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CHANGES IN EQUATION COEFFICIENTS PRODUCED BY TRANSLATION AND ROTATION OF THE COORDINATE SYSTEM

Changes in equation coefficients produced by translation. What changes are produced in the coefficients of the general equation of the second degree

             f(x, y) = ax² + 2bxy + cy² + 2dx + 2ey + f = 0

by a translation of the coordinate system? Let us observe the results of translating the coordinate system to the point (x₀, y₀). Let x'-y' be a coordinate system with origin located at point (x₀, y₀) of the x-y system. We substitute

x = x' + x₀

y = y' + y₀

into equation 1) above.

The equation becomes

f(x' + x₀, y' + y₀) = a(x' + x₀)² + 2b(x' + x₀)(y' + y₀) + c(y' + y₀)² + 2d(x' + x₀) + 2e (y' + y₀) + f

             = ax' ² + 2bx'y' + cy' ² + 2(ax₀ + by₀ + d)x' + 2(bx₀ + cy₀ + e)y' + ax₀² + 2bx₀y₀ + cy₀² + 2dx₀ + 2ey₀ + f

            = ax' ² + 2bx'y' + cy' ² + 2(ax₀ + by₀ + d)x' + 2(bx₀ + cy₀ + e)y' + f(x₀, y₀) = 0

We see that a translation has no effect on the coefficients of the second degree terms (i.e. the coefficients a, b, c). It only changes the first degree coefficients d and e and the constant term f where

Changes in equation coefficients produced by rotation. Under a rotation of the coordinate system about its origin by an angle of θ degrees the general equation of the second degree

            f(x, y) = ax² + 2bxy + cy² + 2dx + 2ey + f = 0

becomes

            a'x' ² + 2b'x'y' + c'y'² + 2d'x' + 2e'y' + f ' = 0

where

            a' = a cos² θ + 2 b sin θ cos θ + c sin² θ

            b' = (c - a) sin θ cos θ + b (cos² θ - sin² θ)

                        = (c - a) sin 2θ + b cos 2θ

            c' = a sin² θ - 2b sin θ cos θ + c cos² θ

            d' = d cos θ + e sin θ

            e' = e cos θ - d sin θ

            f ' = f

where the equations of the transformation are

x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ .

We note that all of the coefficients of the equation change under the rotation of the coordinate system. Only the constant term f remains unchanged.

Invariance of the quantity b² - ac under rotation. The quantity b² - ac is invariant under rotation of coordinates. In other words, if the general equation of the second degree

            f(x, y) = ax² + 2bxy + cy² + 2dx + 2ey + f = 0

is transformed by

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ

to obtain

            a'x' ² + 2b'x'y' + c'y'² + 2d'x' + 2e'y' + f ' = 0

then

            b' ² - a'c' = b² - ac .