In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a translation of axes in two dimensions is a
mapping from an ''xy''-
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
to an ''x'y
'''-Cartesian coordinate system in which the ''x
''' axis is
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
to the ''x'' axis and ''k'' units away, and the ''y
''' axis is parallel to the ''y'' axis and ''h'' units away. This means that the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
''O
''' of the new coordinate system has coordinates (''h'', ''k'') in the original system. The positive ''x
''' and ''y
''' directions are taken to be the same as the positive ''x'' and ''y'' directions. A point ''P'' has coordinates (''x'', ''y'') with respect to the original system and coordinates (''x
''', ''y
''') with respect to the new system, where
or equivalently
In the new coordinate system, the point ''P'' will appear to have been translated in the opposite direction. For example, if the ''xy''-system is translated a distance ''h'' to the right and a distance ''k'' upward, then ''P'' will appear to have been translated a distance ''h'' to the left and a distance ''k'' downward in the ''x'y
'''-system . A translation of axes in more than two dimensions is defined similarly. A translation of axes is a
rigid transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformation ...
, but not a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
. (See
Affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
.)
Motivation
Coordinate systems are essential for studying the equations of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s using the methods of
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and enginee ...
. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, the
foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola,
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, ellipse, etc.) is ''not'' situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a
transformation of coordinates.
The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original ones.
Translation of conic sections
Through a change of coordinates, the equation of a conic section can be put into a
standard form, which is usually easier to work with. For the most general equation of the second degree, which takes the form
it is always possible to perform a
rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
in such a way that in the new system the equation takes the form
that is, eliminating the ''xy'' term. Next, a translation of axes can reduce an equation of the form () to an equation of the same form but with new variables (''x
''', ''y
''') as coordinates, and with ''D'' and ''E'' both equal to zero (with certain exceptions—for example, parabolas). The principal tool in this process is "completing the square." In the examples that follow, it is assumed that a rotation of axes has already been performed.
Example 1
Given the equation
:
by using a translation of axes, determine whether the
locus of the equation is a parabola, ellipse, or hyperbola. Determine foci (or focus),
vertices (or vertex), and
eccentricity.
Solution: To complete the square in ''x'' and ''y'', write the equation in the form
:
Complete the squares and obtain
:
:
Define
:
and
That is, the translation in equations () is made with
The equation in the new coordinate system is
Divide equation () by 225 to obtain
:
which is recognizable as an ellipse with
In the ''x'y
'''-system, we have: center
; vertices
; foci
In the ''xy''-system, use the relations
to obtain: center
; vertices
; foci
; eccentricity
Generalization to several dimensions
For an ''xyz''-Cartesian coordinate system in three dimensions, suppose that a second Cartesian coordinate system is introduced, with axes ''x
''', ''y
''' and ''z
''' so located that the ''x
''' axis is parallel to the ''x'' axis and ''h'' units from it, the ''y
''' axis is parallel to the ''y'' axis and ''k'' units from it, and the ''z
''' axis is parallel to the ''z'' axis and ''l'' units from it. A point ''P'' in space will have coordinates in both systems. If its coordinates are (''x'', ''y'', ''z'') in the original system and (''x
''', ''y
''', ''z
''') in the second system, the equations
hold. Equations () define a translation of axes in three dimensions where (''h'', ''k'', ''l'') are the ''xyz''-coordinates of the new origin. A translation of axes in any finite number of dimensions is defined similarly.
Translation of quadric surfaces
In three-space, the most general equation of the second degree in ''x'', ''y'' and ''z'' has the form
where the quantities
are positive or negative numbers or zero. The points in space satisfying such an equation all lie on a
surface. Any second-degree equation which does not reduce to a cylinder, plane, line, or point corresponds to a surface which is called quadric.
As in the case of plane analytic geometry, the method of translation of axes may be used to simplify second-degree equations, thereby making evident the nature of certain quadric surfaces. The principal tool in this process is "completing the square."
Example 2
Use a translation of coordinates to identify the quadric surface
:
Solution: Write the equation in the form
:
Complete the square to obtain
:
Introduce the translation of coordinates
:
The equation of the surface takes the form
:
which is recognizable as the equation of an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
.
See also
*
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to ev ...
Notes
References
*
*
{{Authority control
Functions and mappings
Euclidean geometry
Linear algebra
Transformation (function)