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Dilations Around a Point, and their Dilation Factor
There Are Three Interactive Dilation Applets below. Scroll down to use them as tools for your homework:
1) Uniform Dilation Centered Around the Origin (0,0) in the XYplane
In our first interactive dilation applet here:
 The dilation is centered around the Origin = (0,0)
 Red points from A_{1} to F_{1} are movable via the cursor. Place them where your dilation homework requires them to be
 Each blue point from A_{2} to F_{2} is the dilated image of the corresponding red point
 You select the dilation factor value using the purple slider near the left margin
 You can reset the applet to its original configuration by clicking on the upper right corner icon
The uniform dilation shown above, centered around the origin (0,0) in the XYplane, is the most widely known type of dilation transformation.
It is a simple example of a linear transformation of the XYplane onto itself, given by the function formula:
where r stands for the dilation factor's numerical value, while x and y are the coordinates of a general point.
In this type of dilation, centered at the origin, the stretching or shrinking indicated by the dilation factor is uniformly and simultaneously applied in all directions of the XYplane, moving each point away from, or towards the Center of Dilation, in this case it being the origin O = (0,0).
There is no difference between centering the dilation on the origin, and centering it on (0,0) because (0,0) is the origin.
In the XYplane, "the origin," and (0,0) are the exact same place, the same point.
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 math, "dilation" may also mean shrinking, depending on a number called the Dilation Factor.
The difference between stretch and dilation in math is that "stretch" is not a technical word, so it keeps its natural meaning that implies some distance is becoming larger.
In contrast, "dilation" in math is a technical word whose meaning is only partially related to its actual, plain English meaning.
The reason is, if the dilation factor has an absolute value smaller than one, then dilation has the effect of shrinking geometric figures, reducing them in size, instead of enlarging them.
2) Uniform Dilation in the XYplane Centered on C, a Movable Point Not the Origin
This is our second interactive dilation applet, for dilation centers other than the origin:
 This dilation is centerd around point C = (h,k). The dilation center is not at the origin.
 You can change the center of this dilation, it is movable via the cursor.
 Red points from A_{1} to A_{6} are also movable via the cursor. Put them where needed for your dilation homework
 Each blue point from B_{1} to B_{6} is the dilated image of its red point handle with respect to point C = (h,k), and a dilation factor = t
 You select the dilation factor value using the purple slider near the left margin
 You can reset the applet to its original configuration by clicking on the upper right corner icon
The dilation shown in our second applet above, is uniform but it is not centered on the origin.
Therefore, it is not a linear transformation of the XYplane, because here the dilated image of the origin is not the origin, and this disqualifies it from being a linear transformation (except when the dilation factor equals 1).
This type of uniform dilation centered at a point that is not the origin is given by the function formula:
where t stands for the dilation factor's numerical value; h and k are the coordinates of point C = (h,k), the center of dilation; while x and y are the coordinates of any general point being dilated with respect to C.
The dilation factor becomes zero at the center of the vertical slider bar.
The value increases when the purple point slides up, and it becomes negative in the lower half of the slider.
 Values larger than 1 for the dilation factor will make shapes grow in size, and move farther away from the center of dilation.
The bigger the dilation factor's value, the bigger the resulting dilated shape.
 When the dilation factor equals 1, the dilation has no effect on the points, it leaves them where they are.
In this case the dilation transformation becomes the identity function.
 A dilation factor smaller than one but bigger than zero shrinks the shapes in size, bringing them closer to the dilation center.
The image of a point in this case will be somewhere in between the original point and the center of dilation.
 When the dilation factor equals zero, it has the effect of shrinking the whole plane into a single point, the center of dilation.
In this case all image points collapse together onto the center of dilation C = (h,k)
 A negative value for the dilation factor makes each point's image to move across the dilation center from the original point.
In this case the dilated image shapes, whether reduced or augmented in size, look like rotated by 180°.
 A dilation factor value of 1 has an effect identical to that of a 180° rotation around the center of dilation.
