Dilation: Definitions and Examples

Dilation: Definitions, Formulas, & Examples

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    Dilation: Understanding the Transformation of Geometric Figures

    Geometric transformations are essential in the study of mathematics. One of these transformations is dilation, which is often used in various applications such as computer graphics, image processing, and architecture. In this article, we will delve into dilation, its definition, examples, and applications.

    Definition of Dilation

    Dilation is a geometric transformation that increases or decreases the size of a geometric figure without altering its shape. In other words, dilation is the transformation that scales an object by a certain factor, but preserves its shape and orientation. The scaling factor used in dilation can be greater than one, which results in an enlargement, or less than one, which results in a reduction.

    Dilation is often represented by the following notation: Dk(P), where P is a point, and k is the scaling factor. Dilation can be performed with respect to a center of dilation, which is a fixed point about which the object is scaled.

    Examples of Dilation

    To better understand dilation, let us look at some examples.

    Example 1: Enlargement

    Consider the rectangle ABCD with vertices A(-1,1), B(3,1), C(3,4), and D(-1,4). We will perform an enlargement of this rectangle with respect to the center of dilation O(0,0) and a scaling factor of 2.

    To perform this dilation, we need to find the image of each point of the rectangle after scaling. The image of point A is A'(Dk(OA)), where OA is the vector from O to A. Therefore, A’ is (2(-1), 2(1)), or (-2, 2). Similarly, we can find the images of the other points of the rectangle: B'(6, 2), C'(6, 8), and D'(-2, 8).

    The resulting figure after dilation is the rectangle A’B’C’D’, which is twice the size of the original rectangle.

    Example 2: Reduction

    Consider the triangle ABC with vertices A(1,2), B(3,6), and C(5,2). We will perform a reduction of this triangle with respect to the center of dilation O(2,2) and a scaling factor of 0.5.

    To perform this dilation, we need to find the image of each point of the triangle after scaling. The image of point A is A'(Dk(OA)), where OA is the vector from O to A. Therefore, A’ is (0.5(1-2)+2, 0.5(2-2)+2), or (1.5, 2). Similarly, we can find the images of the other points of the triangle: B'(2, 4), and C'(4, 2).

    The resulting figure after dilation is the triangle A’B’C’, which is half the size of the original triangle.

    Example 3: Enlargement and Reduction

    Consider the circle with center O(0,0) and radius 2. We will perform an enlargement of this circle with respect to the center of dilation O(0,0) and a scaling factor of 3, followed by a reduction with respect to the center of dilation O(0,0) and a scaling factor of 0.5.

    To perform the first dilation, we need to find the image of each point of the circle after scaling. The image of point A on the circle is A'(Dk(OA)), where OA is the vector from O to A. Therefore, A’ is (3(0), 3(2)), or (0, 6). Similarly, we can find the images of the other points on the circle.

    After performing the first dilation, we obtain a new circle with center O and radius 6, which is three times the size of the original circle.

    To perform the second dilation, we need to find the image of each point of the enlarged circle after scaling. The image of point A’ on the circle is A”(Dk(OA’)), where OA’ is the vector from O to A’. Therefore, A” is (0.5(0-0)+0, 0.5(6-0)+0), or (0, 3). Similarly, we can find the images of the other points on the circle.

    The resulting figure after dilation is the circle O” with center O and radius 3, which is half the size of the enlarged circle.

    Example 4: Dilation in 3D

    Dilation is not limited to two-dimensional figures. It can also be applied to three-dimensional figures. Consider a rectangular prism with vertices A(1,1,1), B(3,1,1), C(3,3,1), D(1,3,1), E(1,1,3), F(3,1,3), G(3,3,3), and H(1,3,3). We will perform a dilation of this rectangular prism with respect to the center of dilation O(0,0,0) and a scaling factor of 2.

