Presentation reflection
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This document discusses 2D transformations through reflection. It defines reflection as producing a mirror image and describes how reflection rotates an object 180 degrees. It then provides examples of reflection in microscopes, telescopes, and overhead projectors. The document outlines the main types of reflection as about the x-axis, y-axis, perpendicular to the xy-plane through the origin, along the line y=x, and along the line y=-x. Graphical representations and matrix transformations are provided for each type of reflection. Examples are worked through to find the coordinates after reflection. The document concludes by asking for any questions.
Presentation reflection
- 1. TOPIC REFLECTION (2D TRANSFORMATION) PRESENTED BY: RITESH VERMA (31401219005) SUNNY KUMAR SAH (31401219002) Slide 1
- 2. CONTENTS What is Reflection? And some of its example. Different kinds of Reflection. Any Questions? References Slide 2
- 3. Q. What is Reflection? Reflection is a transformation that produces a mirror image. • 2D reflection in computer graphics is a kind of rotation where the angle of rotation is 180degree. • The reflected object is always formed on the other side of mirror. • The size of reflected object is same as that of original object. Slide 3
- 4. Examples of Reflection 1 2 3 A microscope uses a mirror to reflect light to the specimen under the microscope. An astronomical reflecting telescope uses a large parabolic mirror to gather dim light from distant stars. A plane mirror is used to reflect the image to the eyepiece. An overhead projector as in figure uses a concave mirror to reflect light from the object to the screen. Slide 4
- 5. Different kinds of Reflection There are generally 4-5 kinds of reflection 1) Reflection of an object about an x-axis or (line y=0). 2) Reflection of an object about an y-axis or (line x=0). 3) Reflection of an object relative to an axis perpendicular to the xy plane and passing through co-ordinate origin. 4) Reflection of an object with respect to the line y=x. 5) Reflection of an object with respect to the line y=-x. Slide 5
- 6. 1) Reflection of an object about an x-axis or (line y=0). x x y -y x-axis y-axis Reflection along X-axis: In this kind of Reflection, the value of X is positive, and the value of Y is negative. We can represent the Reflection along x-axis by following equation- Here, For point P1 X1 = X0 and Y1 = –Y0 We can also represent Reflection in the form of matrix– Homogeneous Coordinate Representation: We can also represent the Reflection along x-axis in the form of 3 x 3 matrix- Original Reflection 0 Slide 6 = =
- 7. Example:- A triangle ABC A(2,2), B(4,2) and C(4,4) is reflected about X-axis. Find its new co-ordinate of A1, B1 & C1. 0 1 2 3 4 1 2 3 4 -4 -3 -2 -1 A(2,2) B(4,2) C(4,4) A1(2,-2) B1(4,-2) C1(4,-4) Graphical Representation X1 Y1 = 1 0 0 -1 Y X 1 0 0 -1 2 2 = 1*2 + 0*2 0*2 + (-1*2) = -2 2 = Solution:- Here the given triangle ABC is reflected about X-axis. For A (2,2) Similarly, For B1 And C1 the points are B1(4,-2) and C1(4,-4) A1(2,-2) y x -x -y Slide 7 A1 =
- 8. x -x y y y-axis Original Reflection x-axis 2) Reflection of an object about an y-axis or (line x=0). Reflection along Y-axis: In this kind of Reflection, the value of X is negative, and the value of Y is positive. We can represent the Reflection along y-axis by following equation- Here, For point P1 X1 = –X0 and Y1 = Y0 We can also represent Reflection in the form of matrix– Homogeneous Coordinate Representation: We can also represent the Reflection along y-axis in the form of 3 x 3 matrix- 0 Slide 8 = =
