Which parameter am I missing in determining a straight line?

[grzz]

Main Question or Discussion Point

A line in 2-dimensional euclidean space can be determined by the coordinates of any two particular points on the line. Hence four "parameters" are required.
Now let’s consider a line given by the usual equation
y = mx + c.
c, being the intercept on the y-axis, is related to the point (0,c) which requires two numbers to be determined.
m, the slope, can be given by the tanθ which requires just one number to be determined.
Hence when the line is given by the equation y = mx + c one requires just three parameters and not four as in the first case.
Which parameter am I missing?
Thanks for any help.

 

[hobbyist]

m and c are not independent.
Consider equation y=mx+c for points (y1,x1) and (y2,x2).
Solving it you will have:
m=(y2-y1)/(x2-x1); c=(y1+y1)/(2m(x2+x1)).
the similar thinking can lead you to understanding that to define a line you need just TWO points, and a number of coordinates for each point depends on space dimension

 

[PeroK]

A line in 2-dimensional euclidean space can be determined by the coordinates of any two particular points on the line. Hence four "parameters" are required.
Now let’s consider a line given by the usual equation
y = mx + c.
c, being the intercept on the y-axis, is related to the point (0,c) which requires two numbers to be determined.
m, the slope, can be given by the tanθ which requires just one number to be determined.
Hence when the line is given by the equation y = mx + c one requires just three parameters and not four as in the first case.
Which parameter am I missing?
Thanks for any help.

In both cases, you have essentially only two parameters.

You can define a line uniquely by specifying m and c.

Or, given x_1 and x_2, you can define it by specifying y_1 and y_2.

The problem with your thinking is that every choice of x_1, x_2, y_1 and y_2 does not define a different line. You get the same line for many choices.

 

[grzz]

... to define a line you need just TWO points, and a number of coordinates for each point depends on space dimension

Thanks for the help.
Yes, in 2-dimensional euclidean space we need four numbers giving the coordinates of the two points. By 'parameter' I meant 'a number'.

 

[hobbyist]

...The problem with your thinking is that every choice of x_1, x_2, y_1 and y_2 does not define a unique line. You get the same line for many choices.

It is obvious that grzz meant a straight line, thus 2 points DEFINE a unique line. The statement about "the same line" although may be true but misleading and irrelevant in the context.

 

[PeroK]

I used the word "unique" when I really meant "different". Edited above.

 

[grzz]

M and C are not independent.

a) Given an intercept c on the y-axis, a straight line can have any slope.
b) A straight line of a given slope m can have any intercept on the y-axis.
Hence the slope and the y-intercept of a straight line are independent.
Is my reasoning correct?

 

[PeroK]

a) Given an intercept c on the y-axis, a straight line can have any slope.
b) A straight line of a given slope m can have any intercept on the y-axis.
Hence the slope and the y-intercept of a straight line are independent.
Is my reasoning correct?

Yes, m and c are clearly independent.

 

[hobbyist]

Yes, m and c are clearly independent.

They are independent in a sense that one can draw different straight lines using different m and c.
But if one has a fixed point (y1, x1) and c, then m is defined, same for (y1,x1) and m, then c defined.
As in original question:

two particular points on the line

 

[grzz]

Let me give the reason for my original post.
How is it that when a straight line is determined by the coordinates of two points one requires four particular numbers and when the straight line is given by the equation
y =mx + c
one requires only three particular numbers i.e.tanθ and the coordinates of c?

 

[PeroK]

Let me give the reason for my original post.
How is it that when a straight line is determined by the coordinates of two points one requires four particular numbers and when the straight line is given by the equation
y =mx + c
one requires only three particular numbers i.e.tanθ and the coordinates of c?

We already answered this question above. You need a minimum of two independent parameters. You can always specify more, depending on how you do it.

For example. Suppose you define a whole number, n, by its prime factorisation. Then, you need to specify the power of every prime in the factorisation of n. In this case, you need an infinite number of parameters to specify n.

Also, if you specify n by its decimal representation, then again you need an unlimited number of parameters. One for every power of 10.

You can always find ways to need more than the minimum.

The key point about a straight line is that you can't do it with 1 parameter. You need at least 2.

 

[hobbyist]

Yes, this is a good question.
But first to clarify that c is just a number, not a point. Therefore the more correct question, why we need 4 numbers (coordinates of 2 SPECIFIC points) and only 2 numbers (m and c).
The answer is quite simple: with equation y=mx+c, you define a line running through those 2 SPECIFIC points, but you are not able to deduce from the equation their SPECIFIC coordinates. However with points coordinates (y1,x1) and (y2,x2) you can define NOT ONLY the line, BUT ALSO points location.
Hope this is clear enough, Say other way, by knowing the line, you do not have enough information to find points, any point on that line will suffice the y=mx+c.

 

[grzz]

Thanks everybody for your help.