This particular dilation does not change the size of geometrical objects.
 The only two dilations that preserve the size of geometrical objects are those with dilation factor = 1, and those with dilation factor = 1.
In the interactive dilation applet above, each time a red segment (either segment A_{2}A_{3}, or one of the sides of triangle A_{4} A_{5} A_{6}) pases through the center of dilation (point C), the blue segment corresponding to its dilated image will pass through C as well.
This is because the center of dilation does not move under the dilation. It is its own image.
3) Two Simultaneous, Independent Dilations, Each with its Own Center and Dilation Factor
In this interactive dilation applet, you can explore the effect of two simultaneous, independent dilations on the same four points:
 One dilation is centered at the origin, with its dilation factor value r being set by the purple slider on the left
 The other dilation is centerd around point Q = (h,k). Its dilation factor t is set by the blue slider near the right margin
 You can move around Q, the center of the second dilation. The center of the first dilation stays fixed at the origin
 Red points from A_{1} to A_{4} are movable via the cursor. Place them anywhere you want them to be for your dilation homework
 Each purple point from B_{1} to B_{4} is the dilated image of its red point handle with respect to the origin, by a dilation factor equal to r
 Each blue point from C_{1} to C_{4} is the dilated image of its red point handle with respect to point Q, by a dilation factor equal to t
 You can reset the applet to its original configuration by clicking on the upper right corner icon
In the interactive dilation applet above, you can see and explore the two different dilation effects on each red point, caused by two different, simultaneous uniform dilations independent from each other.
One dilation, D_{O}, is centered on the origin O = (0,0) and has a dilation factor = r that is separate and independent from the dilation factor = t working in the other dilation, D_{Q}, that is centered around the point Q = (h,k).
D_{O} is a linear transformation, and D_{Q} is not. Their function formulas are given below:
By the differences and similarities in those formulas, you can notice D_{Q} is a variation of D_{O}. They are conceptually related as follows:
1) For D_{Q} we first translate the point Q = (h,k) to the origin, resulting in point P = (x,y) being dragged along to the new location ( x  h, y  k ).
2) Then we aply a dilation from the origin to that point but with a dilation factor = t, sending the point to ( t (x  h), t (y  k) ).
3) Finally we translate the new point using the same translation that would send the origin back to point Q = (h,k), and we end up with the coordinates shown in the formula for D_{Q}.
How to dilate images point by point
All four dilations shown in the three iteractive dilation applets above are examples of uniform dilations.
The term Uniform Dilation refers to a dilation transformation where the same dilation factor is applied to every point on the plane, in every direction from the dilation center C.
In a uniform dilation from a center point C, and given any other point P, if you want to locate the dilated image of P, you simply follow these steps:
1) Draw the straigth line L that passes through C and P.
2) Measure the distance between C and P along this straight line. Let's call this distance d.
3) Multiply d times the dilation factor t, obtaining the number td.
4) Along line L, locate the point that is exactly at a distance td away from dilation center C.
That point will be the dilated image of point P.
At step (4) above, the positive direction away from dilation center C on line L is on the side of L containing point P.
This is important because dilation factors can be negative, sending the image point across C from point P.
The following diagram illustrates the procedure described in steps 1 through 4 above, using a dilation factor value t = 2:
Dilation in the XYplane, and dilation in the Euclidean plane
The two most basic and widely used models of the plane in math are the Cartesian plane (also called the coordinate plane, or the XYplane), and the Euclidean plane. Dilation transformations can be defined, and studied in both models.
The XYplane comes with a square grid, the Xaxis, the Yaxis, and the Origin = (0,0).
The origin is the center of this (x,y) system of coordinates that represents each point with a pair of numbers.
Shapes and figures are represented as sets of points through equations or inequalities.
Geometrical transformations such as dilation are represented via mathematical formulas as vector functions of two variables.
As shown above, a dilation centered on the origin with dilation factor t, is multiplicatively defined by the formula
D(P) = D(x,y) = (tx, ty)
This dilation formula takes as input the two coordinates, x and y, of point P, and it produces as output the point D(P), given by coordinates (tx, ty).
