Difference Between Euclidean And Spherical Geometry

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What Is Euclidean Geometry?

Euclidean geometry sometimes called parabolic geometry is the study of plane and solid figures on the basis of axioms and theorems. Euclidean Geometry is considered as an axiomatic system, where all the theorems are derived from the small number of simple axioms. Since the term ‘’geometry’’ deals with things like points, line, angles, square, triangle and other shapes, the Euclidean geometry deals with the properties and relationship between all these things.

There are two types of Euclidean geometry: plane geometry, which is two-dimensional Euclidean geometry and solid geometry, which is three-dimensional Euclidean geometry.

Euclid’s Five Postulates

  • Assume the three steps from solids to points as solids-surface-lines-points. In each step, one dimension is lost.
  • A solid has 3 dimensions, the surface has 2, the line has 1 and point is dimensionless.
  • A point is anything that has no part, a breadth-less length is a line and the ends of a line point.
  • A surface is something which has length and breadth only.

What You Need To Know About Euclidean Geometry

  • Lines extend indefinitely and have no thickness.
  • A line is the shortest path between two points.
  • In fact, a straight line is infinite.
  • Given three collinear points, notably, one point is always between the other two.
  • To conclude perpendicular lines intersect at one point.
  • Perpendicular lines form four right angles.

What Is Spherical Geometry?

Spherical geometry is the geometry of the two-dimensional surface of a sphere. Spherical geometry works similarly to Euclidean geometry in that there still exist points, lines and angles. For instance, a ‘’line’’ between two points on a sphere is actually a great circle of the sphere, which is also the projection of a line in three-dimensional space onto the sphere. A spherical geometry provides the smallest surface-to-volume ratio of any geometrical shape.

The basic element of spherical geometry is the sphere, a three-dimensional surface made up of the set of all points in space at a given distance from a fixed point referred to as center. Spherical geometry is useful for accurate calculations of angle measure, area and distance on Earth; the study of astronomy, cosmology and navigation and applications of stereographic projection throughout complex analysis, linear algebra and arithmetic geometry.  

What You Need To Know About Spherical Geometry

  • A line is a great circle that divides the sphere into two equal half-spheres.
  • There is a unique great circle passing through any pair of nonpolar points.
  • A great circle is finite and returns to its original starting point eventually.
  • Given three collinear points, each point could be in the middle of the other two.
  • To conclude, perpendicular lines intersect at two points.
  • Perpendicular lines form eight right angles.

Difference Between Euclidean And Spherical Geometry In Tabular Form

No EUCLIDEAN GEOMETRY SPHERICAL GEOMETRY
1 Lines extend indefinitely and have no thickness.   A line is a great circle that divides the sphere into two equal half-spheres.  
2 A line is the shortest path between two points.   There is a unique great circle passing through any pair of nonpolar points.  
3 In fact, a straight line is infinite.   A great circle is finite and returns to its original starting point eventually.  
4 Given three collinear points, notably, one point is always between the other two.   Given three collinear points, each point could be in the middle of the other two.  
5 To conclude perpendicular lines intersect at one point.   To conclude, perpendicular lines intersect at two points.  
6 Perpendicular lines form four right angles.   Perpendicular lines form eight right angles.  

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