select two points, the axis and then the unit circle). What values do you
get?
8. Measure < APD and <APC using “angle†on the Measurement menu (On the
hyperbolic menu). Add them together. Record the value.
9. What does this suggest?
10. Measure < DPB and compare to < APC. What do you notice about the
measurements?
11. Move segment AB. What do you notice about the distance and angle
measurements? What does this demonstrate?
12. Move Point B outside the circle? What happens? Why?
Hyperbolic Inquiry Lesson
Do d-Triangles have circumcenters?
INTRODUCTION
More than two thousand years ago Euclid of Alexandria collected,
compiled, and composed the thirteen volumes of geometry known as the
Elements. This magnum opus would become the quintessential model of the
way in which mathematics is structured, namely, the axiomatic method.
Euclid began by defining his terms and then laying forth his postulates
and common notions, both of which can be viewed as the assumptions he
would work from as his did his geometry. He then set to work in a
proposition-proof format wherein each result was proved using only that
which came before it. Now, it should be noted that Euclid, though his work
was masterful, was not without error. He failed to recognize as we do now
that it is logically futile to define all terms and so there must be
undefined terms; it has also been uncovered that right from his first
proof he made assumptions about things like betweenness and continuity
that were not listed in his postulates and common notions. Nevertheless,
Euclid’s Elements was a logical and mathematical tour de force that was
the standard-bearer of mathematical reasoning and certainty, the standard-
bearer, that is, until it all came crashing down.
The crash occurred when two mathematicians—János Bolyai of Hungary
and Nikolai Lobachevsky of Russia—independently discovered that Euclid’s
famous fifth postulate was independent of the others, leading to a
consistent non-Euclidean geometry. So it was that mathematics’ surest
foundation was shaken. To get a better perspective on this historic event
let us take a moment to consider Euclid’s postulates, giving particular
attention to his famous fifth.
Euclid’s postulates, as recorded in Book I of the Elements, are as
follows:
1. A unique line segment exists between any two distinct points.
2. A line segment can be uniquely extended in a straight manner.
3. A circle exists given any center and radius.
4. A right angle is equal to any other right angle.
5. If a line falling on two other lines makes the interior angles on
the same side less than two right angles, then the two lines, if produced
indefinitely, meet on that side on which are the angles less than the two
right angles.
Many mathematicians felt (and it is hard to blame them!) that the fifth
was too long and complicated to be a postulate and believed that it could
be derived from the first four, all of which were intuitively clear and
acceptable. The many attempts to prove the fifth postulate, however, were
unsuccessful. Then in the early 19th Century Bolyai and Lobachevsky
published their discoveries of hyperbolic geometry, the former’s work
based on replacing the fifth postulate with a parameter and the latter’s
based on the postulate’s negation. No longer was Euclidean geometry the
sole study of shape and space. Eventually it would be proved with the
introduction of hyperbolic models (embedded in Euclidean space) by Klein,
Poincaré and Beltrami that the consistencies of hyperbolic geometry and
Euclidean geometry were logically equivalent. Alas, the proof attempts of
Euclid V were doomed from the start!
A close inspection of the fifth postulate reveals that two
negations exist. One negation is the statement that there exist two lines
such that a transversal forms angles on one side less than two right
angles but, when produced indefinitely, the two lines do not meet on
either side; but to say that the two lines, if produced indefinitely,
also meet on the other side is another negation. The first negation leads
to hyperbolic geometry, which will be the environment of the explorations
to come. The second negation, on the other hand, leads to spherical
geometry which is itself an intriguing world in which to do geometry but,
unfortunately, does not satisfy Euclid’s first postulate (there is more
than one line segment between two distinct points) and will not be
discussed in the remainder of this paper, except for a few comparative
comments in passing.
As indicated above, what follows is a collection of explorations
in the world of hyperbolic geometry. The sections are written in an active
voice, much like Euclid’s own Elements (e.g., he would write “let AC be
drawn through B†rather than “let AC be the segment containing Bâ€). As the
reader, you should envision the paper as documentation of a student’s
investigative excursion into this non-Euclidean landscape, complete with
false starts and modifications.
