Lecture Notes 4: Affine Transformations and 3D Transformations

Review of Affine Transformations

• Affine transformations have limitations when combined.

• Homogeneous coordinates provide a solution to approximate linear transformations.

• This involves representing transformations (including translation) using a single matrix.

• The input vector's dimension increases by one (e.g., 2D becomes 3D).

• Matrix multiplications are efficient for transformations.

Examples of Transformations with Homogeneous Coordinates

• Rotation, scaling, and shear transformations.

• Translation is represented by a specific matrix.

Examples of Combining Transformations

Rotation followed by Translation

• If $R$ is the rotation matrix and $T$ is the translation matrix, then rotation followed by translation is represented as $T \cdot R \cdot x$, where $x$ is the input.

• The resulting matrix combines rotation and translation parameters.

Rotation followed by Shear

• Rotation is applied first, then shear.

• Order matters because changing the order can yield different results.

Translation followed by Rotation

• The result differs from rotation followed by translation.

• Translation followed by rotation is not the same as rotation followed by translation.

Reason for the Difference

• Rotation is performed around the origin by default.

• Translation shifts the origin, so subsequent rotation occurs around a new point.

Commutativity of Transformations

• If $A \cdot B = B \cdot A$, transformations are commutative.

• Translation and rotation are generally not commutative.

• However, translation followed by translation or rotation followed by rotation are commutative.

Properties of Transformations

• Rotations in 2D commute.

• Translations commute.

• Scaling commutes.

• Shear transformations commute.

• In general, transformations are not always commutative

Rotation Around an Arbitrary Point

• To rotate a shape around its center (not the origin):

  • Translate the shape so that the center is at the origin.

  • Perform the rotation.

  • Translate the shape back to its original position.

• This involves $T \cdot R \cdot T^{-1}$, where $T$ is the translation matrix, $R$ is the rotation matrix, and $T^{-1}$ translation inverse.

Applying Transformations to Primitives

Line Segment

• Defined by two points.

• Apply the transformation to both points.

Rectangle

• Defined by two opposite corner points.

• Apply the transformation to these two points.

• This works because affine transformations preserve parallel lines and straight lines.

3D Transformations

• Include scaling, translation, and rotation.

• Also involve the pinhole model, perspective projection, and viewpoint transformation.

Homogeneous Coordinates in 3D

• 3D coordinates become 4D in homogeneous form.

• Linear transformations use 3x3 matrices.

• Affine transformations use 4x4 matrices.

Scaling and Translation in 3D

• Translation: $T<em>x$, $T</em>y$, $T_z$ for three dimensions.

• Scaling: Factors for x, y, and z directions.

Rotation in 3D

• More complex due to axis and order of rotation.

• Rotation around x, y, and z axes.

• Order matters; rotations do not commute.

Rotation Around Z-Axis:

• Similar to 2D rotation in the x-y plane.

Rotation Around X-Axis:

• The y-z plane is rotated

Rotation Around Y-Axis:

• Uses similar parameters but in different matrix positions.

• The matrix element at $[1,1]$ will be equal to one.

Rotation and Inverse Rotation

• Inverse rotation reverses the effect of the original rotation.

• Mathematically, if $R$ is the rotation matrix, the inverse is $R^{-1}$.

• Geometrically, rotating by $\theta$ degrees and then by $-\theta$ degrees.

Properties of Rotation Matrices

• Orthogonal matrices: $R \cdot R^T = I$, where $I$ is the identity matrix.

• $R^{-1} = R^T$

• Transpose is faster to compute than the inverse.

Rotation Around an Arbitrary Vector

• Rotate the vector to align with the x-axis, rotate, and then reverse the rotation.

• Rotate the vector to align with the y-axis, rotate, and then reverse the rotation.

• Multiple rotation and inverse rotations are required which makes this approach complicated.

Rodrigues' Rotation Formula

• Alternative mathematical approach.

• $R = cos(\theta)I + sin(\theta)[u]_x + (1 - cos(\theta))(u \cdot u^T)$

• Rotation around vector $u$, where $u$ is a unit vector.

• Uses sine and cosine of the angle.

• Involves cross product and dot product of vectors.

Normals and Transformations

Definition of Normal

• A vector perpendicular to the surface at a point.

• Changes with curvature.

Transformation of Normals

• If a transformation $A$ is applied.

• The new normal is calculated as $n' = (A^{-1})^T n$.

• Important for non-uniform scaling.

Proof (Key Points)

• If two vectors are perpendicular, their dot product is zero.

• The dot product of A and B can be written as $A^T B$.

• Transformation: $x' = Ax + t$.

• The new transformed normal equals to $n' = (A^{-1})^T n$.

Lecture Summary

• Modeling linear transformations.

• Overcoming the complication of combining multiple affine transformations in 2D using Homogeneous coordinates.

• 3D transformations.

• Rotation around x, y, z axes or a random vector.