analytic geometry
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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:
References: |
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.
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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:
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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Analytic
Geometry
Formulas
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