Euclid’s first four postulates will be cited as axioms, as will a few of
Hilbert’s additional axioms, and there will be conjecturing and proving
that takes place. All the while, though, the geometry will appeal to
intuition and be grounded on the models. So that is where we begin.
EXPLORATION 1: FINDING A MODEL
Euclidean geometry is the study of size, shape, distances, and so
forth, in an ambient space that is in some-sense flat. The most common
manifestation of this is the doing of geometry on a piece of paper on a
desk. The ground on which we walk, run, and generally live is also
perceived to be flat. From such experiences it is natural to assume
several things because they seem to be intuitively true. First, between
any two points we can find a unique line. Second, if we have a segment of
a line then we can extend it in a straight manner. Third, we can construct
a unique circle so long as we know the center and the radius. And fourth,
a right angle is a right angle is a right angle. These assumptions, or
axioms, are based on the familiar “flat†geometry, but also hold on other
surfaces such as surfaces with constant positive curvature (e.g., a
sphere) and surfaces with constant negative curvature (e.g., a
hyperboloid). Let us see what happens if we delve into the latter case,
known as hyperbolic geometry.
Our first order of business is to make sure that we understand
what the axioms are saying in a negatively curved environment. We will
take words like “between,†“onâ€, “point,†“line†and “congruent†to be
undefined terms. This does not mean we are without guidance with regard to
their meaning because intuition plays an important role. For instance, we
can think of two figures as being congruent if we can rigidly move one
precisely onto the other, and a line can be conceptualized as the path
marking the shortest distance between its points.
What is the shortest path between two points A and B in hyperbolic
geometry? Using our model, we can stretch a string tautly along the
surface of the hyperboloid. Based on investigations of this sort we see
that a “straight line†on our model is the intersection of the hyperboloid
with a plane through the central point. The result of such an
intersection can be a hyperbola (of which we would only use half), an
ellipse or a circle. In the latter two cases we run into a problem because
two points can determine more than one line. Specifically, if A and B are
antipodal points of a circle or ellipse, such as the one shown in figure
2, then either arc of the circle or ellipse is a line segment between the
two points. This is a clear violation of Axiom 1.
Perhaps we will not be able to proceed in a way similar to the
explorations of spherical geometry. Perhaps it is not easy to find a
negatively curved surface on which to physically conduct hyperbolic
business. Hence we must return and contemplate what it is we are trying
to accomplish.
We have four axioms in hand and want to explore a geometry in
which the ambient space is not necessarily “flat.†Another way to think
about this is that the lines in the geometry are not necessarily
“straight.†These two ideas are related because a perceived curvature of
lines could really be just a symptom of the curvature of the underlying
space, but rather than try to identify that space we can just accept the
fact that lines appear to be curved. Of course line segments would still
be the shortest path between two points because it could be the case that
what looks like a straight path actually rises or dips through the ambient
curvature, making it longer than it seems. So how can we model this
geometry containing “curved†lines?
Let A and B be two distinct points. We want to define a line l
through A and B, but it has to be unique to satisfy Axiom 1. Thus it
cannot be simply any curve containing the points because there are many of
those. A third point C that was non-collinear with A and B would determine
a unique circle, and we could define l to be the minor arc between A and B
of that circle. Assuming C is fixed, for points D and E that are collinear
with C we could define the line segment between them to be the normal
straight line segment. However, as soon as we fix C there are points, say
F and G, which lie diametrically opposed to each other with respect to
their circle formed with C. In this case there is not a unique line
segment and Axiom 1 is violated. (Axiom 2 also fails—the “lines†are
compact.)
Again, let A and B be distinct points. Instead of fixing a point
we can fix a line l below A and B. If m is the perpendicular bisector of
the Euclidean segment AB, then m either intersects l at a point C or is
parallel to l (again, in the Euclidean sense). In the first case, Axiom 3
gives us a unique circle Γ through A and B with C as the center. In the
second case, we have a ray n emanating perpendicularly from l and
containing A and B. In either case, we have a way to define line segments
for all points lying in the half plane above line l.
