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One Dimension

                                          x      P(x)
             -----+-----+-----+-----+=====+=====+=*---+-----> x
                 -3    -2    -1     0     1     2     3

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One dimension: Points

A point is specified by a single real number called its coordinate. Let the coordinate of P1 be x1, and that of P2 be x2.

The distance from P1 to P2 is

d = sqrt[(x1-x2)2] = |x1-x2|.

The coordinate of the point dividing the line segment P1P2 in the ratio r/s is [rx2+sx1]/[r+s]. As a special case, when r = s, the midpoint of the line segment has coordinate [x2+x1]/2.

The set of all points with coordinate x satisfying a linear equation in x is a single point. Its equation has the general form

Ax + B = 0,
where A is nonzero. The coordinate of the point is -B/A.


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One dimension: General Quadratic Equations

A general quadratic equation can be put into the following form:

ax2 + 2bx + c = 0,

where a is nonzero. If we have an equation of this kind, it can represent one of three different kinds of curve. Which kind depends on the value of the following quantity:

    Delta = 
     a  b 
     b  c 

The cases are as follows:

Case   sign(Delta)   Name               Standard form

 1         -         Real circle        x2/r2 = 1
 2         +         Imaginary circle   x2/r2 = -1
 3         0         Coincident lines   x2 = 0
The equation can be put into standard form by completing the square, and then making a translation to move the center to the origin:

              0 = ax2 + 2bx + c,
              0 = a2x2 + 2abx + ac,
              0 = (ax + b)2 - b2 + ac,
         -Delta = (ax + b)2,
      -Delta/a2 = y2,
      
  where y = x + b/a,

   -sign(Delta) = y2/r2,

where r = sqrt(|Delta|)/a, provided Delta is nonzero. The center of the circle has coordinate -b/a, and the radius is r.


[ Back to Analytic Geometry Formula Contents]

One dimension: Circles

A circle is the set of all points at a distance (or radius) r > 0 from the center P1. Its equation has the form
    (x-x1)2 = r2,
    |x-x1| = r.
    

It consists of just two points, whose coordinates are x1 + r and x1 - r.


Compiled by Robert L. Ward.

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