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1. What is Euclidean Geometry?
Euclidean Geometry is plane geometry, or geometry on a plane (a flat surface). It is the geometry that we study in high school today.
2. What is Non-Euclidian Geometry?
Non-Euclidian Geometry is any geometry that is different than Euclidian Geometry. The basic difference is the surface of which we apply the geometry. There are many types, but the type of Non-Euclidian Geometry presented in the videos is spherical geometry, or geometry on a sphere (ball). Another example of a Non-Euclidian geometry is hyperbolic geometry (briefly mentioned in the video.
3. What is a “line” in spherical geometry? How does it compare to a line in plane geometry?
A line is a great circle. A great circle divides the sphere into two hemispheres. A great circle is finite (has a measureable distance), unlike a line in plane geometry. If the earth were a sphere, then a line of longitude and the equator are examples of great circles.
4. How are line segments alike and different in plane and spherical geometry?
Line segments are parts of a line, defined by their endpoints and all points in between. It is the shortest distance between two points. In plane geometry, a line segment is straight, in spherical geometry it is an arc of a great circle. An example mentioned in the video is that on earth, the shortest distance between Tampa Florida and the Philippines travels over Alaska.
5. What is “betweenness” and how does it apply in both plane and spherical geometry?
Given a line on a plane and three collinear points on the line, one of the points must be between the other two. Not so of three points on a line in spherical geometry (a great circle). On a great circle any of the points can be between the other two, so there is no betweenness in spherical geometry.
6. What is the nature of parallel lines in plane and spherical geometry?
Euclid’s fifth postulate, the most famous postulate of all, states: Given a line and a point not on the line, there exists exactly one line that passes through the given point parallel to the given line. In spherical geometry however, no such lines exist. There are not parallel lines (great circles) in spherical geometry. They all intersect. (Also note, they all intersect at antipodal points – pole points).
7. What are the differences in the nature of circles drawn on a plane to those drawn on a sphere?
On a plane, a circle has one center and one length radius. On a sphere, a circle has two centers. These two centers are antipodal (pole) points. Also, a spherical circle has two different length radii. (Also, the sum of the radii of a spherical circle is always \(180^{\circ}\)).
Euclidean Geometry is plane geometry, or geometry on a plane (a flat surface). It is the geometry that we study in high school today.
2. What is Non-Euclidian Geometry?
Non-Euclidian Geometry is any geometry that is different than Euclidian Geometry. The basic difference is the surface of which we apply the geometry. There are many types, but the type of Non-Euclidian Geometry presented in the videos is spherical geometry, or geometry on a sphere (ball). Another example of a Non-Euclidian geometry is hyperbolic geometry (briefly mentioned in the video.
3. What is a “line” in spherical geometry? How does it compare to a line in plane geometry?
A line is a great circle. A great circle divides the sphere into two hemispheres. A great circle is finite (has a measureable distance), unlike a line in plane geometry. If the earth were a sphere, then a line of longitude and the equator are examples of great circles.
4. How are line segments alike and different in plane and spherical geometry?
Line segments are parts of a line, defined by their endpoints and all points in between. It is the shortest distance between two points. In plane geometry, a line segment is straight, in spherical geometry it is an arc of a great circle. An example mentioned in the video is that on earth, the shortest distance between Tampa Florida and the Philippines travels over Alaska.
5. What is “betweenness” and how does it apply in both plane and spherical geometry?
Given a line on a plane and three collinear points on the line, one of the points must be between the other two. Not so of three points on a line in spherical geometry (a great circle). On a great circle any of the points can be between the other two, so there is no betweenness in spherical geometry.
6. What is the nature of parallel lines in plane and spherical geometry?
Euclid’s fifth postulate, the most famous postulate of all, states: Given a line and a point not on the line, there exists exactly one line that passes through the given point parallel to the given line. In spherical geometry however, no such lines exist. There are not parallel lines (great circles) in spherical geometry. They all intersect. (Also note, they all intersect at antipodal points – pole points).
7. What are the differences in the nature of circles drawn on a plane to those drawn on a sphere?
On a plane, a circle has one center and one length radius. On a sphere, a circle has two centers. These two centers are antipodal (pole) points. Also, a spherical circle has two different length radii. (Also, the sum of the radii of a spherical circle is always \(180^{\circ}\)).