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spherical geometry project

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on 10 April 2014

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Transcript of spherical geometry project

spherical geometry project
by: Justin Barlow

what are some differences between spherical and euclidean geometry?
similarity #2
in both forms of geometry there can be right triangles. a right triangle is a triangle with a right angle.
difference #3
what is euclidean geometry?
euclidean geometry is based on figures in a plane. the system of Euclidean geometry satisfies the parallel postulate. In Euclidean geometrty a line is the shortest path between two points.
what is spherical geometry?
spherical geometry is the study of figures on the surface of a sphere. Spherical geometry
is a form of non-Euclidean geometry. The shortest path between two points on a sphere is along a great circle.
difference #1
parallel lines are in Euclidean geometry but not in spherical geometry.
now for some similarities!!!
"I love math don't you?" - Professor Edward Burger (I don't know if he actually said this but i'm sure he would)
difference # 2
In euclidean geometry the sum of angle measure in any triangle are equal to 180 degrees. In spherical geometry the sum of angle measures in any triangle is always greater than 180 degrees.
similarity #1
source: http://math.stackexchange.com/questions/333683/spherical-geometry-sg-vs-euclidean-geometryeg
source: http://regentsprep.org/regents/math/geometry/GG1/Euclidean.htm
this is the most simple difference out of them all. spherical geometry is on the surface of a sphere for ex. piolots use this to navigate across earth. Euclidean geometry is used in on a flat plane. (a math plane not airplane)
source: https://answers.yahoo.com/question/index?qid=20081101174004AApqpFh
if there are 2 lines in euclidean and spherical geometry and they are not parallel the will eventually intersect
similarity #3
in euclidean geometry and spherical geometry there is a shortest path between two points.
similarity # 4
there are perpendicular lines in spherical and euclidean geometry
discussion question
How are right triangle in spherical geometry different from right triangles in euclidean geometry?
In spherical geometry the sum of a triangle's angles is over 180 degrees. This means that because a triangle has 3 angles, a spherical right triangle can have up to 3 right angles. In euclidean geometry the sum of a triangles angle measures has to equal 180 degrees. While having only 3 angles with out going over 180 degrees a right triangle in euclidean geometry can only have one right angle.
example
what is the parallel postulate?
through a point not on a line there is no line parallel to the given line.
what is hyperbolic geometry?
hyperbolic geometry is a type of non euclidean geometry also called saddle geometry.
differences
the triangle sum theorem: angle measures in a hyperbolic triangle add up to equal anything less than 180 degrees when euclidean triangles angle measures ad up to = 180 degrees.
similarities
they both contain perpendicular lines
geometry video

source: http://regentsprep.org/regents/math/geometry/GG1/Euclidean.htm
source: textbook
source: textbook
source: textbook
source: textbook
Full transcript