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The purpose of this section is to provide students with an introduction to the transformations reflection, rotation, and translation on The Geometer's Sketchpad. It is assumed in these exercises that students are familiar with the Sketchpad.
To perform a reflection with the Sketchpad, you need to identify a line segment (or line) as a mirror, and an object(s) to be reflected.
1) Draw a line segment AB and construct an object like the one below. For the purpose of this exercise, your object should be a closed figure, and there should be at least one point constructed on the side of the figure (See point H below).
2) Select the vertices of the object and define the polygon interior.
3) Select the entire figure, (drag a rectangle around the perimeter of the figure) and choose reflect from the transform menu.
In the figure above, polygon CDEFG was reflected in line segment AB to produce polygon C¢D¢E¢F¢G¢. C¢D¢E¢F¢G¢ is called the image of the reflection and CDEFG is the pre-image.
1) Explore the effect of dragging different parts of the pre-image. Explore what happens if you drag point A and segment AB.
2) What does it mean for a point to be reflected in a line? Try to obtain a definition of a reflection of a point in a line. Write a definition. Hint: You might try to examine the relationship between the segment AB and the segment CC¢. Explain in your own words how you would construct a reflection of an object using paper and pencil.
3) Measure segments, angles, and area. What effect does a reflection have on length, angle measure, and area? Can you make any observations about congruence? How does the image differ from the pre-image?
4) Construct a segment JK || AB. Mark JK as the mirror and reflect the image C¢D¢E¢F¢G¢in JK. What effect does this reflection have on the image? How does the distance between points in the original pre-image and those in the second image compare to the distance between the parallel lines? Can you prove your conjecture?
5) The second image in this figure is also called a translation. Experiment with the translate command in the transform menu and see if you can determine what a translation does to a pre-image.
1) Draw a new sketch in which two line segments AB and CD intersect at a point E. Draw a figure like the one in the previous sketch. Reflect the figure first in AB and then in CD. Move the lines, points or figures. What do you notice about the position of the second image and the original pre-image?
2) When a figure is reflected in two intersection lines, a rotation is obtained. To rotate a figure about a point, you must mark the center of rotation and specify an angle of rotation. Note that positive angles create a counterclockwise rotation. Your task is to find the center and angle of rotation necessary to produce an image that matches the second image that you have just constructed. Note that if you try something that does not work, simply choose undo from the edit menu and try again.
Write your conjectures below.
3) Hide the lines of reflection and their point of intersection. Try to find the center of rotation without using the lines of reflection.
4) Investigate what happens if you reflect a figure in two intersecting lines in the opposite order.
Record your conjectures from these exercises.
Each of the activities below can be tried using combinations of reflections, rotations and translations.
1) Create an arcade shooting gallery like the one below. Wherever possible, make use of reflections, rotations, and/or translations. Create motion by animating pre-images in creative ways.
2) A figure is said to tessellate a plane if you can fill a plane with the figure and copies of the figure so that there are no gaps or empty spaces. See if you can tessellate a plane with an arbitrary triangle. Can you tessellate a plane with an arbitrary quadrilateral?
3) Poolroom math: In figure 1 below, determine the shortest distance from the ball to the pocket if the ball must hit wall AB. Determine by experimentation a way to determine the path the ball must follow. Suppose the ball must hit both AB and CD. What is the shortest path in this case? What would happen if the pool table were rectangular? Experiment with different configurations. Write a paragraph with any conjectures that you obtain from your experiments. Can you find a strategy to find the shortest distance from the ball to the pocket in each case? What if there are barriers that effect certain paths?
4) Miniature golf: Construct a figure like figure 2 and determine a path that will produce a hole-in-one. Will a shot off the diagonal segment RS go in the hole? Design some additional holes to present as a challenge to others in the class.
Exploring Geometry with the Geometer's Sketchpad, Key Curriculum Press, 1992
Arthur Coxford and Zalman Usiskin, Geometry: A Transformational Approach, Laidlaw Brothers, 1971
Richard G. Brown, Transformational Geometry, Dale Seymour Publications, 1973
Abstract: The purpose of this section is to explore the result of the composite of two reflections in two parallel lines or two reflections in two intersecting lines. This section assumes that students have already defined and constructed reflections. The suggested activities begin with student constructions using mira (compass), ruler and protractor and discussion of their conclusions. These results are extended to algebraic mappings for rotations and translations in the coordinate plane.
Students can begin by constructing a composite of two reflections in two parallel lines like the one below which represents:
Students can investigate and define a translation. Part of the discussion should focus on the characteristics that are preserved under a translation. Experimentation should also lead to the conclusion that the distance between a point and its image (2x) is twice the distance between the two parallel lines (x) and in the same direction as the direction of the lines reflected over. Students might try the composite transformation:
in order to discuss the direction of the translation.
The next transformation that students should look at is a composite in two intersecting lines. The drawing below represents the composite transformation:
Students can investigate and define a rotation from this construction. In order to explore the characteristics of a rotation, students should measure the distances of the image and its pre-image from the center of rotation and show that the angle of rotation is twice the angle between the intersecting lines. In order to discuss the direction of rotation, students may want to explore the composite transformation:
MAPPINGS OF REFLECTIONS
We now enter the Cartesian plane to determine the mapping for the two composite transformations discussed above.
