Non-Euclidean Geometry Unit
Euclidean Geometry: The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.
Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land.
Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.
Euclidean Geometry: The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.
Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land.
Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.
Euclid’s First Five Postulates:
1. A unique straight line can be drawn through any two points \(A\) and \(B\)
2. A segment can be extended indefinitely
3. For any two distinct points \(A\) and \(B\), a circle can be drawn with center \(A\) and radius \(\overline{AB}\)
4. All right angles are congruent
5. Given a line \(\ell\) and a point \(P\) not on \(\ell\), there exists a unique line though \(P\) parallel to \(\ell\).
1. A unique straight line can be drawn through any two points \(A\) and \(B\)
2. A segment can be extended indefinitely
3. For any two distinct points \(A\) and \(B\), a circle can be drawn with center \(A\) and radius \(\overline{AB}\)
4. All right angles are congruent
5. Given a line \(\ell\) and a point \(P\) not on \(\ell\), there exists a unique line though \(P\) parallel to \(\ell\).
Euclidean (or Plane) Geometry is Geometry in which the Parallel Postulate is true.
Spherical Geometry: Spherical geometry is a plane geometry on the surface of a sphere. In a plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. Therefore, lines in spherical geometry are Great Circles. A Great Circle is the largest circle that can be drawn on a sphere. The longitude lines and the equator are Great Circles of the Earth. Latitude lines, except for the equator, are not Great Circles. Great Circles are lines that divide a sphere into two equal hemispheres.
Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flying north to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a Great Circle). Another odd property of spherical geometry is that the sum of the angles of a triangle is always greater then \(180^{\circ}\). Small triangles, like those drawn on a football field, have very, very close to \(180^{\circ}\). Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than \(180^{\circ}\).
Hyperbolic Geometry: Hyperbolic geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within the field of Topology.
Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also has many differences from Euclidean geometry.
Spherical Geometry: Spherical geometry is a plane geometry on the surface of a sphere. In a plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. Therefore, lines in spherical geometry are Great Circles. A Great Circle is the largest circle that can be drawn on a sphere. The longitude lines and the equator are Great Circles of the Earth. Latitude lines, except for the equator, are not Great Circles. Great Circles are lines that divide a sphere into two equal hemispheres.
Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flying north to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a Great Circle). Another odd property of spherical geometry is that the sum of the angles of a triangle is always greater then \(180^{\circ}\). Small triangles, like those drawn on a football field, have very, very close to \(180^{\circ}\). Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than \(180^{\circ}\).
Hyperbolic Geometry: Hyperbolic geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within the field of Topology.
Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also has many differences from Euclidean geometry.
Spherical Geometry is geometry in which Euclid’s fifth postulate is replaced with the following:
"Given a line \(g\) and a point \(P\) not on \(g\), every line through \(P\) intersects \(g\); that is, no line through \(P\) is parallel to \(g\)."
"Given a line \(g\) and a point \(P\) not on \(g\), every line through \(P\) intersects \(g\); that is, no line through \(P\) is parallel to \(g\)."
In forming the foundation on which to build plane geometry, certain terms are accepted as being undefined, their meanings being intuitively understood. The units that are presented will accept the following undefined terms.
- Line segment: The \(overline{AB}\), consists of the points \(A\) and \(B\) and all the points on \ (\overleftrightarrow{AB}\) that are between \(A\) and \( B\).
- Circle: The set of all points, \(P\), in a plane that are a fixed distance from a fixed point, \(O\), on that plane, called the center of the circle.
- Parallel lines: Two lines, \(\ell\) and \(m\) on the plane are parallel if they do not intersect.
- Sphere: The set of the points in space that are a given distance from a fixed point, called the center of the sphere.
- Great Circle: A Great Circle is a circle whose center is the center of the sphere and whose radius is equal to the radius of the sphere. The Great Circle in spherical geometry is a line.
- Arc of a Great Circle: The shortest path between two points on the sphere is the arc of a Great Circle.
- Antipodal points (Pole Points): Points that lie at the intersection of a Great Circle and a line through the center of the circle on the sphere.
- Small Triangle: The small triangle is formed by joining three non-collinear vertices with the shorter arc between the vertices. Three vertices then determine only one spherical triangle.
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