91Ó°ÊÓ

A unit square is a specific type of geometric shape characterized by having each of its sides equal to 1 unit. This means the area of the unit square is precisely 1 square unit. In the context of coordinate planes, such as the uv-plane in this exercise, a unit square can be defined by the set of points where each coordinate lies between 0 and 1 inclusive. Thus, the boundaries of the unit square in the uv-plane are:

• 0 ≤ u ≤ 1
• 0 ≤ v ≤ 1

The corners of this square are crucial because they denote the maximum extents along the axes and are at coordinates (0, 0), (0, 1), (1, 0), and (1, 1). These corner points serve as reference points for transformations, helping us understand how shapes morph when plotted on different planes.

The image of a shape upon transformation onto a different plane is essentially how the shape appears after it undergoes a specified transformation. When discussing the transformation of coordinates, the focus is on how the positions of the points in one plane (uv-plane in this instance) reappear in another plane (xy-plane here).
For the given transformation:

• \(x = u^2 - v^2\)
• \(y = 2uv\)

We apply these formulas to each corner point of the unit square to find its new position in the xy-plane. The transformation changes the shape's orientation and the positions of its boundaries, giving a new geometrical figure, which in this case reveals a parallelogram structure in the xy-plane.

Corner points are the vertices of a shape that indicate its size and position in a coordinate plane. For transformations, they are critical since changing these points redefines the entire shape in its new context. In our exercise, the unit square has corner points at:

• (0, 0)
• (0, 1)
• (1, 0)
• (1, 1)

By applying the transformation equations, these points shift to new positions in the xy-plane. Calculating each:

• (0, 0) remains (0, 0) because both coordinates are zero.
• (0, 1) transforms to (-1, 0) as \(x = 0^2 - 1^2 = -1\) and \(y = 2 \times 0 \times 1 = 0\).
• (1, 0) transforms to (1, 0) since \(x = 1^2 - 0^2 = 1\) and \(y = 2 \times 1 \times 0 = 0\).
• (1, 1) changes to (0, 2) as \(x = 1^2 - 1^2 = 0\) and \(y = 2 \times 1 \times 1 = 2\).

These new positions form the corners of the transformed shape in the xy-plane.

A parallelogram is a four-sided figure in which opposite sides are parallel, and it can be viewed as a skewed rectangle. When we transform a unit square on the uv-plane, what results often is not another square. In this specific transformation, the image turns into a parallelogram. The transformed corner points, which help define the shape of this parallelogram in the xy-plane, are (0, 0), (-1, 0), (1, 0), and (0, 2).
Here's how they connect:

• Points (0, 0) and (-1, 0) form one side, and (0, 0) and (1, 0) form another.
• The segment from (-1, 0) to (0, 2) and from (1, 0) to (0, 2) creates the skewed edges.

These connections confirm that the transformed shape is indeed a parallelogram. The transformation distorts the original unit square by applying the formulas for x and y, demonstrating how coordinate transformations affect geometric figures.