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The concept of geometric transformation, examples of geometric transformations
DEFINITION: The geometric transformations are such functions (unambiguous assignments) which assign point to point, i.e. both their domain and their range are point sets.
During a transformation point \latex{ P' } (pronounced \latex{ P } prime) assigned to point \latex{ P } is called the image of point \latex{ P }.
The image of a figure is the whole of collection image points assigned to the points of the figure.
Example 1
Let us take two intersecting straight lines in the plane: \latex{ f } and \latex{ g. } Let us also take triangle \latex{ ABC } in one of the half-planes defined by straight line \latex{ g }.
Let us project the triangle parallel with\latex{ f } onto straight line \latex{ g } (Figure 1). What will be the image of triangle \latex{ ABC } after this parallel projection?
Figure 1
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ A' }
\latex{ B' }
\latex{ g }
\latex{ f }
Solution
The image of the triangle is a line segment of straight line \latex{ g } (it is line segment \latex{ A’B’ } in Figure 1). The domain of this geometric transformation is the set of the points of the triangle; its range is a line segment of straight line \latex{ g }.
As the range has a point which is the image of several points of the triangle, therefore this transformation is not one-to-one.
If \latex{ f } is perpendicular to \latex{ g }, then we are talking about perpendicular projection.
Example 2
Let us take straight line \latex{ g } and point \latex{ O } not lying on the straight line, and also triangle \latex{ ABC } in one of the half-planes defined by the straight line \latex{ g } so that none of the rays starting from point \latex{ O } and passing through the points of the triangle is parallel with \latex{ g }. (Figure 2)
Let us project the triangle from point \latex{ O } onto the straight line \latex{ g. } What will be the image of triangle \latex{ ABC } after this central projection?
Figure 2
\latex{ A }
\latex{ B }
\latex{ B' }
\latex{ A' }
\latex{ C' }
\latex{ C }
\latex{ O }
\latex{ g }
Solution
The image of the triangle is a line segment of straight line \latex{ g }
(it is line segment \latex{ C’B’ } in Figure 2)
(it is line segment \latex{ C’B’ } in Figure 2)
Similarly to example \latex{ 1 } this geometric transformation is not one-to-one either.
Example 3
Let us take two concentric circles (circles with common centre). Let us assign one point of the circle with longer radius to every single point of the circle with shorter radius as can be seen in Figure 3. Is this transformation one-to-one?
Figure 3
\latex{ P }
\latex{ O }
\latex{ Q }
\latex{ Q' }
\latex{ k }
\latex{ k' }
\latex{ P' }
Solution
This geometric transformation is one-to-one, its domain is the set of the points of circle \latex{ k }; its range is the set of the points of circle \latex{ k' }.
⯁ ⯁ ⯁
In the previous three examples the domain of the geometric transformation was the set of the points a figure in each case. Henceforth we are going to deal with geometric transformations
- the domain of which is the set of the points of the plane or the space;
- the domain and the range of which are the same sets;
- which are one-to-one.
In this school year we are going to brush up, extend and apply our knowledge about congruent transformations while proving different theorems and solving different exercises.
DEFINITION: The geometric transformations, in the case of which the image of any line segment is a line segment the length of which is equal to the original length, are called congruent (distance-preserving) transformations. Another term for this type of transformations is isometry.