Let us quickly check the four axioms. Per the paragraph above, we
know that a unique line segment exists for any two points above line l
because we can choose the arc of the circle that lies above l (or else we
have a case of the vertical ray which also presents a unique line
segment). If we use the open half-plane above l then any line segment has
an open neighborhood around it, and thus we can extend the line segment to
include a bit more of the hemisphere. (This suggests, however, that
distances grow exponentially as you get nearer to l.) We can define a
hyperbolic circle as the set of all points a fixed distance away from a
fixed center, which satisfies the third axiom by design. Finally, we can
define hyperbolic angle measures to be the same as the Euclidean angle
measures between the tangent lines of the intersecting arcs; ergo, the
fourth axiom in Euclidean geometry implies the fourth axiom in our model.
Thankfully, we seem to have found a workable model for the geometry that
we wish to investigate (indeed, in finding the model we have already been
investigating quite intensely). A summary seems appropriate.
• Rather than construct an explicit surface on which to do
hyperbolic geometry, we have changed our visual image of “line†and
relegated the ambient curvature to the background.
• The set of points for our hyperbolic plane model is the open upper
half-plane as determined by a line l.
• The line segment between two points is either the arc of the
circle with center on l containing the two points, or is the segment of
the ray perpendicular to l containing the two points.
• As you move closer and closer to l the underlying space curves
more and more, that is to say, the hyperbolic distances do not match the
Euclidean distances present in our model.
EXPLORATION 2: PARALLEL LINES
With a model of hyperbolic geometry at our disposal we can now
examine the nature of lines and line segments in this new world. From past
experience we know that parallel lines in Euclidean geometry are
everywhere equidistant in a certain sense, and in spherical geometry
parallel lines do not exist. One illuminating way to formulate this
distinction is by choosing a line m and a point P not on m. The question
is: how many lines parallel to m contain P? The Euclidean answer is one,
and the spherical answer is zero. Let us seek the hyperbolic answer.
To proceed, it is necessary to make explicit what we mean by
“parallel.â€
Definition. Two lines are parallel if they have no points in common.
Furthermore, it is important to note that in Euclidean geometry two
distinct circles can meet in 0 points, 1 point, or 2 points, and the
single point situation occurs if and only if the circles meet tangentially
at that point. We will use this because the hyperbolic lines of our model
can also be thought of as circles in the traditional Euclidean sense.
Now, let m be a line in the hyperbolic plane and let P be a point not on
m. Label the boundary points of m as A and B. We can construct the
Euclidean line segment PA and then bisect it perpendicularly. If this
perpendicular bisector intersects line l then we can use this intersection
point as the center of a circle and construct the hyperbolic line n that
passes through P and A (though A is not actually in the hyperbolic plane,
this is important!). The Euclidean circles m and n meet at the point A,
and there they are both orthogonal to line l which means that they meet
tangentially. This means that A is the only point at which they meet. But
A is technically off the hyperbolic plane, so necessarily m and n do not
meet in the hyperbolic plane. Thus, by definition, they are parallel
hyperbolic lines. If the perpendicular bisector of PA does not intersect
line l then we can construct the ray from A to P. This is a hyperbolic
line that meets m only at the point A, and so is also parallel to m in
hyperbolic geometry.
The paragraph above has proven the following result in hyperbolic
geometry.
Proposition 1. If m is a line and P is a point not on m, then there exists
a
line through P parallel to m.
So we see that hyperbolic geometry is inherently different than spherical
geometry. Moreover, it is inherently different than Euclidean geometry
because we can repeat the argument above using the point B in place of A,
and this will give us another line through P parallel to m!
Proposition 1 (updated). There exist at least two lines through P parallel
to m.
A bit more examination uncovers infinitely many parallels to m
through P (see figure 8). However, we are seeing that the difference
between parallels in hyperbolic geometry and Euclidean geometry is more
than just a matter of multitude, there is a qualitative difference as
well. In hyperbolic geometry we have some parallel lines (like m and n in
figure 7) that diverge in one direction but converge in the other, and we
have other parallel lines that diverge in both directions.
Definition. Parallel lines are ultraparallel if they diverge in both
directions, and are asymptotically parallel if they converge in one
direction.