First, students should develop and show the basic mappings listed below:
Next students can show that:
COMPOSITES OF TWO REFLECTIONS IN TWO PARALLEL LINES
Students can now consider various variations of the composite of two reflections in two parallel lines.
For example:
Discussion can then focus on single mappings which are equivalent to a translation. For example the mapping
is an example of a translation of -3 in a direction parallel to the x-axis and +2 in a direction parallel to the y-axis.
The mapping
should be proved and as an extension, students could be asked to find a single mapping equivalent to:
COMPOSITES OF TWO REFLECTIONS IN TWO INTERSECTING LINES
The next class of composite transformations includes mappings like the one below:
, a rotation of 90•.
It might be useful to review the composites studied in part one and discuss the characteristics of a rotation. Also students could consider and classify the composites of two reflections over each of the axes.
For example, , a half-turn about the origin.
As an extension of the above, students could first develop the mapping for a reflection over a line which goes through the origin and has an angle of incidence with the x-axis of ß.
Then, it is possible to develop the mapping for a rotation of 2ß by finding the mapping:
There are many variations to these mappings that can be developed by a class. The group can also investigate the equations of functions after they have been transformed by a rotation or translation, by substituting their transformed mappings.
DESCRIPTION OF TRANSFORMATIONS
NOTES TO THE TEACHER
SOME ACTIVITIES FOR STUDENTS
Abstract: Many different approaches can be taken to the concepts involved in transformational geometry. This paper assumes that the reader already has a working knowledge of transformational geometry and of matrix operations. Resources are given at the end of the paper for those who desire more information concerning transformational geometry. The focus will be upon the use of matrices in transformations on the coordinate plane. The ideas presented are probably most appropriate for students in Algebra II or beyond.
THE MATRICES
Transformational geometry is the study of manipulating objects by flipping, twisting, turning and scaling. In the language of transformational geometry, the object which is altered is called a pre-image and the operation which is applied is a transformation. The resulting figure that arises from the operation of the transformation on the pre-image is called the image. Transformations and their effects on pre-images can take many notational forms, but only the matrix form of notation will be discussed here.
Before transformations can actually be discussed, we need to determine how to represent coordinatized objects in matrix form. In general, we will create a matrix with dimension 2xn to represent any figure where n is the number of points needed to uniquely
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determine the figure. The first row of these matrices will always represent the x-coordinates of these points and the second row will hold the y-values. Unless an order is specifically required, the coordinates of the object may be entered in any desired order. This matrix for any object is called a point matrix. In the case of Figure One, a 2x3 point matrix is needed because all three of the vertices of the triangle are necessary to define the figure. The point matrix representing is: |
Notice that the matrix was defined as , and therefore the columns of the matrix had to correspond to the specific order of the vertices given in the name of the object. Had the triangle not been named specifically, any arrangement of the columns in the matrix would have been another acceptable form for the point matrix.
Geometrically, if our is reflected over the x-axis, then we realize that the y-coordinates of the points in our triangle would be multiplied by negative one, but the x-coordinates would not be changed. Notationally, we say that the image of upon reflection over the x-axis (rx) is . In matrix notation we have:
Up to this point, nothing has been shown to make matrices anything more than an alternate notational form. The power of this form becomes evident when the transformational operations are represented in matrix form and matrix operations are utilized to perform the transformations. The notational form of transformations by matrices is:
where the pre-image and the image matrices are in the 2xn form that was discussed earlier and, in most cases, the transformation matrix is represented by a 2x2 matrix.
The four basic types of geometric transformations are reflections, rotations, translations and dilations. Of these, all but translations are easily represented by a general form of 2x2 matrix. To determine the format for these 2x2 matrices, think about the following matrix problem:
From one point of view, this is a very dull problem. All we have said is that any matrix multiplied by the identity matrix is equal to the original matrix. The true significance of this statement is seen when the product is viewed in terms of the transformation product given above. In this light, the first matrix is identified as representing some transformation, T. The second matrix is a pre-image matrix for the points (1,0) and (0,1) and the third matrix is the image of (1,0) and (0,1) under T. Think about the power of this statement! Given any 2x2 matrix representing some transformation, T, the images of the points (1,0) and (0,1) are represented by a matrix which is identical to the transformation matrix for T. This means that if you want to determine a 2x2 matrix for a transformation, then all you have to do is determine the images of (1,0) and (0,1) under that transformation. These two seemingly insignificant points determine all basic transformation matrices!
THE PROCESS
Let's return to our transformation of from earlier. Recall that we wanted to reflect the triangle over the x-axis. We just showed that to determine a transformation matrix, we need to know the images of (1,0) and (0,1), and by the properties of reflections:
By our previous discovery, this says that the matrix representing rx is given by . Now, if we multiply the rx matrix by the point matrix for , we get:
where the image matrix is identical to the pre-image matrix we found earlier. This verifies our procedure.
