Points, Lines and Curves

Summary

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The most practical branch of mathematics is geometry. The term 'geometry' is derived from the Greek word 'geometron'. It means Earth's measurement. The fundamental elements of geometry are given below :

Point :
In geometry, dots are used to represent points. A point is used to represent any specific location or position. It neither has any size, nor dimensions such as length or breadth. A point can be denoted by a capital letter of the English alphabet. Points can be joined in different ways.


 
Line segment :
A line segment is defined as the shortest distance between two points. A line segment has two end points and a hence a definite length. For example, if any two points, M and N are marked on a sheet of paper, then the shortest way to join M to N is a line segment. It is denoted by  MN _  and NM _ .   Points M and N are the end points of the line segment.



Line :
A line is made up of an infinite number of points that extend indefinitely in either direction. Two points determine a line. For example, if a line segment from M to N is extended beyond M in one direction and beyond N in the other, then we get a line, MN. It is denoted by  MN . A line can also be represented by small letters of the English alphabet.


 
Ray :
A ray is a portion of a line. It starts at one point and goes on endlessly in one direction. For example, if a line from M to N is extended endlessly in the direction of N, then we get a ray, MN. It is denoted by   MN and can be read as ray MN.


 
Plane :
A plane is said to be a very thin flat surface that does not have any thickness, and is limitless. A plane is always represented by a minimum of three points.
For example, the sheet shown below is said to be plane ABC. An infinite number of points can be contained within a plane.
 

 
Intersecting lines :
If two lines pass through a point, then we say that the two lines intersect at that point. Thus, if two lines have one point in common, then they are called intersecting lines. Intersecting lines always intersect, cut or cross each other. For example, If two lines pass though a point P, then the two lines are called intersecting lines.


 
Parallel lines or non- intersecting lines :
In a plane, if two lines have no point in common, then they are said to be parallel or non- intersecting lines. Parallel lines never meet, cut or cross each other.


 
Parallel lines never meet, cut or cross each other. If two lines m1 and m2 are parallel, we write m1 || m2.
 
Curves:
Curves can be defined as figures that flow smoothly without a break. A line is also a curve, and is called a straight curve.


 
Simple curves :
Curves that do not intersect themselves are called simple curves.


 
Open curves
Curves whose end points do not meet are called open curves.



Closed curves :
Curves whose end points join to enclose an area are called closed curves.


 
For a closed curve, we can identify three regions:
 

 
The interior of the curve: Green points  are in the interior of the closed curve.
Boundary of the curve: Blue points are on the boundary of the closed curve.
Exterior of the curve: Red points are in the exterior of the closed curve.
The interior of a curve together with its boundary is called its “region”.

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