We call this point D(P) = (tx, ty), the "dilated image" of point P = (x,y), with respect to the origin, and dilation factor t.
In the Euclidean plane model there is no (x,y) grid. This model is all drawings on a blank sheet of paper.
The Euclidean plane is the model where middle school students work out doublecolumn geometry proofs based mostly on geometry postulates, and properties of triangles, straight lines, angles, and circles.
The diagram above is an example of how the dilation transformation may be illustrated in the Euclidean plane model, without any coordinates or mathematical formulas, using instead drawings with straight lines, or triangles.
Uniform dilations produce similar figures
Whether in the Euclidean plane, or in the XYplane, uniform dilation is a geometric transformation that preserves the angles and shapes of geometrical objects but almost always changes their size.
Because it generally does not preserve the size of geometrical figures, dilation is not an isometry.
Dilated images of geometrical objects are similar to the original objects, meaning they may be reduced, or enlarged, and even look rotated in comparison to the original object but they always keep the same shape, and the same angles.
The dilated image of a straight line is a straight line. The dilated image of a square is a square. The dilated image of a circle is a circle. The dilated image of a triangle is a similar triangle. The dilated image of a rectangle is a similar rectangle. The size and location of the dilated image usually changes but the shape remains the same.
Zoomin and Zoomout are dilations
To explain or understand the meaning of the dilation transformation, probably the easiest way is to refer to some of its applications you may have seen in reallife, like the Zoomin and Zoomout buttons in internet maps or image drawing or photo editing applications.
In such applications, when you click on the Zoomin button, it produces the effect of a dilation transformation with a dilation factor greater than 1, because it enlarges a part of the image around the center of the screen, that acts as the dilation center. It may also add some finer detail that was not visible before.
When you click on the Zoomout button, it produces the effect of a dilation transformation with a positive dilation factor smaller than 1, because it causes the image on the screen to shrink into a smaller area.
Dilation in the movies
Have you seen any of these movies?
 AntMan (2015)  Paul Rudd, Michael Douglas, Corey Stoll
 Honey, I Shrunk the Kids (1989)  Rick Moranis, Matt Frewer, Marcia Strassman
 Innerspace (1987)  Dennis Quaid, Martin Short, Meg Ryan
 Fantastic Voyage (1966)  Stephen Boyd, Raquel Welch, Edmond O'Brien
 The Phantom Planet (1961)  Dean Fredericks, Coleen Gray, Anthony Dexter
 The Incredible Shrinking Man (1957)  Grant Williams, Randy Stuart, April Kent
The common theme linking these movies is that people get miniaturized in them.
Miniaturizing people is an excellent example of uniform dilation transformation with a dilation factor between 0 and 1.
What does dilation factor of 2/3 mean?
Suppose you take a photo, and you order prints of it in the following sizes:
 4R (4 inch by 6 inch)
 8R (8 inch by 12 inch)
 12R (12 inch by 18 inch)
Because the 8R print dimensions are double than those of the 4R print, these two print sizes are in the ratio of 2:1
Similarly the 12R print dimensions are triple than those of the 4R print, so these two sizes are in the ratio of 3:1
Now, comparing the 8R print size to the 12R print size, you can apply a dilation with dilation factor of 2/3 to reduce the 12R size, and make it match the 8R size. So a dilation factor of 2/3 means a reduction to 67% of the original size.
Similarly, you can apply a dilation with dilation factor 3/2 to enlarge the 8R size, and make it match the 12R size.
Thus, a dilation factor of 3/2 means an enlargement by 50% of the original size, meaning to 150% of the original size.
Other types of dilation transformations:
In more general terms, the main geometric cases of dilation transformations are:
 Dilation around a point called the Center of Dilation.
 Dilation with respect to a line called the Axis of Dilation.
 Dilation with respect to two intersecting lines, with two different dilation factors, one per axis line.
 Dilation with respect to a plane ( in 3D space ).

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