Translation of Axes

Translation of Axes considers a transformation in which the new axes are parallel and directed the same way as the original axes. Axes translation entails completing two geometric transformations: a horizontal and vertical shift. As a result, the new axes can be generated by relocating the old axes horizontally and vertically by h units while keeping their directions the same. Let x and y represent the coordinates of any point P on the old axes, while x’ and y’ represent the coordinates of P on the new axes. Then x equals x’ + h, and y equals y’ + k.

Translation of Axes

Coordinate systems are required when applying analytic geometry methods to study curve equations. The axes are conveniently positioned concerning the curve under consideration when using the coordinate geometry method. For instance, while studying elliptic and hyperbolic equations, the foci are typically located on one of the axes and are symmetrically positioned with regard to the origin.

Suppose the curve (hyperbola, parabola, ellipse, etc.) is not located conveniently concerning the axes. In that case, the coordinate system should be adjusted to place the curve in a familiar and convenient location. A transformation of coordinates is the process of producing this shift. Similarly, many maths problems can be solved more quickly if the coordinates are translated.

Shifting of Origin

Translation of axes consists of mapping from an XY-Cartesian coordinate system to an x’y’-Cartesian coordinate system in which:

• y’ axis is parallel to the y axis and h units away
• the x’ axis is parallel to the x-axis and k units away

These are known as a translation of axes in two dimensions.

This means that the new coordinate system’s origin O’, has the original system’s coordinates (h, k). The positive x’ and y’ directions are identical to the positive x’ and y’ directions. In the original system, a point P has coordinates (x, y), while in the new system, it has coordinates (x’, y’).

The point P appears translated in the opposite direction in the new coordinate system. P appears to have been translated h to the left and k downward in the x’y’-system if the xy-system is translated h to the right and k upward. A translation of axes in more than two dimensions is defined in the same way.

Generalisation of Dimensions

Consider an xyz-Cartesian coordinate system in three dimensions with axes x’, y’, and z’ parallel to the x-axis and h units from it, y’ axis parallel to the y axis and k units from it, and the z’ axis parallel to the z-axis and l units from it. In both systems, a point P in space will have coordinates. The equations are (x, y, z) in the original system and (x’, y’, z’) in the second system if the coordinates are (x, y, z).

Translation of Quadratic Surfaces

The most general second-degree equation in x, y, and z has the form in three-dimensional space.

Ax² +By2+Cz² +Dxy + Exz + Fyz + Gx + Hy + Kz + L = 0

The points A, B, C, until L in space that satisfies this equation are on a surface. A quadric surface corresponds to any second-degree equation that does not reduce to a cylinder, plane, line, or point.

The method of axes translation can be used to simplify second-degree equations, revealing the nature of certain quadric surfaces, just as it can in plane analytic geometry. “Completing the square” is the most crucial instrument in this process.

Conic Sections Translation

The equation of a conic section can be transformed into a standard form by changing the coordinates, which is usually easier to work with. It is always possible to rotate the axes in such a way that the equation takes the form,

Ax² + Cy² + Dx + Ey + F = 0

 In the new system for the most generic equation of the second degree.

Then, with certain exceptions—for example, parabolas—a translation of axes can convert an equation of the form to an equation of the same form but with new variables (x’, y’) as coordinates and D and E both equal to zero. “Completing the square” is the most important instrument in this process.

Removal of the xy word without affecting its origin

We get x=(XcosYsin) and y=(Xsin+Ycos) when the axes are rotated through an angle.

Put this into the formula ax²+2hxy+by²=0

Set the XY coefficient to zero and calculate the angle of rotation required to remove the XY term.

Conclusion

When the coordinate axes are relocated while keeping the new axes parallel to the previous axes, the coordinates of a point are detailed. The axes are conveniently positioned concerning the curve under consideration when using the coordinate geometry method. The equation of a conic section can be transformed into a standard form by changing the coordinates, which is usually easier to work with. By translating coordinate axes, finding new equations if the origin is translated to a given point, finding new coordinates after translation, we can simplify the equations and graph by the transformed equation.