Chapter 5 - Geometry
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|Geometry - Geometry Basics - Contents|
Geometry is the mathematical examination of shapes. The study of geometry came from ancient Greece and many of its original symbols are still used today. Geometric shapes from one-dimensional to three-dimensional can be represented with equations using one to three variables. There are three real geometric spaces used to examine shapes in geometry. The number line is an infinite line that represents the set of real numbers. A geometric plane, such as the Cartesian Plane, is two perpendicular (90 degrees) number lines that both represent the set of real numbers. In this geometric space an equation of functions with two variables x and y can be used to represent a point, a line or a polygon, such as a triangle, square or circle. A three-dimensional geometric space is three perpendicular number lines (the third line is shown with dashes at an angle to the first two) and can represent the set of real numbers for three variables. The origin is the point in any geometric space where all the variables equal zero. There are four basic shapes that can be examined in these spaces, a point has no dimension, a line is one-dimensional, a plane is a two-dimensional flat shape and a three-dimensional object. In a three-dimensional geometric space an equation with three variables x, y and z are used to represent a point, a line, a plane or a solid shape, such as a polyhedra. The z variable is dependent and the function can be written z equals a function of x and y.
|Geometry - Geometry Basics - Examples|
Three dimensional equation form:
z = f(x, y)
z equals a function of x and y
On a number line:
The definition of a point, x = 3,
is the point at positive three.
The definition of a line,
for all x such that -3 <= x <= 3,
is a line from negative three to positive three.
A definition of a sphere is x² + y² + z² = 1
This is the unit sphere.