Line (Geometry)

Line (Geometry)



The word "line" is open to many different interpretations, as in:

"That horse is descended from a great line of thoroughbreds." or

"Toe the line, or else!"


The concept of "line" in geometry is so basic that a definition may not be necessary. It might be better to say that a definition may not be possible (or adequate.)

Here are some possible definitions:


A line has length but no breadth. This "line" could not be seen under the most powerful microscope.

A line is the shortest possible distance between two fixed points. In astronomy the shortest possible distance between two fixed points might be curved. Some wit might argue that there is no such thing as a fixed point. After all every point on the surface of the earth is always moving.

A line is the locus of a point that moves from one fixed point to a second fixed point so that the distance traveled is the minimum possible. With current technology the minimum possible distance between New York City and Rome follows the curvature of the earth.

The word "point" has been mentioned but not defined. Can we define a point?

If you draw a "line" on a piece of paper and then crumple the paper, what does this do to the line?


In this page the "line" is described in the context of Cartesian geometry in exactly two dimensions.


Line in Cartesian GeometryEdit

 
Figure 1. Cartesian plane illustrating many different lines.


How many lines do you see in Figure 1? The answer takes us into a mixture of philosophy and geometry.


One answer might be: "None. I see many images, each with the appearance of a rectangle and each containing a line such as   or  ."


"You can't see the whole line  . At best all you can see is a segment of the line between  "


"Are the limits   included or excluded?"


"Do you see the red line?"


"I see a red image with the appearance of a rectangle (or trapezoid, I can't be sure), probably representing the line  , but it could represent the line  ."


A second answer might be: "Too many for the current discussion. After all, the character   contains 5 lines."


Let's go back to the original question: "How many lines do you see in Figure 1?"


While I see many more than 3, it seems that there are 3 of interest to this discussion and I answer: "Three."


"Describe them."


"A red line with equation  , a blue line with equation   and a green line with equation  "


Despite the possibility of endless limitations and diversions such as those mentioned above, we accept this answer as satisfactory for the current discussion.


The line defined.



In figure 1, the blue line may be defined as just that: "the blue line." However, if we are to answer profound questions about the blue line, such as "How far is the blue line from the red?" or "Where do the blue and green lines intersect?" we need to define the blue line in algebraic terms.


The blue line is the line that passes through points   and it has equation  .

Calculate the values of  :

 

 

 


 

 


 

 

and the equation of the blue line becomes:  . This equation has the form   where:

  slope of blue line   and  the  , the value of   at the point   where the line and the   intersect.


The red and green lines both intersect the   at the point  . The   intercept is  .

The red line has equation   The green line has equation  


Slope of line


 
Figure 2. Slope of line illustrated.
When   increases by   units,   increases by   units.
Slope of oblique line  
When   decreases by   units,   decreases by   units.
Slope of oblique line  


See Figure 2. The oblique line passes through points  


  The oblique line has equation   and it passes through the point   The   intercept is   and:

Oblique line has equation  


Back to Figure 1. By inspection,   and the red line has equation   and the green line has equation  






Parallel lines


 
Figure 3. Parallel lines in the Cartesian plane.
The 3 colored lines all have slope  .
Each colored line has equation:  
The 3 colored lines are parallel.


See Figure 3. The three colored lines are parallel because they all have the same slope  


Remember that the lines   are all parallel, as are the lines  























Lines with same   intercept


 
Figure 4. Lines with same   intercept.
The 3 colored lines all pass through point  .
They have the same   intercept.
Each colored line has equation:  


See Figure 4. The colored lines represent the family of lines that pass through the point  . There is one exception. The line   passes through the point   but it cannot be represented by the equation  

Note the red line. As   increases,   decreases, and the line goes down from left to right. Slope of the line is   and line has equation  






















Go to top of section "Line in Cartesian Geometry."

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Line as locus of pointEdit



 
Figure 1. Line as locus of point equidistant from 2 fixed points.
Any point on line   is equidistant from points  
 

In this section the line is defined as the locus of a point that is always equidistant from two fixed points. In Figure 1 the two fixed points are   and the length   is non-zero. By definition length   = length  


Let points   have coordinates  


Length  


Length  


  Therefore:


 

 

 


Expand and the result is:

 


This equation has the form:  where:

 


In Figure 1 the line through points   has equation  


If defined as the locus of a point equidistant from points   the calculation of   produces the equation  


If defined as the locus of a point equidistant from points   the calculation of   produces the equation  


Distance from point to line



Length  .