The asymptotically parallel lines (of which there are two, based
on our proof of Proposition 1) seem to be the bounds of a region that
contains m, and any hyperbolic line through P contained in that region
will necessarily intersect m. Conversely, any hyperbolic line through P
outside of that region will be ultraparallel.
We have defined parallel as non-intersecting. There is another
notion, however, related to parallelism that is worth consideration—the
parallel transport.
Definition. Two lines are parallel transports of one another if there
exists a transversal that creates equal corresponding angles.
In Euclidean geometry two lines are parallel transports if and only if
they are parallel. Does such a result hold in hyperbolic geometry?
Proposition 2. If two lines are ultraparallel, then they are parallel
transports.
Let m and n be ultraparallels. Our task is to find a third line p
that creates equal angles in corresponding positions with regard to m and
n. Recall that the hyperbolic angles in our model are conformal to the
Euclidean angles.
Intuitively, if we think of a very small (in the sense of
Euclidean circles) transversal, this will create an angle with respect to
m that is nearly zero and a corresponding angle with respect to n that is
nearly two right angles (see figure 9). Now, we let the radius of the
transversal circle (i.e., the hyperbolic line) grow until it is nearly the
largest transversal possible. In this case, the angle in the same position
as before is nearly two right angles with respect to m and is nearly zero
with respect to n. They have switched the inequality! Since this process
of growth was continuous, by the intermediate value theorem, there exists
some transversal p that creates equal corresponding angles. Thus m and n
are parallel transports along p.
It is important to note that the argument for Proposition 2 fails
for asymptotically parallel lines.
web posting, November, 20062 Exponential, Logarithmic and Hyperbolic
Functions
Definition 3:
If , the logarithm to the base a of x is .
Definition 4:
The number e is defined by
.
Note:
.
Theorem 5:
is differentiable for and
.
[justifications of theorem 5:]
Theorem 6:
is differentiable for all x and
.
[justifications of theorem 6:]
Note:
.
Definition 5 (The hyperbolic cosine and sine functions):
The hyperbolic cosine function is defined as
,
while the hyperbolic sine function is defined as
.
Theorem 7:
.
Note:
.
Note:
since
Definition 6 (other hyperbolic functions):
The hyperbolic cosine function is defined as
Hyperbolic Functions ï€ ï€
Hyperbolic cosine of x:
(vi) Sector Area
Example For the cardioid , the total area enclosed by the curve will be
given by
Example For one of the leaves of the four leaved rose
(vii) Intersections of curves expressed in polar form.
Example Find the points of intersection of and . Find also the area
enclosed between the two graphs, outside the cardioid.
For intersections ïœ
ïœ
ïœ Points of intersection are in the form
and
The sketch below helps.
Shaded area = area of sector of circle
Hyperbolic sine of x:
Hyperbolic tangent:
Hyperbolic cotangent:
Hyperbolic secant:
Hyperbolic cosecant:
Identities
Derivatives
Integrals
Useful Identities
Derivatives of Inverse Logarithm Formulas for Evaluating
Hyperbolic Functions Inverse Hyperbolic Functions
Integrals of Inverse Hyperbolic Functions
Lesson Summary:
Students will measure distances and angles in Euclidean and Hyperbolic
space on intersecting line segments, circles and triangles to discover the
character of hyperbolic space. Students will use this knowledge to
construct a triangle and determine whether triangle in hyperbolic space
have circumcenters.
Key words:
H2, unit disc, d-line, d-segment, d-circle, boundary
Background Knowledge:
Students should be familiar with Cabri software. The hyperbolic menu
should be downloaded from the internet. The lab should be completed after
the students have studied the axioms and theorems in absolute and
Euclidean geometry. Specifically, students should be familiar with the
linear pair axiom, vertical pair theorem, properties of circles,
properties of triangles, isosceles triangle theorem and circumcenter of
triangles.
Learning Objectives
1. To become familiar with the concept of distance and angle measure
in hyperbolic space to test axioms and theorems which are true in
Euclidean space.
2. Understand the existence of perpendicular and parallel lines in
hyperbolic space.
