Line (in the works)

Description

Line as Scalable a Point

Consider the point {a,b}. If we multiply it by t, it would scale the point. That is, if t is greater than 1, {a,b} gets farther from origin. If t is 1, {a,b} remains the same. If t is greater than 0 but less than one, then {a,b}*t shrinks {a,b} towards the {0,0}. If t is negative, {a,b}*t is reflected thru the origin. Thus, if we let t vary, {a,b}*t is a line passing the origin. If x denote the x-coordinate component, and y denote the y-coordinate, in a rectangular coordinate system, then the parametric formula for a line is: x:=a*t, y:=b*t.

line vector
A point V scaled by 2 becomes C, and scaled by -1 becomes B. line parametric

Slope

The inclination of a line is measured by the ratio of its rise divided by its extend. This measurement is called the line's slope. For example, for a line {a,b}*t, its inclination is b/a, because for every movement by “a” distance to the right, it rises or falls by b. We can see that if b is large and a is small, we have a near vertical line. Vertical lines thus have slope ∞. Horizontal lines has slope 0. Very logical. Also, if the ratio b/a is positive, than the inclination is from lower bottom to upper right, like a slash symbor “/”. If b/a is negative, than the inclination is from upper left to bottom right, like a backslash symbol “\”.

Parametric Formula

If we add a vector to the formula {a,b}*t, we obtain {a,b}*t+{c,d}. In traditional notation, we write x:=t*a+c, y:=t*b+d. We can easily see that this is a line passing the point {c,d}. So, the line passing {c,d} and with slope b/a is {a,b}*t+{c,d}.

Equation of a Line

To find the algebraic equation, we write out the parametric formula as x:=a*t+c, y:=b*t+d. Isolate t in one equation and substitute into the other, simplify, we get b/a*x-y+d-b*c/a==0. This is a line with slope b/a and passing the point {c,d}.

Suppose we want to find the parametric formula of a line passing two ponts {a,b} and {c,d}. For that, we note that the vector from {a,b} to {c,d} is {c,d}-{a,b}. So, a line with that slope is ( {c,d}-{a,b})*t. To make the line pass {c,d}, just add it. So, a line passing two points {a,b} and {c,d} is then ({c,d}-{a,b})* t +{c,d} .

To find the equation of a line passing thru two points {a,b} and {c,d}, we expand ({c,d}-{a,b})* t +{c,d} to get {c + (-a + c) t, d + (-b + d) t} then write it out as

x:= c + (-a + c) t,
y:= d + (-b + d) t.

Eliminate t by isolating t in one equation and substitute it in another, we get (d - b) x + (a - c) y + (b c - a d) ==0. This is the equation of a line passing {a,b} and {c,d}.

History

A line is probably the most ancient geometric figure of all. The concept of a straight line probably precedes any concept or development of measurement, precursor to metrology and later, geometry.

In Euclid's Elements, the greatest text book of mathematics in history in terms of duration and influence, book I gives 17 definitions and 5 postulates (which today we'd call axioms), both mentioned the straight line. Here's the quote.

Definition 4. A straight line is a line which lies evenly with the points on itself.

Postulates:
1. [It is possible] to draw a straight line from any point
   to any point.
2. [It is possible] to extend a finite straight line continuously 
   in a streight line.
3. [It is possible] to describe a circle with any center 
   and distance [radius].
4. That all right angles are equal to one another.

5. That if a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles, the two
straights lines, if produced indefinitely, meet on that side on which
the angles are less than the two right angles.

A equivalent statement for postulate number 5 is that if a point lies outside of a line, then there are one and only one line passing this point and parallel to the given line. This postulate is knows as the Parallel Axiom. This axiom has always been seen as unintuitive and mathematicians have tried to derive it from the other axioms from the Greek times to as late as 1700s, all without sucess. Only around early 1800s it gradually became clear to mathematicians that axiom 5 is independent to the other axioms, and consistent geometries can be developed where axiom 5 is different, and non-Euclidean geometries were born. As a result, it gave birth to a new era in the philosophy of mathematics where math in particular geometry is no longer considered as the reality of the physical world, but a logical system independent of reality.

The concept of a line is also later refined to mean the shortest distance between two points in a given space. In the differential geometry era started by Gauss, where surfaces are considered as a space by themselves, a “line” in such a space is thus the shortest distance between two points, all remaining on the surface. Such concept is called geodesics. For example, if we consider a sphere as a space by itself, a line from pole to pole would not be a “straight line” going through “inside” of the sphere, because such line would be out of the sphere as a space. Instead, a straight line (shortest distance) on a sphere to a higher-dimensional space observer would be the great arc.

In projective geometry, developed in 1700s and 1800s, there emerges the concept of a line at infinity. Also, when projective geometry developed into a axiomatic system, it became apparent that lines and points are mirroring concepts in the framework of projective geometry. Later, the abstract projective plane is represented as topologically equivalent cross-cap surface.

Lines also share relation with circles. A line can be considered as the limit of a circle as its radius grows to infinity. This is particularly apparent in the so-called “inversive geometry”, where inversion transformation is used often to solve problems.

It turns out, that lines are the graphs of first degree polynomials. That is, equations of the form a*x+b*x+c==0. Often we have the problem of two such equations and we want to solve for x and y. We call it systems of 2 linear equations with 2 unknowns. Geometrically, it is finding the intersection of two lines. When solving multiple of such equations with multiple unknowns, it developed into a complete branch of theories called Linear Algebra, and its usefulness and abstraction have progressed far beyond having to do with lines.

Formula

A parametric formula for a line passing {a,b} and {c,d} is ({c,d}-{a,b})* t +{c,d}, with -∞ ≤ t ≤ ∞. In terms of coordinate equation, it is y ==(b*c - a*d - b*x + d*x)/(-a + c)

Properties

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

line.

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2006-05