Length   is non-zero. Therefore, at least one of   must be non-zero.


Length   distance from point   to line.


Consider the expression   and substitute   for  


We show that   or  


If you make the substitutions and expand, you will see that the equality is valid.


Therefore   distance from point   to line.


Similarly we can show that   distance from point   to line.


If the equation of the line has form:   then


 


  coefficient of   coefficient of    


If the equation of the line   has  

the distance from point   to the line is  

the distance from point   to the line is   and  

Length   and length   have the same absolute value with opposite signs.



Use of multiplier K



Consider the equation   If   this doesn't make sense.


To make sense of the relationship introduce a multiplier   become   and the relationship is:


 


 


If  


Consider the line   and the point  


 


If the equation of the line is expressed as  

the equation makes sense, but the   doesn't change.


 


  the distance from point   to the line as calculated above with equation  


Calculation of the equation of the line equidistant from points   initially produces:   Calculation of the equation of the line equidistant from points   initially produces:  


Go to top of this section "Line as locus of point."

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Normal form of lineEdit



 
Figure 1. Graph illustrating normal form of straight line.
  ω.
  ω   ω  
At point   

See Figure 1. The green line through point   has equation   or  


The normal   to the line from the origin has length  


Let   ω  ω   ω  


The normal   to the line is in the quadrant where cosine   is negative and sine   is positive.


The normal form of the equation is   ω   ω   or  

This puts the origin on the negative side of the line.


Directed distance from line to origin   Directed distance from origin to line  


Components of the normal form


 
Figure 2. Composition of normal form with point   on line.
  ω;   ω.
 


See Figure 2. This example shows line   with point   on the line.


The normal to the line is in the quadrant where cosine   is positive and sine   is positive.


  ω  


  ω   ω   is the normal with length   Point   has coordinates  


  ω   ω  


  ω   ω  


 


  The point   is on the line.


 
Figure 3. Composition of normal form with point   not on line.
  ω;   ω.
 
Points   are   from green line through  

See Figure 3. This example shows line   with point   not on the line.


The normal to the line is in the quadrant where cosine   is positive and sine   is negative.


  ω.

  ω   ω   is the normal with length   Point   has coordinates  

  ω   ω  


  ω   The negative value for   establishes direction towards the origin.


  The points   are   from the line indicating   units toward the origin.


Normal form in practice




 
Figure 4. Three lines in normal form.

See Figure 4. The green line has equation   and point   is on the line.

The brown line has equation   and point   is on the line.

The brown and green lines are in the quadrant where cosine   is negative and sine   is positive.


Distance from brown line to origin  

Distance from green line to origin  


Distance from green line to point     toward the origin.

Distance from brown line to point   away from the origin.


Distance from green line to brown line   toward the origin.

Distance from brown line to green line   away from the origin.


The purple line has equation   and point   is on the line. The purple line is in the quadrant where cosine   is positive and sine   is negative. The purple and brown lines are parallel, but in opposite quadrants.

Distance from brown line to point   toward the origin.

Distance from purple line to point   toward the origin.

When calculating distance between brown and purple lines, it is important to see that they are in opposite quadrants. If direction is not important, you can say that the brown line has equation   and the distance between them is  


Go to top of this section "Normal form of line."

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Direction NumbersEdit

 
Figure 1. Direction numbers.
 
 
Both sets of direction numbers are equivalent to  


See Figure 1.


The red line has   intercept   and   intercept   Slope of red line  


Red line has equation   When   increases by   increases by   The line has   where   represents a change in   and   represents a corresponding change in  


This section shows that the line may be defined using known points and  


Consider the points   The value of   at   is the value of   at   The change in   between   is represented by arrow   with length   The value of   at   is the value of   at   The change in   between   is represented by arrow   with length  


  are the direction numbers of the red line at point  

The red line passes through point   with direction numbers  


Consider the points  

The red line passes through point   with direction numbers  


The direction numbers   for the red line are consistent because they represent a ratio, the slope of the line.


Given the point   and direction numbers   the red line can be defined as   and the equation of the red line is

 


Given the point   and direction numbers   the red line can be defined as   and the equation of the red line is

  the same as that calculated above.


For convenience, both sets of direction numbers   can be expressed as  


The equation of the red line is given as:   hence direction numbers  


Direction numbers are valid only if the distance between the two points of reference is non-zero. Therefore, at least one of   must be non-zero.