3. Determine if isosceles triangle theorem holds in hyperbolic space.
4. Use understanding of the nature of hyperbolic space to determine
if hyperbolic triangles have circumcenters.
Materials
Cabri
Access to lab via the internet
Assessment
A lab report with answers to the questions and constructions illustrating
completion of the lab
Circmcenter of d-Triangles
Lab Goal: To determine if the hyperbolic triangles have circumcenters.
Activity:
Part I. Comparing Euclidean and Hyperbolic Space (open Hyperbol.men)
Euclidean Space (this section should confirm what you already know)
1. Create two intersecting lines AB, and CD. (Use line
tool)
Label the point of intersection P.
(intersection point tool )
2. Create segment AP, PB, CP, and PD.
(Use segment tool)
Hide the lines.
(Use hide/show tool)
3. Measure <APD and <APC. Use the calculator function to add the angles.
Record the value. What theorem or axiom have you just illustrated? Do you
predict that this will hold in hyperbolic geometry? Why?
(use the angle measurement tool)
4. Measure <CPB. Record the value. Compare to <APD. What theorem or
axiom have you just demonstrated? Do you predict that this will hold in
hyperbolic geometry? Why?
Hyperbolic Space – The hyperbolic menus appears on the last four buttons
on the toolbar. These will be referred to as:
12. Figure Menu
13. Construction Menu
14. Reflection Menu
15. Measurement Menu
5. Create your hyperbolic plane. (On the Measurement menu – Button 15,
create a unit disc by choosing a center point and a point on the x-axis
which will represent 1 unit. All constructions made here have the
properties of H2. Create a d-line (on the Figure Menu) by choosing two
points A and B and then choosing the unit circle. (Label A and B using the
Euclidean label tool.)
6. Create a d-segment AB on the d-line (Figure Menu) by choosing A and B
and then the unit circle. This segment is also called an “arcâ€. Create
d-line CD and d-segment CD such that AB and CD intersect. Label the
intersection P.
7. Measure the non-E distance of AB and CD (Using the Measurement menu,
select two points, the axis and then the unit circle). What values do you
get?
8. Measure < APD and <APC using “angle†on the Measurement menu (On the
hyperbolic menu). Add them together. Record the value.
9. What does this suggest?
10. Measure < DPB and compare to < APC. What do you notice about the
measurements?
11. Move segment AB. What do you notice about the distance and angle
measurements? What does this demonstrate?
12. Move Point B outside the circle? What happens? Why?
Hyperbolic Inquiry Lesson
Do d-Triangles have circumcenters?
INTRODUCTION
More than two thousand years ago Euclid of Alexandria collected,
compiled, and composed the thirteen volumes of geometry known as the
Elements. This magnum opus would become the quintessential model of the
way in which mathematics is structured, namely, the axiomatic method.
Euclid began by defining his terms and then laying forth his postulates
and common notions, both of which can be viewed as the assumptions he
would work from as his did his geometry. He then set to work in a
proposition-proof format wherein each result was proved using only that
which came before it. Now, it should be noted that Euclid, though his work
was masterful, was not without error. He failed to recognize as we do now
that it is logically futile to define all terms and so there must be
undefined terms; it has also been uncovered that right from his first
proof he made assumptions about things like betweenness and continuity
that were not listed in his postulates and common notions. Nevertheless,
Euclid’s Elements was a logical and mathematical tour de force that was
the standard-bearer of mathematical reasoning and certainty, the standard-
bearer, that is, until it all came crashing down.
The crash occurred when two mathematicians—János Bolyai of Hungary
and Nikolai Lobachevsky of Russia—independently discovered that Euclid’s
famous fifth postulate was independent of the others, leading to a
consistent non-Euclidean geometry. So it was that mathematics’ surest
foundation was shaken. To get a better perspective on this historic event
let us take a moment to consider Euclid’s postulates, giving particular
attention to his famous fifth.
Euclid’s postulates, as recorded in Book I of the Elements, are as
follows:
1. A unique line segment exists between any two distinct points.
2. A line segment can be uniquely extended in a straight manner.
3. A circle exists given any center and radius.
4. A right angle is equal to any other right angle.
5. If a line falling on two other lines makes the interior angles on
the same side less than two right angles, then the two lines, if produced
indefinitely, meet on that side on which are the angles less than the two
right angles.
