3 d transformations
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This document discusses 3D transformations in computer graphics including translation, rotation, scaling, and shearing. Translation simply moves an object along the x, y, and z axes and can be represented by a 4x4 matrix. Rotation is more complex in 3D and involves rotating around the x, y, or z axes. Scaling enlarges or shrinks an object along the x, y, and z axes. Shearing skews an object by changing the coordinates of one axis based on the values of another axis. The document provides formulas and examples for performing each type of 3D transformation.
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3 d transformations
This document discusses 3D transformations in computer graphics. It covers 3D translation, rotation, and scaling. Translation moves an object by adding offsets to the x, y, and z coordinates. Rotation in 3D can occur around any axis and is represented using 4x4 matrices. Scaling enlarges or shrinks an object along the x, y, and z axes by multiplying coordinates by scale factors. More complex transformations can be achieved by combining multiple simple transformations.
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3D transformations are represented by 4x4 matrices. They include translation, scaling, and rotation. Rotation can be about coordinate axes or arbitrary axes. For arbitrary axis rotation, the process involves translating the axis to the origin, rotating the axis to align with an axis, rotating about that axis, then applying the inverse transformations to return the axis to its original orientation.
3 d transformation
Here are the key steps to solve this problem:
1. Write the rotation matrix for 45 degree rotation about the x-axis:
R = [1 0 0]
[0 cos(45) -sin(45)]
[0 sin(45) cos(45)]
2. Write the scaling matrix with factors of 4 in each direction:
S = [4 0 0]
[0 4 0]
[0 0 4]
3. Pre-multiply the original coordinates by the rotation matrix, then scale the results by pre-multiplying the rotation matrix by the scaling matrix:
A' = SR(0,1,0)
= S(0,0.
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3 d transformation
Here are the key steps to solve this problem:
1. Write the rotation matrix for 45 degree rotation about the x-axis:
R = [1 0 0]
[0 cos(45) -sin(45)]
[0 sin(45) cos(45)]
2. Write the scaling matrix with factors of 4 in each direction:
S = [4 0 0]
[0 4 0]
[0 0 4]
3. Pre-multiply the original coordinates by the rotation matrix, then scale the results by pre-multiplying the rotation matrix by the scaling matrix:
A' = SR(0,1,0)
= S(0,0.
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3. 3D rotation is more complex than 2D and can be done around any arbitrary axis by constructing an orthonormal axis system and deriving rotation matrices.
Transformations in Computer Graphics
Transformations are used to move and manipulate 3D objects in computer graphics. The key types of 2D transformations are:
1. Translation moves objects by adding offsets to the x and y coordinates.
2. Scaling enlarges or shrinks objects by multiplying the x and y coordinates by scale factors.
3. Rotation spins objects around the origin by applying trigonometric functions to the coordinates.
Transformations can be represented using matrices, allowing multiple transformations to be combined through matrix multiplication. Common 2D transformations like translation, scaling, rotation, and shearing can be encoded concisely using 2×2 transformation matrices.
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3 d transformations
- 1. Computer Graphics 3D Transformations Total Slides 29
- 2. From 2D to 3D • Translation is simple as in 2D • Use of Homogeneous coordinate in 3D – In 3D transformation always use Matrix: 4x4 • All transformation in 3D is simple but only Rotation transformation is complex in 3D transformation.
- 3. 3D Translation Translate using (tx, ty,tz): x’=x+ tx, y’=y+ ty , z’=z+ tz or x y P P+T T z = = = += z y x t t t z y x z y x TPP TPP' en, ' ' ' ' met,
- 4. 3D Translation 2 In 4D homogeneous coordinates: x y P P+T T z 11000 100 010 001 1 ' ' ' or, = = z y x t t t z y x z y x MPP'
- 5. 3D Rotation 1 z P x y P’ α 1000 0100 00cossin 00sincos )( with,)( Or ' cossin' sincos' :as-aroundangleoverRotate − = = = += −= αα αα α α αα αα α z z zz yxy yxx z R PRP'
- 6. 2D Rotation about the origin. y x r r P’(x’,y’) P(x,y) θ φ y φ φ sin. cos. ry rx = = x θφθφφθ θφθφφθ cos.sin.sin.cos.)sin(. sin.sin.cos.cos.)cos(. rrry rrrx +=+=′ −=+=′
- 8. 3D Rotation 2 z x y Rotation around axis: - Counterclockwise, viewed from rotation axis z x y z x y
- 9. 3D Rotation 3 z x y z x y z x y zy yx xz → → → xz zy yx → → → Rotation around axes: Cyclic permutation coordinate axes xzyx →→→
- 11. 11000 000 000 000 1 ' ' ' or, = = z y x s s s z y x z y x SPP' 3D scaling Scale with factors sx, sy,sz: x’= sx x, y’= sy y, z’= sz z or
- 12. 3D scaling
- 13. April 2010 13 ( ), ,f f fx y z x y z Scaling with respect to a fixed point (not necessarily of object) ( ), ,f f fx y z x y z ( ), ,f f fx y z x y z ( ), ,f f fx y z x y z ( ) ( ) ( ) 1 0 0 1 0 0 1 0 0 1 0 0 0 1 x x f y y f z z f S S x S S y S S z − − − × × = − T S T
- 14. 3D Shearing about X,Y,Z Axis • Shearing about X axis- X coordinate will change but no change in Y and Z axis coordinate. • Shearing about Y axis- Y coordinate will change and changes are made only in X and Z axis. • Shearing about Z axis- Z coordinate will change but no change in X and Y axis. 1 0 0 0 1 0 0 0 1 0 0 0 0 1 shy shz 1 0 0 0 1 0 0 0 1 0 0 0 0 1 Shx Shz 1 0 0 0 1 0 0 0 1 0 0 0 0 1 Shx Shy
- 22. Question • Translate a Pyramid whose coordinates are (0,0,0), (1,0,0) (0,0,1) and (0,1,0). Translate this pyramid 3 unit x, y, z direction and find out final transformed matrix. • Find the rotation matrix along x, y, z axis using derivation of rotation in 2D.
- 23. Thank You!!