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Using Matrices to Perform Geometric Transformations

Using Matrices to Perform Geometric Transformations. Kendalyn Paulin. Review of Basic Transformations. Translation Reflection Rotation Dilation. How do Matrices apply to Transformations?. Remember we can translate a figure up, down, left and right.

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Using Matrices to Perform Geometric Transformations

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  1. Using Matrices to Perform Geometric Transformations Kendalyn Paulin

  2. Review of Basic Transformations • Translation • Reflection • Rotation • Dilation

  3. How do Matrices apply to Transformations? • Remember we can translate a figure up, down, left and right. • When we do that we are changing the x and y coordinates of the original figure

  4. Translating a Figure • Say we have a triangle with coordinates: A(0,0), B(2,5) and C(7,-1) shown below. The Matrix form would look like this:

  5. Translate • Say you want to translate the figure 4 units to the left and 3 units up. You can do this by adding the translation matrix to the original matrix. The result is the final coordinates of the new figure.

  6. What is a Matrix? A matrix is a 2D array of numbers which can have any width and height. The one below had a height and width of 2. So it is called a 2x2 matrix (said “two-by-two”).

  7. cont They are usually stated by their height first, then their width. The one below would be a 4x3 matrix.

  8. Translation Matrices • Add these matrices to translate figure…. Up x units Down x units Right x units Left x units

  9. Adding Matrices Add the values of the corresponding positions to each other. Ex:

  10. Adding Matrices • Can you add two matrices that are different sizes? = ? +

  11. Subtracting Matrices • How do you think we can subtract two matrices? • Is it the same process as addition? • Why or why not?

  12. Subtracting Matrices • Same as addition, but subtracting instead. Once again, matrices must be of the same size. Ex:

  13. Original Triangle Dilated by a Factor of 2

  14. Dilate a figure • In order to dilate a figure, scalar multiplication is used. To dilate the triangle by a factor of 2, just multiply the matrix by 2.

  15. Scalar Multiplication • In the scalar multiplication, every entry is multiplied by a number, called a scalar. In this example the number being multiplied by is 2. Ex:

  16. Other Dilations • You can also dilate the figure by a fraction, this will make the triangle smaller. If you dilate by a factor ½, the triangle will be half as big as it originally was. You can investigate this on your own.

  17. Multiplying Matrices • Multiplying matrices will be investigated in a later course. This lesson will only briefly show multiplication. • Here is what a resulting matrix looks like. • We will use excel to do our multiplication matrices.

  18. Example (2X3) (3X1) (2X1) *Don’t worry about being able to do this procedure. We will use excel!

  19. Reflection and Rotation • These transformations will be investigated using Microsoft Excel. • We will review our findings in the next slides.

  20. What transformation matrices to you multiply to do what? • Image stays the same • Reflect over x axis • Reflect over the y axis

  21. What transformations? • Image dilates by 2 • Rotates image 90 degrees clockwise • Dilates the image by a factor of 2 then rotates the image 90 degrees clockwise

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