HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a translation of axes in two dimensions is a mapping from an ''xy''-
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
to an ''x'y'''-Cartesian coordinate system in which the ''x''' axis is parallel 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 ''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 p ...
. (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 general ...
.)


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 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 engineering, and als ...
. 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 focus (geometry), 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 ty ...
s and
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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, called connected component ( ...
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 Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
, 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 ar ...
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 : 9x^2 + 25y^2 + 18x - 100y - 116 = 0 , 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. Solution: To complete the square in ''x'' and ''y'', write the equation in the form : 9(x^2 + 2x) + 25(y^2 - 4y) = 116 . Complete the squares and obtain : 9(x^2 + 2x + 1) + 25(y^2 - 4y + 4) = 116 + 9 + 100 : \Leftrightarrow 9(x + 1)^2 + 25(y - 2)^2 = 225 . Define : x' = x + 1 and y' = y - 2 . That is, the translation in equations () is made with h = -1, k = 2 . The equation in the new coordinate system is Divide equation () by 225 to obtain : \frac + \frac = 1 , which is recognizable as an ellipse with a = 5, b = 3, c^2 = a^2 - b^2 = 16, c = 4, e = \tfrac . In the ''x'y'''-system, we have: center (0, 0) ; vertices (\pm 5, 0) ; foci (\pm 4, 0) . In the ''xy''-system, use the relations x = x' - 1, y = y' + 2 to obtain: center (-1, 2) ; vertices (4, 2), (-6, 2) ; foci (3, 2), (-5, 2) ; eccentricity \tfrac .


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 A, B, C, \ldots , J are positive or negative numbers or zero. The points in space satisfying such an equation all lie on a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. 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 : x^2 + 4y^2 + 3z^2 + 2x - 8y + 9z = 10 . Solution: Write the equation in the form : x^2 + 2x \qquad + 4(y^2 - 2y) + 3(z^2 + 3z) = 10 . Complete the square to obtain : (x + 1)^2 + 4(y - 1)^2 + 3(z + \tfrac)^2 = 10 + 1 + 4 + \tfrac . Introduce the translation of coordinates : x' = x + 1, \qquad y' = y - 1, \qquad z' = z + \tfrac . The equation of the surface takes the form : x'^2 + 4y'^2 + 3z'^2 = \tfrac , which is recognizable as the equation of an
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
.


See also

*
Change of basis In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are conside ...
*
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 geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the ...


Notes


References

* * {{Authority control Functions and mappings Euclidean geometry Linear algebra Transformation (function)