This document discusses 3D transformations including reflection and shearing. It begins with an overview of 3D reflection about the XY, XZ, and YZ planes. It then describes how to perform reflection about any plane using a series of translation, rotation, reflection, and inverse transformations. Examples of 3D shearing about the X, Y, and Z axes are also provided. The document concludes with a numerical example of finding the reflection matrix for a plane through the origin with normal vector I+J+K.

1. 3D TRANSFORMATION
(Reflection & Shearing)
By:
Arvind Kumar
Assistant Professor
(Vidya College of Engg.)

2. Contents
By: Arvind Kumar
3D
Reflection
Shearing

3. 3D REFLECTION
By: Arvind Kumar
3D
3D Reflection about XY-Plane
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MXY
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4. 3D REFLECTION
By: Arvind Kumar
3D
3D Reflection about XZ-Plane
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5. 3D REFLECTION
By: Arvind Kumar
3D
3D Reflection about YZ-Plane
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6. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Steps to be Follows
1. Translate, that lies in the reflection plane to the origin.
2. Rotate the Normal Vector to the reflection plane at the
origin until coincident with z axis.
3. Reflect the object through xy Plane.
4. Perform inverse Rotation transformation
5. Perform inverse Translation transformation
Rxy-1
T-1
Rxy
T
Mxy
     TRRT  xyxy xyM M11 


7. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Step-1 Translation
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tz
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T

8. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Step-2 Rotation about xy plane
let the normal vector :
N= 𝒏𝟏𝒊 + 𝒏𝟐 𝒋 + 𝒏𝟑 𝒌
N= 𝒏𝟏 𝟐 + 𝒏𝟐 𝟐 + 𝒏𝟑 𝟐 ,
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N
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Rxy 
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2322 nn 

9. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Step-3 Reflection about XY Plane
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Mxy

10. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Step-4 Inverse Rotation
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11. 3D REFLECTION wrt ANY PLANE
By: Arvind Kumar
3D
Step-5 Inverse Translation
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12. Numerical
By: Arvind Kumar
3D
Q1. Find the matrix for the mirror reflection wrt the
plane passing through the origin and having a
normal vector whose direction is N= I+J+K.
Solution:
Here p0 (0,0,0) and plane passes through the origin .
Hence translation is not necessary.
the normal vector : N= I + J + K
Here, n1 = 1, n2= 1, n3=1
𝑵 = 𝟑 , λ = 𝟐


13. Numerical
By: Arvind Kumar
3D
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xyR
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14. Numerical
By: Arvind Kumar
3D
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Resultant Matrix:
      xyxy xyM M RR 1


15. Numerical
By: Arvind Kumar
3D
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1000
03/13/23/2
03/23/13/2
03/23/23/1
M
      xyxy xyM M RR 1

After Reflection points are:
X’ = (1/3, -2/3, -2/3) Y’ = ( -2/3, 1/3, -2/3),
Z’ = (-2/3, -2/3, 1/3)

16. 3D Shearing
By: Arvind Kumar
3D
3D Shearing about X-Axis
3D
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17. 3D Shearing
By: Arvind Kumar
3D
3D Shearing about Y-Axis
3D
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18. 3D Shearing
By: Arvind Kumar
3D
3D Shearing about Z-Axis
3D
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