    To perform this dilation, we need to find the image of each point of the rectangular prism after scaling. The image of point A is A'(Dk(OA)), where OA is the vector from O to A. Therefore, A’ is (2(1), 2(1), 2(1)), or (2, 2, 2). Similarly, we can find the images of the other points of the rectangular prism.

    The resulting figure after dilation is the rectangular prism A’B’C’D’E’F’G’H’, which is twice the size of the original rectangular prism.

    Example 5: Dilation of a Fractal

    Dilation can also be applied to fractals, which are complex patterns that repeat themselves at different scales. Consider the Sierpinski triangle, which is a fractal that is created by repeatedly dividing an equilateral triangle into four smaller triangles, and removing the central triangle. The Sierpinski triangle is an example of a self-similar fractal, which means that it has the same shape at different scales.

    We can perform a dilation of the Sierpinski triangle with respect to the center of dilation O(0,0) and a scaling factor of 2. To perform this dilation, we need to find the image of each point of the Sierpinski triangle after scaling. We can use the fact that the Sierpinski triangle is self-similar, and that each smaller triangle is similar to the original triangle.

    The resulting figure after dilation is the Sierpinski triangle with larger triangles at each scale.

    Applications of Dilation

    Dilation has numerous applications in various fields, including computer graphics, image processing, architecture, and engineering. In computer graphics, dilation is used to scale and transform images to create visual effects. In image processing, dilation is used to sharpen edges and remove noise from images. In architecture and engineering, dilation is used to create larger or smaller models of buildings and structures.

    Quiz: Test Your Knowledge of Dilation

    1. What is dilation? a. A geometric transformation that preserves shape and orientation. b. A geometric transformation that alters shape and orientation. c. A geometric transformation that preserves shape but alters orientation. d. A geometric transformation that alters shape but preserves orientation.
    2. How is dilation represented? a. Dk(P) b. b. Dk(O,P) c. Dk(O,P,k) d. Dk(P,k)
    3. What is the center of dilation? a. The point that is being dilated. b. The point that remains fixed during dilation. c. The point that moves during dilation. d. The point that determines the scale factor of dilation.
    4. What is the scale factor of dilation? a. The distance between the original point and its image after dilation. b. The ratio of the distance between the original point and the center of dilation, and the distance between the image and the center of dilation. c. The distance between the center of dilation and its image after dilation. d. The ratio of the distance between the center of dilation and the original point, and the distance between the center of dilation and its image after dilation.
    5. What is the effect of a dilation with a scale factor greater than 1? a. The figure is enlarged. b. The figure is reduced. c. The figure is reflected. d. The figure is rotated.
    6. What is the effect of a dilation with a scale factor between 0 and 1? a. The figure is enlarged. b. The figure is reduced. c. The figure is reflected. d. The figure is rotated.
    7. What is the effect of a dilation with a scale factor of 1? a. The figure is enlarged. b. The figure is reduced. c. The figure remains the same size. d. The figure is reflected.
    8. What is the effect of a dilation with a negative scale factor? a. The figure is enlarged. b. The figure is reduced. c. The figure is reflected. d. The figure is rotated.
    9. What is the relationship between similar figures and dilation? a. Dilations preserve similarity. b. Dilations do not preserve similarity. c. Dilations preserve congruence. d. Dilations do not preserve congruence.
    10. Can dilation be applied to three-dimensional figures? a. Yes, dilation can only be applied to three-dimensional figures. b. No, dilation can only be applied to two-dimensional figures. c. Yes, dilation can be applied to both two-dimensional and three-dimensional figures. d. No, dilation cannot be applied to any type of figure.

    Answers:

    1. d
    2. c
    3. b
    4. b
    5. a
    6. b
    7. c
    8. c
    9. a
    10. c

    Conclusion

    Dilation is a powerful geometric transformation that has numerous applications in various fields. It is a useful tool for scaling and transforming figures to create visual effects and remove noise from images. Dilations can be applied to both two-dimensional and three-dimensional figures, and they preserve similarity. Understanding dilation is essential for anyone interested in geometry, computer graphics, or image processing.

     

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