[3] A translation of axes is a rigid transformation, but not a linear map.
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.
The process of making this change is called a transformation of coordinates.
[5] 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 such a way that in the new system the equation takes the form that is, eliminating the xy term.
"[7] In the examples that follow, it is assumed that a rotation of axes has already been performed.
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 That is, the translation in equations (2) is made with
[9] Equations (6) define a translation of axes in three dimensions where (h, k, l) are the xyz-coordinates of the new origin.
[10] A translation of axes in any finite number of dimensions is defined similarly.
In three-space, the most general equation of the second degree in x, y and z has the form where the quantities
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.
[11] 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.
"[12] 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.