What about other types of transformations? The table below follows our critical points throughout various transformations and determines the matrices that represent those transformations.
|
TRANSFORMATION |
POINTS |
MATRIX |
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Reflection over the y-axis ry | ||
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Reflection over y = x ry=x | ||
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90• Rotation R90 | ||
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180• Rotation R180 | ||
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Scale Change, SA,B (A is the horizontal stretch) (B is the vertical stretch) |
A further extension of the matrix idea is the verification of various transformation properties. Recall that in transformational geometry, a rotation is defined as a double reflection over each of two intersecting lines where the magnitude of the rotation is twice the acute angle between the lines. A 90• rotation should then be the result of a composition of two reflections over lines 45• apart. Two transformations from above that match this requirement are ry=x followed by ry. In matrix notation, this is:
While matrix multiplication is not commutative, it certainly is associative and therefore we can multiply the our two transformation matrices and obtain a single matrix representing the overall composition of transformations. Performing this operation, we get:
Notice from our chart that the resulting transformation matrix is the same as the matrix for R90.
Any other composition of matrix transformations or geometric properties can be verified in this manner. You are encouraged to try several different transformation combinations using matrices.
THE PROOFS
By again following the points (1,0) and (0,1) and this time using some trigonometry and geometry, we can verify that the general matrix for a rotation of any angle Q is given by:
RQ =
A proof for this can be found in UCSMP's Functions, Statistics and Trigonometry.
Matrix operations can also be used to quickly verify various trig identities that previously took long class hours to complete. A favorite example is the simultaneous proof of both the sin(a+b) and the cos(a+b) formulas.
Recall that the polar definition of sine and cosine says that RQ(1,0) = (cosQ,sinQ). This means that if you were to rotate the point (1,0) Q degrees (or radians) about the origin, then the coordinates of the image point are (cosQ,sinQ). Since the angle a+b can be thought of as a single angle, we know by definition that Ra+b(1,0) = (cos(a+b),sin(a+b)). By transformational geometry, this is also equivalent to taking the point (1,0) and applying a rotation of a followed by a rotation of b. This is represented by the following matrix equation:
We can use matrix multiplication to simplify the product on the right to obtain:
We now have two different representations of the same ordered pair. This means that the x- and y-coordinates must be identical. This completes the proof of the identity.
THE TIME SAVER
While all of the above properties have been documented in many current textbooks, the following quality of matrix transformations is presented in only a few.
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We know what happens to points when a transformation is applied to them, but how is the area of an object affected? Given the triangle in Figure Two, the area is easily computed to be 6. What would happen to the triangle if the scale change S2,3 were applied? By definition, all of the x-coordinates on the triangle are doubled and the y-coordinates are tripled. If you were to graph the resulting image triangle then you could determine that the area of the new triangle was 36. How does this relate to the matrix transformations? The key this time lies in the determinant of the matrices. Recall that the matrix representing the scale change |
S2,3 is and from matrix operations we know that . The value obtained for det[S2,3] is exactly the scale change for the area of the object under the transformation. The area of the image of triangle ABC under S2,3 must therefore be the product of the original area and the determinant of the transformation matrix, or 36.
While the area of the ABC's image was easy to compute, what would happen if the pre-image had undergone some bizarre transformation? Placing random numbers into a 2x2 matrix to use as a transformation, we can represent the following equation as some random transformation of triangle ABC:
The transformation represented by our random matrix involves a sheer in addition to a few possible other manipulations and the images of the points (0,4), (0,0), and (3,0) are respectively (20,16), (0,0), and (6,-18). While the original figure was a right triangle, the image is not and by traditional techniques, finding the area becomes a bit more difficult. Using matrices, however, the process is exactly the same as before and we find that the new area is equal to the old area multiplied by the determinant of the transform matrix. Performing this computation, we find that the area is 6*(8+30) or 228.
Another nifty application involves finding the area of any ellipse. First realize that any ellipse can be transformed to a congruent ellipse centered on the origin with its major axis along either the x- or y-axis. This new ellipse can also be considered the image of the unit circle under some scale change, SA,B where A is the horizontal stretch applied to the unit circle and B is the vertical stretch. If you were to list enough points from the unit circle in a point matrix to perform the transformation into the ellipse, we would have in matrix form:
Since the area of the unit circle is p and the determinant of the transformation matrix is AB, then the image of our transformed ellipse is pAB. By the arguments of transformational geometry, this argument holds for all ellipses and we have a relatively simple argument for the formula of the area of an ellipse.
CONCLUSION
The approaches and processes developed in transformational geometry provide a particularly valuable insight to many branches of mathematics beyond the purely geometrical. This paper has presented some of the more powerful applications of transformations via matrices. The study of the various forms of transformational geometry has dramatically altered and sharpened this author's view of the world and of mathematics and it is hoped that the beauty of this form will be of value to you also.
REFERENCES
Geometry. University of Chicago School Mathematics Project. Scott, Foresman Publishers.
Advanced Algebra. University of Chicago School Mathematics Project. Scott, Foresman Publishers.
Functions, Statistics, and Trigonometry. University of Chicago School Mathematics Project. Scott, Foresman Publishers.
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