Many mathematicians felt (and it is hard to blame them!) that the fifth
was too long and complicated to be a postulate and believed that it could
be derived from the first four, all of which were intuitively clear and
acceptable. The many attempts to prove the fifth postulate, however, were
unsuccessful. Then in the early 19th Century Bolyai and Lobachevsky
published their discoveries of hyperbolic geometry, the former’s work
based on replacing the fifth postulate with a parameter and the latter’s
based on the postulate’s negation. No longer was Euclidean geometry the
sole study of shape and space. Eventually it would be proved with the
introduction of hyperbolic models (embedded in Euclidean space) by Klein,
Poincaré and Beltrami that the consistencies of hyperbolic geometry and
Euclidean geometry were logically equivalent. Alas, the proof attempts of
Euclid V were doomed from the start!
A close inspection of the fifth postulate reveals that two
negations exist. One negation is the statement that there exist two lines
such that a transversal forms angles on one side less than two right
angles but, when produced indefinitely, the two lines do not meet on
either side; but to say that the two lines, if produced indefinitely,
also meet on the other side is another negation. The first negation leads
to hyperbolic geometry, which will be the environment of the explorations
to come. The second negation, on the other hand, leads to spherical
geometry which is itself an intriguing world in which to do geometry but,
unfortunately, does not satisfy Euclid’s first postulate (there is more
than one line segment between two distinct points) and will not be
discussed in the remainder of this paper, except for a few comparative
comments in passing.
As indicated above, what follows is a collection of explorations
in the world of hyperbolic geometry. The sections are written in an active
voice, much like Euclid’s own Elements (e.g., he would write “let AC be
drawn through B†rather than “let AC be the segment containing Bâ€). As the
reader, you should envision the paper as documentation of a student’s
investigative excursion into this non-Euclidean landscape, complete with
false starts and modifications.
Euclid’s first four postulates will be cited as axioms, as will a few of
Hilbert’s additional axioms, and there will be conjecturing and proving
that takes place. All the while, though, the geometry will appeal to
intuition and be grounded on the models. So that is where we begin.
EXPLORATION 1: FINDING A MODEL
Euclidean geometry is the study of size, shape, distances, and so
forth, in an ambient space that is in some-sense flat. The most common
manifestation of this is the doing of geometry on a piece of paper on a
desk. The ground on which we walk, run, and generally live is also
perceived to be flat. From such experiences it is natural to assume
several things because they seem to be intuitively true. First, between
any two points we can find a unique line. Second, if we have a segment of
a line then we can extend it in a straight manner. Third, we can construct
a unique circle so long as we know the center and the radius. And fourth,
a right angle is a right angle is a right angle. These assumptions, or
axioms, are based on the familiar “flat†geometry, but also hold on other
surfaces such as surfaces with constant positive curvature (e.g., a
sphere) and surfaces with constant negative curvature (e.g., a
hyperboloid). Let us see what happens if we delve into the latter case,
known as hyperbolic geometry.
Our first order of business is to make sure that we understand
what the axioms are saying in a negatively curved environment. We will
take words like “between,†“onâ€, “point,†“line†and “congruent†to be
undefined terms. This does not mean we are without guidance with regard to
their meaning because intuition plays an important role. For instance, we
can think of two figures as being congruent if we can rigidly move one
precisely onto the other, and a line can be conceptualized as the path
marking the shortest distance between its points.
What is the shortest path between two points A and B in hyperbolic
geometry? Using our model, we can stretch a string tautly along the
surface of the hyperboloid. Based on investigations of this sort we see
that a “straight line†on our model is the intersection of the hyperboloid
with a plane through the central point. The result of such an
intersection can be a hyperbola (of which we would only use half), an
ellipse or a circle. In the latter two cases we run into a problem because
two points can determine more than one line. Specifically, if A and B are
antipodal points of a circle or ellipse, such as the one shown in figure
2, then either arc of the circle or ellipse is a line segment between the
two points. This is a clear violation of Axiom 1.
