Essay About Unique Line L And Hyperbolic Geometry

What Is Hyperbolic Geometry?
Essay Preview: What Is Hyperbolic Geometry?
Report this essay
What is Hyperbolic Geometry? Why should high school
students study Hyperbolic Geometry? What role does technology
play in the study of Hyperbolic Geometry? This paper will
address these questions. In chapter 1, Hyperbolic Geometry will
be described as will support for mathematics education regarding
the subject. Chapter 2 will focus on the similarities and
differences between Euclidean Geometry and Hyperbolic Geometry.
A study conducted on teaching Hyperbolic Geometry to high school
geometry students will be discussed in Chapter 3.
“Hyperbolic geometry is, by definition, the geometry you get
by assuming all the axioms for neutral geometry and replacing
Hilberts parallel postulate by its negation, which we shall call
the hyperbolic axiom”(Greenberg, 1993, p. 187). A look at the
history of Hyperbolic Geometry will help provide understanding of
the definition. Euclidean Geometry gives the foundation for
Hyperbolic Geometry. Euclidean Geometry began around 300 BC in
Euclids book Elements. Euclidean Geometry was based on five
axioms, which were:
1. For every point P and for every point Q not equal to P
there exists a unique line l that passes through P and
2. For every segment AB and for every segment CD there
exists a unique point E such that B is between A and E
and segment CD is congruent to segment BE.
3. For every point O and every point A not equal to O,
there exists a circle with center O and radius OA.3
4. All right angles are congruent to each other.
5. For every line l and for every point P that does not lie
on l there exists a unique line m through P that is
parallel to l (Greenberg, 1993).