Geometric Transformation of a Hyperbola into a Circle Through Complex Dilation

To understand the transformation of the hyperbola x^2 - y^2 1 into a circle through the dilation of y by a factor of i, let's break it down step by step.

Original Hyperbola

The equation x^2 - y^2 1 represents a hyperbola centered at the origin with its transverse axis along the x-axis. This hyperbola opens to the left and right, with asymptotes given by the lines y plusmn; x.

Dilation by a Factor of i

When we dilate y by a factor of i, we are essentially transforming the y-coordinate into the complex plane. The dilation can be expressed mathematically as:

y iy

Substituting y into the hyperbola equation gives:

x^2 - (iy)^2 1

Simplifying this results in:

x^2 - -y^2 1 quad Rightarrow quad x^2 y^2 1

This is the equation of a circle with radius 1 centered at the origin.

Geometric Interpretation

Transformation to Complex Plane

The dilation by i suggests a rotation in the complex plane. Specifically, multiplying by i corresponds to a 90-degree counterclockwise rotation. Therefore, the hyperbola is being rotated into a circular shape.

From Hyperbola to Circle

Geometrically, this transformation means that the set of points described by the hyperbola is mapped to a set of points that form a circle in the xy-plane.

Complex Coordinates

When dealing with y iy, we can think of y as a real number and iy as a complex number. This highlights the relationship between real and imaginary components, illustrating how transformations in the complex plane can yield different geometric shapes.

Conclusion

In summary, dilating the y-coordinate of the hyperbola x^2 - y^2 1 by a factor of i leads to a transformation that rotates the hyperbola into a circle, specifically the unit circle described by x^2 y^2 1. This illustrates the connection between hyperbolic and circular geometries through complex transformations.