Explanation of Mathematics

PIVOT! History & Background Explanation of Mathematics Significance & Application References

Explanation of Mathematics

Translations:


A translation is a rigid motion in which our pre-image is shifted around the coordinate plane. Thus there is no change in size or shape, translations only change the position of the image. For a formula of translations, we have (x,y) -> (x+a, y +b), where a is the amount we are shifting to the right (positive a) or left (negative a), and b is the amount we are shifting up (positive b) or down (negative b).

As seen in the image above, our pre-image (triangle ABC) shifts down 3 and to the left 8, thus for each of our points we can find the new point by applying our formulas. A = (4,6) -> A' = (4-8,6-3)=(-4,3), B =(6,4)->B'=(6-8,4-3)=(-2,1), C = (1,2)-> C' = (1-8,2-3)=(-7,-1).
*Notice how the orientation of the coordinates has not changed. Also notice that the result of performing two translations can be done using a singular translation.

Rotations:


Rotations rotate each point in our pre-image a certain number of degrees around a point. When our degrees are positive the pre-image will rotate counter clockwise and when the degrees are negative the pre-image will rotate clockwise. The rule of rotation about the origin are as follows: rotating 90 deg will have (x.y) -> (-y,x), rotating 180 deg will have (x,y)->(-x,-y), and rotating 270 deg will result in (x,y)->(y,-x). Notice that rotating 360 deg will result in a complete circle and the pre-image will end up in its original position, thus not changing.

An example of a rotation is given above and we can use our given rules to determine our new points. K=(-4,-4)->K'=(4,-4), L=(0,4)->L'=(-4,0), M=(0,2)->M'=(-2,0), N=(-4,-2)->N'=(-2,4).
*Notice how the image is now on its side, but the coordinates are still in the same order if we look at them clockwise. Also notice that performing two rotations is equivalent to performing one singular rotation.

Reflections:


Reflections mirror our pre-image over a line to result in our new image. When points are reflected off the line both the pre-image and the image will be the same distance from the line, but on the opposite side of the line. While there are an infinite number of reflection lines that could be made, we will focus on the basic rules. Reflection over the x-axis will result in (x,y)->(x,-y), reflection over the y-axis will result in (x,y)->(-x,y), reflection over the line f(x)=x will result in (x,y)->(y,x), and reflection over f(x)=-x will result in (x,y)->(-y,-x).

In the above image we can see that our pre-image is being reflected over the line f(x)=x, thus referring back to our rules we can determine the points. A=(-2,2)->A'=D=(2,-2), B=(-6,5)->B'=E=(5,-6), and C=(-3,6)->C'=F=(6,-3).
*Notice that the orientation and order of our shape changed with a reflection.

Isometries:


All of the above mentioned rigid motions are isometries. The one transformation that is not an isometry is dilation. Dilation is the matter of shrinking or growing an image. There are two different kinds of isometries. Direct isometries preserve distance and orientation, rotations and translations are direct isometries. Opposite isometries are isometries that preserve distance, but change the order, or orientation, from clockwise to counterclockwise, or vice versa, reflection is an opposite isometry. (mathwarehouse.com) As we have discussed before, translations and rotations preserve orientation and distance, and reflection does not preserve orientation.

Consider the following example: suppose you have a quarter sitting on your dresser. In the morning you pick it up and put it in your pocket. You go to school, hang out at the mall, flip it to see who gets the ball first in a game of touch football, return home exhausted and put it back on your dresser. Although your quarter has had the adventure of a lifetime, the net result is not very impressive; it started its day on the dresser and ended its day on the dresser. Oh sure, it might have ended up in a different place on the dresser, and it might be heads up instead of tails up, but other than those minor differences it's not much better off than it was at the beginning of the day. From the quarter's perspective there was an easier way to end up where it did. The same effect could have been accomplished by moving the quarter to its new position first thing in the morning. Then it could have had the whole day to sit on the dresser and contemplate life, the universe, and everything.

If two isometries have the same net effect they are considered to be equivalent isometries. With isometries, the “ends” are all that matters, the “means” don't mean a thing.” (infoplease.com) We know that performing any two isometries results in any one isometry. Let’s come up with some rules for those. Practice each of the following sets of translations on this applet:
Translation/Rotation
Translation/Reflection
Translation/Translation
Rotation/Rotation
Rotation/Reflection
Reflection/Reflection


Isometry Applet




3D Geometric Transformations



While isometries are generally pretty understood on the two dimensional scale. When discussing three dimensional geometric transformations can get just a little bit more complicated.

Here is a video explaining those transformations


"Rotation, translation, scaling, and shear compose the set of the elementary 3D geometrical transformations. The corresponding matrices, MT, following the convention of a righthanded coordinate system...Two successive transformations may be combined into a new transformation, the application of which gives the same result as the result given by the successive application of the two transformations."
(Marinakis, 2021, p. 150)