Using direction numbers



1. Format of any point.

Given a line defined as   any point on the line has format:   For example, if the red line in Figure 1 is defined as   any point on the line is   If   the point is   or point  


2. Normal to the line.

Refer to the section "Line as locus of a point" above. If the line has equation   the normal to the line has direction numbers  


3. Point at specified distance.

Given a line defined as   calculate the two points on the line at distance   from  

Let one point at distance   from   have coordinates  

Then  

 

For example, given a line defined as   calculate the two points on the line at distance   from  

 


The points are:   or  


4. Point at intersection of two lines.

Let one line have equation   and let the other be defined as  

Any point on the second line has format   The point   satisfies the first equation. Therefore:

 

 

 

 

If   the lines are parallel.


5. Angle between two lines.

 
Figure 2. Angle of intersection using direction numbers.
Line   has direction numbers  
Line   has direction numbers  

See Figure 2.

Line   has direction numbers  

Line   has direction numbers  

The aim is to calculate the angle between the two lines,  


 

 

 


Using the cosine rule  

 

 

 


If   and the lines are perpendicular.


if  ,

             ° or  ° and the lines are parallel.


Go to top of this section "Direction Numbers."

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Direction CosinesEdit

Let the line   have direction numbers  


The value  

The value  

 .


 
Figure 1. Angle of intersection using direction cosines.
 
All values   are direction cosines.
 

When a set of direction numbers   has   the direction numbers are direction cosines. In Figure 1   All values   are direction cosines. From 5 above,   This statement is equivalent to:


      or

 


For example, two lines have direction numbers   Calculate the angle between them.

Convert to direction cosines:  

 °.


If  ° or  °, and the lines are parallel.


If  ° or  °, and the lines are perpendicular.








1.   using direction cosines


 
Figure 2. Cos(2α) using direction cosines.
 
 
 


In Figure 2 line   has direction numbers   line   has direction numbers   and   The values   are direction cosines.


   


         


This statement is equivalent to:

  or  












2.   using direction cosines


 
Figure 3. cos(3α) using direction cosines.
 
All values   are direction cosines
 


See Figure 3.


Lines   are defined as       respectively.

 


  therefore:  


           

 


             


 


3.   using direction cosines


 
Figure 4. sin(α-β) using direction cosines.
 
All values   are direction cosines
 


See Figure 4.


Lines   are defined as     respectively.

 


     


Line   has equation  


Point   has coordinates  


Length      




4. Bisect angle between lines using direction cosines


 
Figure 5. Bisect an angle using direction cosines.
Line   bisects  
Line   has direction numbers  


See Figure 5. The aim is to produce line   the bisector of  


Lines   are defined as   and   respectively.


Ensure that all values   are direction cosines. At least one of   and one of   are non-zero.

Let line   have direction numbers  


 

 

 

 

Slope of line  

Let line   have direction numbers  


if     equals  

 if     equals  

  Both lines are parallel and in same direction.

  Three lines   are colinear.

 

 else :  equals 

  The   axis.

elif   equals equals 

  The   axis.

elif   equals 

  The lines are parallel and in opposite directions.

 

  The normal to line   or line  

else :

 

 


   


5.   using direction cosines.


 
Figure 6. sin(α+β) using direction cosines.
 
 
 
 
 


See Figure 6.


 

Point   has coordinates  

Line   is defined as   Length  

Point   has coordinates  

 

 





Go to top of this section "Direction Cosines."

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Intercept form of LineEdit



 
Figure 1. Intercept form of line.
Intercepts are  
Green line has equation  

The general equation of the straight line is:   where at least one of   is non-zero.

The intercept form of the line requires that all of   be non-zero.


 

 

 

 


 

If  

If  


Let   The line passes through the points   where   is the   intercept and   is the   intercept.


The equation   becomes   the intercept form of the equation.


See Figure 1. In this example  

The green line has equation  


Go to top of this section "Intercept form of Line."

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Angle of IntersectionEdit

 
Figure 1. Intersection of two Lines.
Given lines   calculate  


See Figure 1.


Given line   with equation   in which at least 1 of   is non-zero and line   with equation   in which at least 1 of   is non-zero, the aim is to calculate  


Let line   have slope   and

Let line   have slope  


Using  


  which can never have the value  


If   and the two lines are parallel. Also:

 


If  ° and the two lines are perpendicular. Also:

 If   and each line is parallel to an axis, else:

  If   is non-zero and   the lines are perpendicular.


Go to top of this section "Angle of Intersection."

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