Translation of axes#Translation of conic sections

Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. 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 ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola, 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.{{harvtxt|Protter|Morrey|1970|pp=314–315}}

The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original ones.{{harvtxt|Anton|1987|p=107}}

{{Main|Conic section}}

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

{{NumBlk|:|$Ax^2\; +\; Bxy\; +\; Cy^2\; +\; Dx\; +\; Ey\; +\; F\; =\; 0$ {{spaces|4}} ($A$, $B$ and $C$ not all zero);|{{EquationRef|3}}}}

it is always possible to perform a rotation of axes in such a way that in the new system the equation takes the form

{{NumBlk|:|$Ax^2\; +\; Cy^2\; +\; Dx\; +\; Ey\; +\; F\; =\; 0$ {{spaces|4}} ($A$ and $C$ not both zero);|{{EquationRef|4}}}}

that is, eliminating the xy term.{{harvtxt|Protter|Morrey|1970|p=322}} Next, a translation of axes can reduce an equation of the form ({{EquationNote|3}}) 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."{{harvtxt |Protter|Morrey|1970|p=316}} In the examples that follow, it is assumed that a rotation of axes has already been performed.

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.

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

:$9(x^2\; +\; 2x\; +\; 1)\; +\; 25(y^2\; -\; 4y\; +\; 4)\; =\; 116\; +\; 9\; +\; 100$

:$\backslash Leftrightarrow\; 9(x\; +\; 1)^2\; +\; 25(y\; -\; 2)^2\; =\; 225\; .$

Define

:$x\text{'}\; =\; x\; +\; 1$ {{spaces|4}} and {{spaces|4}} $y\text{'}\; =\; y\; -\; 2\; .$

That is, the translation in equations ({{EquationNote|2}}) is made with $h\; =\; -1,\; k\; =\; 2\; .$ The equation in the new coordinate system is

{{NumBlk|:|$9x\text{'}^2\; +\; 25y\text{'}^2\; =\; 225\; .$|{{EquationRef|5}}}}

Divide equation ({{EquationNote|5}}) by 225 to obtain

:$\backslash frac\{x\text{'}^2\}\{25\}\; +\; \backslash frac\{y\text{'}^2\}\{9\}\; =\; 1\; ,$

which is recognizable as an ellipse with $a\; =\; 5,\; b\; =\; 3,\; c^2\; =\; a^2\; -\; b^2\; =\; 16,\; c\; =\; 4,\; e\; =\; \backslash tfrac\{4\}\{5\}\; .$ In the x'y'-system, we have: center $(0,\; 0)$; vertices $(\backslash pm\; 5,\; 0)$; foci $(\backslash pm\; 4,\; 0)\; .$

In the xy-system, use the relations $x\; =\; x\text{'}\; -\; 1,\; y\; =\; y\text{'}\; +\; 2$ to obtain: center $(-1,\; 2)$; vertices $(4,\; 2),\; (-6,\; 2)$; foci $(3,\; 2),\; (-5,\; 2)$; eccentricity $\backslash tfrac\{4\}\{5\}\; .${{harvtxt |Protter|Morrey|1970|pp=316–317}}

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

{{NumBlk|:|$x\text{'}\; =\; x\; -\; h,\; \backslash qquad\; y\text{'}\; =\; y\; -\; k,\; \backslash qquad\; z\text{'}\; =\; z\; -\; l$|{{EquationRef|6}}}}

hold.{{harvtxt|Protter|Morrey|1970| pp=585–586}} Equations ({{EquationNote|6}}) define a translation of axes in three dimensions where (h, k, l) are the xyz-coordinates of the new origin.{{harvtxt|Anton|1987|p=107}} A translation of axes in any finite number of dimensions is defined similarly.

{{Main|Quadric surface}}

In three-space, the most general equation of the second degree in x, y and z has the form

{{NumBlk|:|$Ax^2\; +\; By^2\; +\; Cz^2\; +\; Dxy\; +\; Exz\; +\; Fyz\; +\; Gx\; +\; Hy\; +\; Iz\; +\; J\; =\; 0\; ,$|{{EquationRef|7}}}}

where the quantities $A,\; B,\; C,\; \backslash ldots\; ,\; J$ 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.{{harvtxt |Protter|Morrey|1970|p=579}}

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."{{harvtxt |Protter|Morrey|1970|p=586}}

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

:$x^2\; +\; 2x\; \backslash qquad\; +\; 4(y^2\; -\; 2y)\; +\; 3(z^2\; +\; 3z)\; =\; 10\; .$

Complete the square to obtain

:$(x\; +\; 1)^2\; +\; 4(y\; -\; 1)^2\; +\; 3(z\; +\; \backslash tfrac\{3\}\{2\})^2\; =\; 10\; +\; 1\; +\; 4\; +\; \backslash tfrac\{27\}\{4\}\; .$

Introduce the translation of coordinates

:$x\text{'}\; =\; x\; +\; 1,\; \backslash qquad\; y\text{'}\; =\; y\; -\; 1,\; \backslash qquad\; z\text{'}\; =\; z\; +\; \backslash tfrac\{3\}\{2\}\; .$

The equation of the surface takes the form

:$x\text{'}^2\; +\; 4y\text{'}^2\; +\; 3z\text{'}^2\; =\; \backslash tfrac\{87\}\{4\}\; ,$

which is recognizable as the equation of an ellipsoid.{{harvtxt |Protter|Morrey|1970|p=586}}

• Change of basis
• Translation (geometry)

• {{ citation | first1 = Howard | last1 = Anton | year = 1987 | isbn = 0-471-84819-0 | title = Elementary Linear Algebra | edition = 5th | publisher = Wiley | location = New York }}
• {{ citation | first1 = Murray H. | last1 = Protter | first2 = Charles B. Jr. | last2 = Morrey | year = 1970 | lccn = 76087042 | title = College Calculus with Analytic Geometry | edition = 2nd | publisher = Addison-Wesley | location = Reading }}

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