2D Geometric Transformations and Matrices

Transcription

1 Background: Objects are drawn and moved in 2D space and 3D space on a computer screen b multipling matrices. Generall speaking, computer animation is achieved as follows b repeating steps 1, 2, and 3 below. Basic 2D Matrices: The basic matrices listed below are the matrices that computer programmers would use to construct a geometric transformation which would cause a figure to be translated in a specific direction, rotated about a point, reflected over a line, stretched about a point, shrunk about a point or sheared. All 2D affine geometric transformations can be described as a product of two or more of the basic 3 x 3 transformation matrices. Each of these matrices has an inverse which makes it eas for a computer programmer to reverse or undo a transformation of a figure. Mapping of Preimage Points to Image Points: The mapping of a preimage point in the x- coordinate plane to an image point in the x- coordinate plane is accomplished b multipling the preimage point b a geometric transformation. In the listing below, preimage points (x, ) are represented b a 3 x 1 to the right of the transformation. The resulting image point is represented b the 3 x 1 to the left of the equal sign. The Basic Matrices Translation or Slide Reflection over x-axis Reflection over -axis x + j 1 0 j x k 0 1 k D Geometric s and Matrices 1) Use the coordinates of the current geometric object or shape to calculate the coordinates of a new geometric object. The coordinates of a new object are calculated b multipling the coordinates of the current object b a so that the product will give us the coordinates of an object to that has been translated, rotated about a point, reflected over a line or plane, horizontall stretched/shrunk, or verticall stretched/shrunk. 2) Erase the object from the screen. 3) Use the coordinates of the new object to draw the ogject on the screen and then make the coordinates of the current object equal to the coordinates of the new object. Size from (0, 0) jx j 0 0 x k 0 k x x x x (x + j, + k) <----- (x, ) (x, - ) <----- (x, ) (- x, ) <----- (x, ) (jx, k) <----- (x, ) Rotation about (0,0) Shear x + j 1 j 0 x kx k (x + j, + kx) <----- (x, ) xcos( q ) - Sin( q ) Cos( q ) -Sin( q ) 0 x xsin( q ) Cos( q ) Sin( q ) Cos( q ) ( xcos( q ) - Sin( q ), xsin( q ) + Cos( q) ) < ( x, ) If j > 1, figure is expanded horizontall b a factor of j. If j < 1, figure is shrunk horizontall b a factor of j. If k > 1, figure is expanded verticall b a factor of k. If k < 1, figure is shrunk verticall b a factor of k. If θ > 0 0, the figure is rotated CCW θ degrees about (0,0). If θ < 0 0, the figure is rotated CW θ degrees about (0,0).

2 Matrix Multiplication and Geometric Linear s A linear geometric transformation of a figure is accomplished b performing a series of basic transformations to the figure in some specific order. The order in which two transformations are performed is important because geometric transformation operations and multiplication are generall non commutative. In order to achieve desired results, one needs to pa attention to the order in which the operations are performed. A linear transformation is constructed b calculating the product of basic geometric transformation matrices. Each in the product is a basic geometric transformation which corresponds to a basic geometric transformation. Due to how multiplication works and the fact that the geometric transformation will be multiplied b a 3-row b n-column preimage of points, the matrices must be listed in right-to-left order in the product of transformation matrices. Because multiplication is associative, it is not necessar to include an grouping smbols in the product expression. The following examples illustrate how to use based transformations to do a geometric transformation on a preimage figure. Example 1: Rotate polgon ABCDE counterclockwise about the origin Preimage of nonconvex polgon to the right. Rotation (120 0 CCW rotation about (0,0). Sin(120 0 ).866, Cos(120 0 ) 0.5 Image of nonconvex polgon to the left. A E D B C Example 2: Horizontal shear transformation with horizontal shear factor j Preimage of rectangle Shear transformation Image of parallelogram.

