Line (Geometry)

From Wikiversity
Jump to navigation Jump to search

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 Geometry[edit]

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."

Go to top of page ["Line (Geometry)."]

Line as locus of point[edit]



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."

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

Go to top of page ["Line (Geometry)."]

Normal form of line[edit]



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."

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

Go to top of page ["Line (Geometry)."]

Direction Numbers[edit]

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."

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

Go to top of page ["Line (Geometry)."]

Direction Cosines[edit]

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 equalsequals

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."

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

Go to top of page ["Line (Geometry)."]

Intercept form of Line[edit]



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."

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

Go to top of page ["Line (Geometry)."]

Angle of Intersection[edit]

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."

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

Go to top of page ["Line (Geometry)."]