Homothety

Homothety is a geometric transformation that maps the points of a figure relative to a fixed point O, altering the distances between the points proportionally by a factor k, while keeping the shape of the figure unchanged.

The point O is called the center of homothety, and the factor k is known as the homothety ratio.

In simpler terms, homothety can shrink or enlarge a figure while maintaining the alignment of the points (collinearity) and preserving the relative distances between them.

For instance, consider a figure ABC and select point O as the center of homothety, with a value of k as the homothety ratio.

Now, take the vector $ \overrightarrow{OA} $ and multiply it by the homothety ratio k, resulting in another vector $ \overrightarrow{OA'} $ that shares the same origin and direction as the original (O) but with a different length.

$$ \overrightarrow{OA'} = k \times \overrightarrow{OA} $$

This helps us identify another point A' on the plane that lies along the same line as OA.

So, if A is a point on the original figure and A' is its corresponding point in the homothetic figure, then the points O, A, and A' are aligned, and the segment OA' is k times the length of the segment OA.

Note: I’m using vectors (e.g., $ \overrightarrow{OA} $) instead of segments (e.g., $ \overline{OA} $) because in homothety, it’s crucial to consider the direction of the geometric transformation. This is because the homothety ratio k can be negative (k<0), which would indicate an "inverse homothety."

If I repeat this process for the other points B and C of the original figure, I end up with another figure A'B'C' that is similar to the original one.

The figure A'B'C' retains the same shape as the figure ABC but differs in size.

In this case, the transformed figure is enlarged.

Depending on the homothety ratio (k), different effects can be observed:

• If |k| > 1, the homothety results in an enlargement.
• If |k| < 1, the homothety results in a reduction.
• If k = 1, the homothety is an identity transformation, meaning the figure remains unchanged.
• If k = -1, the homothety is equivalent to central symmetry.

It's important to remember that the homothety ratio can also be negative (k<0).

The sign of the homothety ratio determines the type of homothety: direct or inverse.

• Direct Homothety
  If k>0, the homothety is referred to as direct because the corresponding points remain in the same quadrant.
  
• Inverse Homothety
  If k < 0, the homothety is referred to as inverse, as it not only causes a reduction or enlargement but also a reflection with respect to the homothety center O.
  

What’s the purpose? Homothety is used to enlarge or reduce geometric figures while maintaining their proportions and original shapes. It is especially useful for studying relationships between similar figures.

A Practical Example

Let's consider a square with a side length of a and a homothety center at its center O.

If I apply a homothety with a ratio of k=2, I get a square with a side length of 2a.

The result is an enlargement.

If I apply a homothety with a ratio of k = 0.5, I get a square with a side length of a/2.

In this case, the result is a reduction.

Obviously, with the same ratio k, the result changes depending on the chosen homothety center.

For example, consider the same square but with the homothety center at O(2,3).

If I apply a homothety with a ratio of k=2, I get a square with a side length of 4a.

The result is completely different from the previous one.

The Equations of Homothety

These equations provide a simplified representation of homothety in Cartesian coordinates when the homothety center is at the origin O(0,0) of the Cartesian plane.

$$ \begin{cases} x' = k \cdot x \\ \\ y' = k \cdot y \end{cases} $$

Where k is the homothety ratio, (x, y) are the coordinates of a point in the original figure, and (x', y') are the coordinates of the corresponding point in the homothetic figure.

If the homothety center O is not at the origin but at another point O(x₀,y₀), the equations become slightly more complex.

$$ \begin{cases} x' = x_0 + k \cdot (x - x_0) \\ \\ y' = y_0 + k \cdot (y - y_0) \end{cases} $$

These equations describe how each point of the original figure is mapped to the homothetic figure with respect to the center O(x₀, y₀).

Example: In this case, the homothety ratio is k=2, and the homothety center is at the coordinates O(2,3), so x₀=2 and y₀=3.

Point A is located at the coordinates (x, y) = (3, 2). I use the equations to calculate the coordinates (x', y') of the corresponding point A' after the geometric transformation: $$ \begin{cases} x' = x_0 + k \cdot (x - x_0) \\ \\ y' = y_0 + k \cdot (y - y_0) \end{cases} $$ Applying the homothety ratio k=2: $$ \begin{cases} x' = x_0 + 2 \cdot (x - 2) \\ \\ y' = y_0 + 2 \cdot (y - 3) \end{cases} $$ Substituting x₀=2 and y₀=3: $$ \begin{cases} x' = 2 + 2 \cdot (x - 2) \\ \\ y' = 3 + 2 \cdot (y - 3) \end{cases} $$ Now, using the coordinates of point A (x=3, y=2): $$ \begin{cases} x' = 2 + 2 \cdot (3 - 2) \\ \\ y' = 3 + 2 \cdot (-1) \end{cases} $$ $$ \begin{cases} x' = 4 \\ \\ y' = 1 \end{cases} $$ So, the coordinates of the corresponding point A' are x'=4 and y'=1. $$ \begin{cases} x' = 4 \\ \\ y' = 1 \end{cases} $$ Using the same method, you can calculate the corresponding points B', C', and D'.

Key Observations

Here are some key observations and properties of homothety:

• Homothety is not an isometric transformation
  Since the absolute distances between points change, it’s clear that, except when k=1, homothety is not an isometric transformation. However, like isometries, homothety preserves certain invariant properties of figures, such as angle magnitudes, collinearity (alignment of points), and proportions between distances of points.
  
• Proportionality of distances
  The distances between points in the homothetic figure are proportional to the corresponding distances in the original figure. Thus, the relative distances between points remain unchanged in homothety (even though absolute distances do not).
  
  

  For example, the segment AB is to segment AC as segment A'B' is to segment A'C'. This proportion holds: $$ \overline{AB}:\overline{AC} = \overline{A'B'}: \overline{A'C'} $$

• Conservation of angles
  The angles in the original figure and in the homothetic figure (the transformed figure) are congruent, meaning they have the same measure.
  
• Invariance of the center
  The center of homothety (O) remains unchanged by the transformation and is the only fixed point in the homothety.
  

  

• When the homothety ratio is k=-1, homothety equals central symmetry
  When the homothety ratio is k=-1, the transformation vectors have the same origin and direction but opposite orientation. In this specific case, the final result is identical to that of a central symmetry with the same center O.
  

  Note: In this particular case, the sides of the original figure and those of the transformed figure are congruent. $$ \overline{AB} \cong \overline{A'B'} \\  \overline{AC} \cong \overline{A'C'} \\ \overline{BC} \cong \overline{B'C'} $$

And so forth.