Lines

1. Lines in Cartesian coordinates

The general equation of a line in cartesian coordinates (orthogonal axes and equal units) is:
                                                                             ax+by+c = 0.
The coefficients (a,b) determine the normal direction n of the line consequently rotating this vector gives the direction of the line n'=Jn=(-b,a). Here J denotes the positive rotation by a right angle about the origin of coordinates.
The function  f(x,y) = ax+by+c  gives for every point X=(x,y) of the plane a multiple of the distance from the line. Namely:
                                                                             f(x,y) = ax+by+c = k*distance(X,e), where k² = a²+b².

The line divides the plane in two half-planes characterized by the sign of f(x,y). The positive part is on the side pointed by the normal vector n=(a,b). The other half-plane is the negative part.
The normal form of the line results by dividing with the coefficient k and is equivalent by taking the normal n to be a unit vector:
n = (cos(w), sin(w)) => f(x,y) = cos(w)x + sin(w)y + c.
Then for every point X=(x,y) the value of the function f(x,y) gives the signed distance of X from the line.
Where the sign is positive for the positive half-plane and negative otherwise.
Dividing by k one can always pass to the normal form of the equation determining the same line with the original one.

2. Parametric form

The parametric form of a line is given by:
a(t) = X₀ + tv.
Here X₀=(x₀,y₀) is a point on the line and v=(r,s) is its direction, so that Jv = (-s,v) is a normal of the line.

The transition from the equation   ax + by + c = 0  is done by setting  x = t and solving for y:
                                                                       x = t,      y = -(1/b)(at+c).
Special cases (a=0 or/and b=0) are handled in the obvious way.
The transition from parametric form to the equation  f(x,y) = 0 is done by "eliminating" t from the equations:
             x = x₀ + t*r ,   y = y₀ + t*s  =>   (x-x₀)/r  =  (y-y₀)/s <=>   sx - ry + (ry₀-sx₀) = 0.

3. Line formularium

Below is a line formularium listing equations resulting by the various basic conditions defining a line.

The list could be extended ad infinitum, especially if one allows conditions comming from relations with other curves and not only lines or/and points. Standard books in analytic Geometry such as Salmon, Emch, Looney contain a wealth of relative formulas.

4. Two lines

Among many others there are four principal constructs defined by a pair of lines:
1) Their intersection point,
2) Their (directed) angle,
3) Their pencil.
4) Their quadratic form.
1) Given the lines through their equations: f(x,y) = ax + by + c = 0, g(x,y) = a'x + b'y + c' = 0,
their intersection point results by solving this system and gives:

The lines are parallel when the denominator ab'-a'b =0, i.e. the normals of the lines are collinear.
2) The directed angle is found by the inner product of the normals n=(a,b) and n'=(a',b'):

3) Their pencil is the set of all lines passing through their intersection point. These can be written in the form:

Here the variables (s,t) are considered as parameters taking arbitrary values and defining the various members of the pencil. The subject is quite rich and is discussed in some detail in the file LinePencils.html .
4) The quadratic form of the two lines results by multiplying their equations and forming the quadratic one:
h(x,y) = f(x,y)*g(x,y) = 0.
This represents a so-called degenerate conic consisting of the union of two lines. This is discussed in ConicsDegenerate.html .
Quite interesting is also the configuration of three lines (triangles) discussed in ThreeLines.html .

1. Lines in Cartesian coordinates

2. Parametric form

3. Line formularium

4. Two lines

See Also