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analytic geometry   -

branch of geometry in which points are represented with respect to a coordinate system, such as cartesian coordinates. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th century. Its most common application - the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions - allows problems in algebra to be treated geometrically and geometric problems to be treated algebraically. The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry.
One Dimension: Points || General Quadratic Equations || Circles
Two Dimensions: Points || Directions || Lines || Triangles || Polygons || 
Conic Sections [hyperbolas | parabolas | ellipses | circles]
General Quadratic Equations || 
Three Dimensions:


Points || Directions || Lines || Planes || Triangles || Tetrahedra || 
General Quadratic Equations and Quadric Surfaces || Spheres

1. Gellert, W., S. Gottwald, M. Hellwich, H. Kästner, H. Küstner, eds., K. A. Hirsch
and H. Reichardt, Scientific Advisors, The VNR Concise Encyclopedia of Mathematics,
2nd edition, Van Nostrand Reinhold, New York, NY, 1989, pp. 282-319, 530-547.

2. Zwillinger, Daniel, CRC Standard Mathematical Tables and Formulae,
30th Edition, CRC Press, Boca Raton, FL, 1996, pp. 249-319.

One Dimension
[Back to Contents]
                                                    x      P(x)
                       -----+-----+-----+-----+=====+=====+=*---+-----> x
                           -3    -2    -1     0     1     2     3

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.

[Back to Contents]

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 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.

[Two Dimensions]   [Back to Contents]   [Three Dimensions]

Compiled by Robert L. Ward.

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