The English noun "dilation" means dilatation, expansion, or stretch.
As a verb, "to dilate" means to expand, or cause to expand, to enlarge, to make wider or larger, or to become wider or larger.
The adjective "dilated" means widened, distended, expanded, extended, or stretched.
In geometry, there are a few possible different interpretations (specific implementations) of the above, very general "dilation" concept.
First, let's notice that the indicated stretching, or expansion, can uniformly and simultaneously be applied in all directions of the XY-plane, by moving each point away from, or towards a fixed point called the "center of dilation."
As an alternative, we could possibly consider stretching / expanding the XY-plane in only one direction (horizontally, or vertically, for example), by moving each point away from, or towards a straight line called the "axis of dilation."
These interpretations of Dilation are specific examples of Linear Transformations, a much more general concept (for example, rotations around the origin are not dilations but another type of linear transformations).
All dilations shown in the interactive worksheet at the top of this page are examples of uniform dilation from a center, the first interpretation mentioned above.
They are based on the point (0,0), the origin of the XY-plane. We can call them "uniform," or "central" dilations because under these transformations, each red point, as part of the dilated image of the original shape (shown in blue) moves on a straight line directly away from, or towards the origin (0,0).
In math, we use coordinates (x,y) to represent points in the XY-plane.
To uniformly dilate a shape with respect to the origin, we multiply both coordinates of each of its points by a constant number k called "dilation factor."
Therefore, the dilations shown in this page are multiplicatively defined by the formula Q = F(P) = F(x,y) = (kx, ky)
This dilation formula takes as input the two coordinates, x and y, of point P, and it produces as output the point Q = F(P), given by coordinates (kx, ky).
We call point Q = (kx, ky), the "dilated image" of point P = (x,y), with respect to the center of dilation (0,0), and dilation factor k.
In this uniform sense, multiplicatively defined with the same dilation factor in both coordinates, Dilation is a geometric transformation that preserves the angles and shapes of geometrical objects but it usually changes their size.
Because it generally does not preserve the size of geometrical objects but only their shape, dilation is not an isometry.
Even though the word Dilation originally means stretch, or expansion, its geometric, technical definition - multiplicatively given by the formula F(x,y) = (kx, ky), allows for dilation to actually mean "contraction" or "shrinking," in some cases. This happens when the dilation factor k is less than 1 in absolute value, like a proper fraction, or a decimal between -1 and 1.
Notice when you set the purple slider button for the dilation factor k to the value 0.5, the red shape becomes smaller than the blue shape, as each red point gets located exactly mid-way between the origin and the corresponding blue point.
Therefore, if k = 0.5, and you move each blue point to a location point on the grid marked with two even number coordinates, like (4,6) for example, then the red points will automatically move to other points on the grid with whole number coordinates (maybe odd numbers but whole numbers anyway) because the corresponding blue point has two even coordinates and the dilation factor k is set to 0.5 = 1/2.
A dilation with factor k=0 sends all points to the origin (0,0). It "collapses" every shape into a single point.
A dilation with factor k=1 keeps each point in its very same place. It does not move any point. It does not change the size of a geometric shape. It does not "do" anything. This particular dilation is also known as the "Identity transformation."
A dilation with factor k = -1 has an effect identical to that of a 180-degree rotation around the origin (0,0). This particular dilation does not change the size of geometrical objects. The only two dilations that preserve the size of geometrical objects are the dilation with factor k=1, and the dilation with factor k = -1.
When one of the sides of the original polygon A1B1C1D1 (the blue one) pases through the origin (0,0), the corresponding side of its dilated image A2B2C2D2 (the red polygon) will pass through the origin as well.
In the interactive GeoGebra worksheet applet presented above, you can move the blue points around the screen, not only one at a time but also two at a time by segments, meaning, you can move each one of A1B1, B1C1, C1D1, and D1A1, the sides of the blue polygon.
I want to thank Kenneth Frank for preparing a Turkish version of this Dilation transformation Geogebra applet.
In this YouTube video Geometry - Dilations the instructor explains the difference between Dilation, and other geometric transformations, as well as Dilation related concepts like Dilation Factor, and Center of Dilation.