- 9. 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 A(2,2) B(4,2) C(4,4) B1(-4,2) A1(-2,2) C1(-4,4) Example:- A triangle ABC A(2,2), B(4,2) and C(4,4) is reflected about Y-axis. Find its new co-ordinate of A1, B1 & C1. X1 Y1 = -1 0 0 1 Y X -1 0 0 1 2 2 = (-1*2) + 0*2 0*2 + 1*2 = 2 -2 = Similarly, For B1 And C1 the points are B1(-4,2) and C1(-4,4) A1(-2,2) Solution:- Here the given triangle ABC is reflected about Y-axis. For A (2,2) y x -x -y Slide 9 A1 = Graphical Representation
- 10. x-axis y-axis x y -y -x Original Reflection -y-axis -x-axis 0 Reflection perpendicular to XY plane: In this kind of Reflection, the value of both X and Y is negative. We can represent the Reflection along origin by following equation- Here, For point P1 X1 = –X0 and Y1 = –Y0 We can also represent Reflection in the form of matrix– Homogeneous Coordinate Representation: We can also represent the Reflection though the origin in the form of 3 x 3 matrix- 3) Reflection of an object relative to an axis perpendicular to the xy plane and passing through co-ordinate origin. Slide 10 = =
- 11. 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 A(2,2) B(4,2) C(4,4) B1(-4,-2) A1(-2,-2) C1(-4,-4) Example:- A triangle ABC A(2,2), B(4,2) and C(4,4) is reflected about co-ordinate origin. Find its new co-ordinate of A1, B1 & C1. X1 Y1 = -1 0 0 -1 Y X -1 0 0 -1 2 2 = (-1*2)+ 0*2 0*2 +(-1*2) = -2 -2 = Similarly, For B1 And C1 the points are B1(-4,-2) and C1(-4,-4) A1(-2,-2) Solution:- Here the given triangle ABC is reflected to Origin. For A (2,2) -y -x x y Slide 11 A1 = Graphical Representation
- 12. 0 x-axis y-axis 45 x x y y Original Reflection 4) Reflection of an object with respect to the line y=x. Reflection along with the line: In this kind of Reflection, the value of X is equal to the value of Y. We can represent the Reflection along the line Y=X by following equation- Here, For point P1 X1=Y0 and Y1=X0 We can also represent Reflection in the form of matrix– Homogeneous Coordinate Representation: We can also represent the Reflection along the line y=x in the form of 3 x 3 matrix- Slide 12 = =
- 13. 5 8 9 10 1 2 3 4 0 -y y 6 7 2 1 5 3 4 x 6 7 8 9 10 -x A(7,2) B(10,2) C(10,5) C1(5,10) B1(2,10) A 1 (2,7) Example:- A triangle ABC A(7,2), B(10,2) and C(10,5) is reflected with the respect to the line Y=X. Find its new co-ordinate of A1, B1 & C1. X1 Y1 = 0 1 1 0 Y X 0 1 1 0 2 7 = (0*7) + (1*2) (1*7) + (0*2) = 7 2 = B1(2,10) and C1(5,10) A1(2,7) For A (7,2) Similarly, For B1 And C1 the points are Solution:- Here the given triangle ABC is reflected through line Y=X. Slide 13 A1 = Graphical Representation
- 14. 0 5) Reflection of an object with respect to the line y=-x. -x y -y x y-axis x-axis -x-axis -y-axis p0(x0 , y0) p1(x1 , y1) Reflection along with the line: In this kind of Reflection, the value of Y is equal to the value of -X. We can represent the Reflection along the line Y=-X by following equation- Here, For point P1 X1=-Y0 Y1=-X0 We can also represent Reflection in the form of matrix– x1 y1 = 0 -1 -1 0 x0 y0 y1 = y0 0 -1 0 -1 0 0 0 0 1 x0 x1 1 1 Homogeneous Coordinate Representation: We can also represent the Reflection along the line y=-x in the form of 3 x 3 matrix- Slide 14
- 15. 5 8 9 10 1 2 3 4 0 -y y 6 7 -9 -10 -6 -8 -7 -5 -4 -3 -2 -1 -x 2 1 3 4 -2 -6 -5 -4 -3 -1 x C(4,6) A(2,9) B(2,6) C1(-6,-4) B1(-6,-2) Example:- A triangle ABC A(2,9), B(2,6) and C(4,6) is reflected with the respect to the line Y=-X. Find its new co-ordinate of A1, B1 & C1. X1 Y1 = 0 -1 -1 0 Y X 0 -1 -1 0 9 2 = (0*7) +(-1*2) (-1*7)+ (0*2) = -2 -9 = B1(-6,-2) and C1(-6,-4) A1(-9,-2) For A (2,9) Similarly, For B1 And C1 the points are Solution:- Here the given triangle ABC is reflected through line Y=-X. Slide 15 A1 = Graphical Representation