Perhaps we will not be able to proceed in a way similar to the
explorations of spherical geometry. Perhaps it is not easy to find a
negatively curved surface on which to physically conduct hyperbolic
business. Hence we must return and contemplate what it is we are trying
to accomplish.
We have four axioms in hand and want to explore a geometry in
which the ambient space is not necessarily “flat.†Another way to think
about this is that the lines in the geometry are not necessarily
“straight.†These two ideas are related because a perceived curvature of
lines could really be just a symptom of the curvature of the underlying
space, but rather than try to identify that space we can just accept the
fact that lines appear to be curved. Of course line segments would still
be the shortest path between two points because it could be the case that
what looks like a straight path actually rises or dips through the ambient
curvature, making it longer than it seems. So how can we model this
geometry containing “curved†lines?
Let A and B be two distinct points. We want to define a line l
through A and B, but it has to be unique to satisfy Axiom 1. Thus it
cannot be simply any curve containing the points because there are many of
those. A third point C that was non-collinear with A and B would determine
a unique circle, and we could define l to be the minor arc between A and B
of that circle. Assuming C is fixed, for points D and E that are collinear
with C we could define the line segment between them to be the normal
straight line segment. However, as soon as we fix C there are points, say
F and G, which lie diametrically opposed to each other with respect to
their circle formed with C. In this case there is not a unique line
segment and Axiom 1 is violated. (Axiom 2 also fails—the “lines†are
compact.)
Again, let A and B be distinct points. Instead of fixing a point
we can fix a line l below A and B. If m is the perpendicular bisector of
the Euclidean segment AB, then m either intersects l at a point C or is
parallel to l (again, in the Euclidean sense). In the first case, Axiom 3
gives us a unique circle Γ through A and B with C as the center. In the
second case, we have a ray n emanating perpendicularly from l and
containing A and B. In either case, we have a way to define line segments
for all points lying in the half plane above line l.
Let us quickly check the four axioms. Per the paragraph above, we
know that a unique line segment exists for any two points above line l
because we can choose the arc of the circle that lies above l (or else we
have a case of the vertical ray which also presents a unique line
segment). If we use the open half-plane above l then any line segment has
an open neighborhood around it, and thus we can extend the line segment to
include a bit more of the hemisphere. (This suggests, however, that
distances grow exponentially as you get nearer to l.) We can define a
hyperbolic circle as the set of all points a fixed distance away from a
fixed center, which satisfies the third axiom by design. Finally, we can
define hyperbolic angle measures to be the same as the Euclidean angle
measures between the tangent lines of the intersecting arcs; ergo, the
fourth axiom in Euclidean geometry implies the fourth axiom in our model.
Thankfully, we seem to have found a workable model for the geometry that
we wish to investigate (indeed, in finding the model we have already been
investigating quite intensely). A summary seems appropriate.
• Rather than construct an explicit surface on which to do
hyperbolic geometry, we have changed our visual image of “line†and
relegated the ambient curvature to the background.
• The set of points for our hyperbolic plane model is the open upper
half-plane as determined by a line l.
• The line segment between two points is either the arc of the
circle with center on l containing the two points, or is the segment of
the ray perpendicular to l containing the two points.
• As you move closer and closer to l the underlying space curves
more and more, that is to say, the hyperbolic distances do not match the
Euclidean distances present in our model.
EXPLORATION 2: PARALLEL LINES
With a model of hyperbolic geometry at our disposal we can now
examine the nature of lines and line segments in this new world. From past
experience we know that parallel lines in Euclidean geometry are
everywhere equidistant in a certain sense, and in spherical geometry
parallel lines do not exist. One illuminating way to formulate this
distinction is by choosing a line m and a point P not on m. The question
is: how many lines parallel to m contain P? The Euclidean answer is one,
and the spherical answer is zero. Let us seek the hyperbolic answer.
To proceed, it is necessary to make explicit what we mean by
“parallel.â€
Definition. Two lines are parallel if they have no points in common.
