Simple Geometry Figures

 

Motivation:

                        The study of Geometry could be originated from that people would like to describe what they saw from their eyes:  for example, the shapes of objects,  the path to a specific place, the territory of a tribe, or the face of a person. So,  we just start with some basic elements in order for describing objects more precisely.

 

 

 

 

 

 

Basic Elements of Geometry Figures

                 To describe the shape of an object,  what could be the best way to do that?  We need a consistent approach that can be used to describe the shapes of most objects.  It is naturally to think from the way how people draw the picture of an object: they start from a point, and some lines are extended from that point.  Furthermore,  people might notice some places are high and some places are low so that it is reasonable to imagine a place that is completely smooth.  Those concepts are refined in mathematics so that they can be applied to many other things.  In this section, we just introduce some of them from the aspect of mathematics.

 

 

 

 

 

Point    

           Mathematically,  a point only indicates the position. It does not have length, area, or volume.

 

 

 

 

 

 

Line,  Line Segment, and Ray

          Conceptually, a line is formed by linking two points with ruler.   However, we call the one with two endpoints as “line segment”; and the one extending in two directions to infinite far places as “line”.  Ray is that you start with a line segment by extending it in one direction. So, ray only has one endpoint.  A line, line segment or ray does not have “thickness” or occupy any area. 

 

 

 

 

 

Plane

          Conceptually, you can think a plane is a smooth paper sheet without thickness. And it extends to infinitely far places.

 

           A point is known as 0-dimesional space;  Line is 1-dimensional;  a plane is 2-dimensional.  And an object with length, width, and height is 3-dimensional.

 

 

 

 

 

Angle

          An angle is formed by two rays sharing the same endpoint.

 

                             

 

          Thus, a line can also be considered as “an angle” because it can be formed by two rays sharing the same endpoint.  One of the conventions to measure the magnitude of an angle is known as “degree” .  The angle of a line is known as 180 degrees, denoted as 180⁰ .    When the angle is 90⁰ , we say that it is a right angle and the two rays are perpendicular to each other . 

 

 

 

 

Parallel Lines

            If two lines on the same plane do not cross, we say that the two lines are parallel to each other.  Please notice that  it is possible that two lines never cross to each other and they are not parallel in 3-dimensional space.  When we speak of two parallel lines, we require the two lines are on the same plane.