College Geometry - neutral geometry and Euclidean geometry Essay

College Geometry - neutral geometry and Euclidean geometry - Essay Example It is also termed as neutral geometry because it is neutral in reference to parallel postulate. Other geometries related to hyperbolic geometry, neutral geometry, ordered geometry, among others (Ball, 2008). Differences between neutral geometry and Euclidean geometry Euclidean geometry tends to be axiomatic system; in this case, all is theorems in other terms â€Å"true statements† of derivatives of smaller axioms. In reference to the book by the name Elements, Euclid illustrates five axioms (postulates) as far as plane geometry is concerned. The followi9ng is some of the postulations: 1) The first line should be drawn from any point 2) A finite line should be produced and it should be straight and continuous 3) Right angles are equal 4) A circle is described in reference to the center and its radius 5) In parallel lines, it a straight line meets other two straight lines, and makes an interior angle of which both are less than 900, the two lines can meet if extended on those s ides that they make angles less than right angles (the difference between lines in Euclidean and those of spherical are illustrated in spherical geometry). On the other hand, absolute geometry is more the same as ordered geometry. ... If the intersection points forms interior angles of less than 90, they form the basis for both hyperbolic and spherical geometry. They are both under non-Euclidean geometry where they are attained through the parallel postulates in Euclid. In addition, the postulate can be still be defined that â€Å"in every line 1 and each external point q, there exist unique lines through q that are parallel to 1.† this gives a basis for spherical constructions (Ball 2008). Spherical geometry This is a plane geometry that is on a sphere’s surface; its basic elements are lines and points but are defined in a different way. They are defined in such a way that the shortest distance between any two points runs along the same two points. The sum the angles in this triangle is more than 1800 but small triangles of the same kind are slightly larger than 1800; an ideal example being those in football fields. This is because the base length is not a perfect straight line. See a practical exam ple below. The diagram illustrates the structure of a spherical triangle It lines are also continuous and ends up forming a circle. Lines that run around the geometry making the longest distance are known as â€Å"Great Circles.† Below is an illustration of spherical geometry. The diagram above illustrates how lines are different from those in Euclidean and thus forming Great Circles. The sum of the angles in any triangle is 180 degrees In reference to Euclidean geometry, the parallel postulate when extended makes a three sided diagram that is interior angles sum up to 1800. This is because the subsequent angles are reflections of the points of origin (Eves, 1990). (a) (c) (b) From the diagram above, we can prove that the exterior angle at