Explain Introduction to Non-Euclidean Geometry?
Up to this point, the type of geometry we have been studying is known as Euclidean geometry. It is based on the studies of the ancient Greek mathematician Euclid. Euclidean geometry was a way to explain or describe the basic layout of the universe. Hundreds of years after him, a few mathematicians developed geometries that are not based on Euclid's axioms. In this chapter, we will explore some concepts of non-Euclidean geometry.
A line, according to Euclid, is perfectly straight and extends infinitely in both directions. Keep in mind that Euclid lived in a world that believed the Earth was flat. But now we know that Earth is a sphere, a line of the Euclidean postulate, perfectly straight and infinitely long, could not exist on the surface of the Earth. A "line" on a spherical surface must follow a curved path. The geometry based on a sphere is called sphere geometry.
Definition
A great circle of a sphere is the circle determined by the intersection of the spherical surface and a secant plane that contains the center of the sphere.
Definition
Lines are great circles in sphere geometry.The equator and longitudinal lines on a globe are great circles. Latitudes on a globe are not great circles.
You already know that on a plane, the shortest distance between any two points is a line segment joining these two points. The shortest distance between any two points on a sphere is measured along a curved path that is a segment of a great circle. The length of a line segment depends on the size of the sphere. Polar points are the points created by a line passing through the center of a sphere intersecting with the sphere. The North and South Poles on Earth are polar points.
Postulate
For any given pair of points on a sphere, there is exactly one line containing them. Conversely, it is also true that a line contains at least two points. But consider now the parallel postulate on a flat plane, "Through a given point not on a given line there is exactly one line parallel to the given line." On a sphere, every line intersects with all other lines.
Postulate
On a sphere, through a given point not on a given line there is no line parallel to the given line.
Definition
A biperpendicular quadrilateral is a quadrilateral with two sides perpendicular to a third one.
The legs are the two sides perpendicular to the same side.
The base is the side to which the two legs are perpendicular.
The base angle is an angle between base and leg.
The summit is the side opposite the base.
The summit angle is an angle between summit and leg.
Definition
An isosceles birectangular quadrilateral, or a Saccheri quadrilateral is a biperpendicular quadrilateral with congruent legs.
An eighteenth century priest named Saccheri, for whom the Saccheri quadrilateral is named, studied the figure. He tried to use it to prove that the Euclidean parallel postulate was true. Instead he came across something remarkable in the field of non-Euclidean geometry. Using the new postulate on parallel lines, we can prove that a Saccheri quadrilateral is not a rectangle and its two summit angles are not right angles.
Theorem
If the two summit angles of a biperpendicular quadrilateral are unequal, then the larger angle is adjacent to the shorter leg.
Theorem
The summit angles of a Saccheri quadrilateral are congruent.
Theorem
In a Saccheri quadrilateral, the bisector of the base and the summit is perpendicular to both of them.