3 Example 3: Expand b a factor of 1.5 and rotate 90 0 clockwise about (-4, -6) [T1] [T2] [T3] Preimage of smaller flag to the left Translation [T2][T0][T3][T1] [T2][T0][T3][T1] Rotation : (90 0 CW rotation About (0,0). Notice that Sin(-90 0 ) -1 and Cos(-90 0 ) 0 [T0] Translation Size transformation with scale factor of 1.5 and expanded about (0, 0). Geometric transformation that causes the figure to be enlarged b a factor of 1.5 and rotated 90 0 CW about (-4, -6) Comments: 1. [T1] and [T2] are inverse matrices. You should verif that [T1][T2] [T2][T1] equals the identit. 2. Let [I0] equal the inverse of [T0]. [I0] is a 90 0 counterclockwise rotation about (0, 0). Use the facts that Sin(90 0 ) 1 and Cos(90 0 ) 0 to find [I0]. Then verif that [T0][I0] [I0][T0] the identit. 3. Let [I3] equal the inverse of the size transformation T3. [I3] is a size transformation with scale factors j and k both set to 2/3. Find [I3] and verif that [T3][I3] [I3][T3] the identit. 4. The inverse of is the product of inverse matrices, but multiplied in reverse order. The inverse of equals [I1][I3][I0][I2]. This can be easil verified b observing the following: [I1][I3][I0][I2] [T2][T0][T3][T1] [T2][T0][T3][T1] [I1][I3][I0][I2] the identit. Like multiplication of real numbers, multiplication is associative. When we multipl middle pairs of matrices, the product collapses to the identit. Consider the following product of real numbers: 5/4(3/2(5*1/5)2/3)4/5 5/4(3/2 (1)2/3)4/5 5/4(1)4/5 1

4 Example 4: Reflect the upper nonconvex polgon over the line x + 4. Preimage of upper nonconvex polgon. Translation [T0] First translation 45 0 CW rotation Translation [T1] [T4] Reflection over x-axis Rotation (45 0 CW rotation about (0,0). Sin(-45 0 ) -.707, Cos(-45 0 ).707 Rotation (45 0 CCW rotation about (0,0). Sin(45 0 ).707, Cos(45 0 ) [T2] [T3] [T1] [T3][T4][T2][T0] Reflection over x-axis after 45 0 CW rotation 45 0 CCW rotation Geometric transformation that causes the polgon to be reflected over the line x [T1][T3][T4][T2][T0] Comments: 1. [T0] and [T1] are inverse matrices. You should verif that [T0][T1] [T1][T0] equals the identit. 2. Show wh matrices [T2] and [T3] are inverse matrices. 3. What is the inverse of [T4]? 4. The inverse of is the product of inverse matrices, but multiplied in reverse order. The inverse of [I0][I2][I4][I3][T1]. This can be easil verified b observing the following: [I0][I2][I4][I3][I1] [T1][T3][T4][T2][T0] [T1][T3][T4][T2][T0] [I0][I2][I4][I3][I1] the identit. (Refer to the comments on previous page.)

5 Geometric Exercises Name Exercise 1: Multipl the transformation b the preimage. Use to draw the transformation image. Image Translate Preimage Exercise 2 : Reflect over x-axis Translate The figure will be first translated and then reflected over the x-axis. matrices must be listed in right to left order. Multipl the transformation b the preimage. Use to draw the transformation image. Image Preimage

6 Exercise 3: Translate Reflect over x-axis The figure will be first reflected over the x-axis and then translated. Matrices must be listed in right to left order. ) Multipl the transformation b the preimage. Use to draw the transformation image. Image Preimage Exercise 4 : CCW rotation of the figure about the point (8, -2) Translate back to center of rotation Rotate 90 0 CCW about (0,0) Translate to center of rotation to (0, 0) Multipl the transformation b the preimage. Use to draw the image. matrices must be listed in right to left order. Image Preimage

7 5. Write a translation that would cause a figure to be slid 10 units to the left and 5 units up. 6. Write the inverse of the from exercise (5) above. Verif that it is the inverse. 7. The to the right could be used to rotate a figure about (0,0). Write the inverse of this. Verif that it is the inverse Write a transformation that would cause a figure to be reflected over the line x. Hint: Recall that reflecting the point (x,) over the line x is equivalent to mapping (x,) to (,x). 9. Write the inverse of the from exercise (8) above and verif that it is the inverse. 10. Write a transformation that would cause a figure to be stretched horizontall b a factor of 7/4 and shrunk or squeezed verticall b a factor of 2/ Write the inverse of the from exercise (10) above. Verif that it is the inverse.

8 12. Write a transformation that would cause a figure to be slide 7 units to the right and then reflected over the x-axis. 13. Write the inverse of the from exercise (12) above. Verif that it is the inverse. 14. Write a transformation that would cause a figure to be reflected over the -axis and then slide 5 units down. 15. Matrix [F] below is a 72 0 CCW rotation. Find the product [F] [F] [F] [F] [F] or [F] 5. [F] Write a transformation that would cause a figure to be rotated counterclockwise about (0,0). Use the Sin and Cos kes on our graphing calculator. Round elements to nearest Warning! Make sure that our calculator's angle mode is set to degree and not radian. 17. Write a transformation that would cause a figure to be rotated clockwise about (0,0). Use the Sin and Cos kes on our graphing calculator. Round elements to nearest Warning! Make sure that our calculator's angle mode is set to degree and not radian. 18. Write a transformation that would cause a figure to be rotated 90 0 counterclockwise about the point (-3,2).