Furthermore, it is important to note that in Euclidean geometry two
distinct circles can meet in 0 points, 1 point, or 2 points, and the
single point situation occurs if and only if the circles meet tangentially
at that point. We will use this because the hyperbolic lines of our model
can also be thought of as circles in the traditional Euclidean sense.
Now, let m be a line in the hyperbolic plane and let P be a point not on
m. Label the boundary points of m as A and B. We can construct the
Euclidean line segment PA and then bisect it perpendicularly. If this
perpendicular bisector intersects line l then we can use this intersection
point as the center of a circle and construct the hyperbolic line n that
passes through P and A (though A is not actually in the hyperbolic plane,
this is important!). The Euclidean circles m and n meet at the point A,
and there they are both orthogonal to line l which means that they meet
tangentially. This means that A is the only point at which they meet. But
A is technically off the hyperbolic plane, so necessarily m and n do not
meet in the hyperbolic plane. Thus, by definition, they are parallel
hyperbolic lines. If the perpendicular bisector of PA does not intersect
line l then we can construct the ray from A to P. This is a hyperbolic
line that meets m only at the point A, and so is also parallel to m in
hyperbolic geometry.
The paragraph above has proven the following result in hyperbolic
geometry.
Proposition 1. If m is a line and P is a point not on m, then there exists
a
line through P parallel to m.
So we see that hyperbolic geometry is inherently different than spherical
geometry. Moreover, it is inherently different than Euclidean geometry
because we can repeat the argument above using the point B in place of A,
and this will give us another line through P parallel to m!
Proposition 1 (updated). There exist at least two lines through P parallel
to m.
A bit more examination uncovers infinitely many parallels to m
through P (see figure 8). However, we are seeing that the difference
between parallels in hyperbolic geometry and Euclidean geometry is more
than just a matter of multitude, there is a qualitative difference as
well. In hyperbolic geometry we have some parallel lines (like m and n in
figure 7) that diverge in one direction but converge in the other, and we
have other parallel lines that diverge in both directions.
Definition. Parallel lines are ultraparallel if they diverge in both
directions, and are asymptotically parallel if they converge in one
direction.
The asymptotically parallel lines (of which there are two, based
on our proof of Proposition 1) seem to be the bounds of a region that
contains m, and any hyperbolic line through P contained in that region
will necessarily intersect m. Conversely, any hyperbolic line through P
outside of that region will be ultraparallel.
We have defined parallel as non-intersecting. There is another
notion, however, related to parallelism that is worth consideration—the
parallel transport.
Definition. Two lines are parallel transports of one another if there
exists a transversal that creates equal corresponding angles.
In Euclidean geometry two lines are parallel transports if and only if
they are parallel. Does such a result hold in hyperbolic geometry?
Proposition 2. If two lines are ultraparallel, then they are parallel
transports.
Let m and n be ultraparallels. Our task is to find a third line p
that creates equal angles in corresponding positions with regard to m and
n. Recall that the hyperbolic angles in our model are conformal to the
Euclidean angles.
Intuitively, if we think of a very small (in the sense of
Euclidean circles) transversal, this will create an angle with respect to
m that is nearly zero and a corresponding angle with respect to n that is
nearly two right angles (see figure 9). Now, we let the radius of the
transversal circle (i.e., the hyperbolic line) grow until it is nearly the
largest transversal possible. In this case, the angle in the same position
as before is nearly two right angles with respect to m and is nearly zero
with respect to n. They have switched the inequality! Since this process
of growth was continuous, by the intermediate value theorem, there exists
some transversal p that creates equal corresponding angles. Thus m and n
are parallel transports along p.
It is important to note that the argument for Proposition 2 fails
for asymptotically parallel lines.Kline, M. (1982). Mathematics: The Loss
of Certainty. Oxford: Oxford University Press.
3.4 Exterior Angle Inequality
3.5 The Inequality Theorems
3.6 Additional Congruence Criteria
3.7 Quadrilaterals
3.8 Circles
Homework 5
-- Ticket URL: <https://svn.boost.org/trac10/boost/ticket/9672#comment:11> Boost C++ Libraries <http://www.boost.org/> Boost provides free peer-reviewed portable C++ source libraries.
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