In Figure 25.3, you must be careful in the second drawing. With isometries, the distance between points has to stay the same, so they are all kind of stuck together. That's because where one point goes, the rest follow, so to speak. With translations, for example, you only need to know the initial and final positions of one point. How many points is “a few” depends on the type of motion. It turns out that you only need to know where a few points go in order to know where all of the points go. You might be tempted to think that in order to understand the effects of an isometry on a figure, you would need to know where every point in the figure is moved. The isometries that move it around in the plane. So all congruent triangles stem from one triangle and Isometry (or sequence of isometries) that transforms one triangle into the other. The same plane, it turns out that there exists an If you have two congruent triangles situated in Angle measure is also invariant under an isometry. Collinearity and between-ness are invariant under an isometry. If a property doesn't change during a transformation, that property is said to be invariant. If a point is between two other points before an isometry is applied, it will remain between the two other points afterward. In other words, if three points are collinear before an isometry is applied, they will be collinear afterwards as well. An isometry will not affect collinearity of points, nor will it affect relative position of points. The image of an object under an isometry is a congruent object. I can phrase this in more precise mathematical language. An isometry will not change the size or shape of a figure. With isometries, the “ends” are all that matters, the “means” don't mean a thing.Īn isometry can't change a geometric figure too much. If two isometries have the same net effect they are considered to be equivalent isometries. Then it could have had the whole day to sit on the dresser and contemplate life, the universe, and everything. The same effect could have been accomplished by moving the quarter to its new position first thing in the morning. From the quarter's perspective there was an easier way to end up where it did. 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. 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. 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. In the morning you pick it up and put it in your pocket. It doesn't matter what happens in between.Ĭonsider the following example: suppose you have a quarter sitting on your dresser. In studying isometries, the only things that are important are the starting and ending positions. In this case P = P´ and P is called a fixed point of the isometry. It is possible for a point to end up where it started. Under an isometry, the image of a point is its final position.Īn isometry in the plane moves each point from its starting position P to an ending position P´, called the image of P. These will be discussed in more detail as the section progresses.Īn isometry is a transformation that preserves the relative distance between points. The four types of rigid motion (translation, reflection, rotation, and glide reflection) are called the basic rigid motions in the plane. These transformations are also known as rigid motion. There are many ways to move two-dimensional figures around a plane, but there are only four types of isometries possible: translation, reflection, rotation, and glide reflection. Whenever you transform a geometric figure so that the relative distance between any two points has not changed, that transformation is called an isometry. But the location of the ants relative to each other has not. The location of the ants will change relative to the plane (because they are on the triangle and the triangle has moved). Imagine two ants sitting on a triangle while you move it from one location to another. The word isometry is used to describe the process of moving a geometric object from one place to another without changing its size or shape.
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