Notes in Week 2 - Rotations and Transformations

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Published	10/15/2024	The general 2D rotation matrix is given by {{c1::\(R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pm…
Published	10/15/2024	In 3D, a rotation about the z-axis is represented by the matrix {{c1::\(R_z(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \…
Published	10/15/2024	The 3D rotation about the x-axis is represented by the matrix {{c1::\(R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\s…
Published	10/15/2024	The 3D rotation about the y-axis is represented by the matrix {{c1::\(R_y(\theta) = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1…
Published	10/15/2024	A rotation matrix must satisfy {{c1::\(R^T R = I\)}}, meaning its transpose is equal to its inverse.
Published	10/15/2024	The determinant of a rotation matrix is always {{c1::1}}. This ensures that the matrix represents a proper rotation.
Published	10/15/2024	Euler angles represent a rotation as a sequence of three angles: {{c1::\phi}} (yaw), {{c2::\theta}} (pitch), and {{c3::\psi}} (roll).
Published	10/15/2024	Euler angles can be represented as the product of three rotation matrices: {{c1::\(R_z(\phi) R_y(\theta) R_x(\psi)\)}}.
Published	10/15/2024	The {{c1::axis-angle representation}} describes a 3D rotation using a unit vector and an angle.
Published	10/15/2024	The homogeneous transformation matrix is of the form {{c1::\(H = \begin{pmatrix} R & d \\ 0 & 1 \end{pmatrix}\)}}, where \(R\) is the rotation…
Published	10/15/2024	The inverse of a homogeneous transformation matrix is computed as {{c1::\(H^{-1} = \begin{pmatrix} R^T & -R^T d \\ 0 & 1 \end{pmatrix}\)}}.
Published	10/15/2024	In 2D, a point rotated by \(\theta\) is given by {{c1::\(p' = R(\theta) p\)}}, where \(R(\theta)\) is the 2D rotation matrix.
Published	10/15/2024	The determinant of a 3D rotation matrix must be {{c1::1}} for it to represent a proper rotation.
Published	10/15/2024	The composition of rotations is achieved by multiplying their respective rotation matrices: {{c1::\(R_3 = R_1 R_2\)}}.
Published	10/15/2024	In 3D, a rigid body transformation combines {{c1::rotation}} and {{c2::translation}}.
Published	10/15/2024	A {{c1::skew-symmetric matrix}} is used in Rodrigues' formula to represent the cross-product operation.
Published	10/15/2024	In homogeneous transformations, the translation vector \(d\) is applied after the rotation matrix \(R\). This is represented by {{c1::\(H = \begin{pma…
Published	10/15/2024	The rotation matrix and translation vector in homogeneous coordinates are combined into a single {{c1::4x4}} matrix.
Published	10/15/2024	The inverse of a 3D rotation matrix is equal to its {{c1::transpose}}: \(R^{-1} = R^T\).
Published	10/15/2024	A {{c1::homogeneous transformation matrix}} combines both rotation and translation in a 4x4 matrix.
Published	10/15/2024	The {{c1::right-hand rule}} is used to determine the direction of positive rotation around an axis.
Published	10/15/2024	A {{c1::rotation matrix}} is orthogonal and its inverse is equal to its transpose.
Published	10/15/2024	{{c1::Rodrigues' formula}} is used to compute the rotation matrix from an axis-angle representation: \(R = I + \sin \theta K + (1 - \cos \theta) …
Published	10/15/2024	The {{c1::composition of rotations}} involves multiplying two rotation matrices to apply multiple rotations sequentially.
Published	10/15/2024	In 3D transformations, a {{c1::rigid body motion}} includes both rotation and translation.
Published	10/15/2024	{{c1::Euler angles}} describe a sequence of three rotations about the z, y, and x axes.
Published	10/15/2024	A {{c1::skew-symmetric matrix}} represents the cross-product operation for a given vector.
Published	10/15/2024	A 3D rotation matrix for a rotation about the x-axis is given by {{c1::\(R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & …
Published	10/15/2024	A 3D rotation matrix for a rotation about the y-axis is given by {{c1::\(R_y(\theta) = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 &amp…
Published	10/15/2024	A 3D rotation matrix for a rotation about the z-axis is given by {{c1::\(R_z(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin …
Published	10/15/2024	The {{c1::determinant}} of a rotation matrix must be 1 to ensure it represents a proper rotation.
Published	10/15/2024	The {{c1::transpose}} of a rotation matrix is equal to its inverse: \(R^{-1} = R^T\).
Published	10/15/2024	In homogeneous transformations, the {{c1::translation vector}} represents the displacement of an object in space, while the {{c2::rotation matrix}} ha…
Published	10/15/2024	The inverse of a {{c1::homogeneous transformation matrix}} is calculated by taking the transpose of the rotation matrix and negating the translation v…
Published	10/15/2024	The {{c1::special orthogonal group SO(3)}} represents all proper rotations in 3D space.
Published	10/15/2024	A rigid body transformation is represented by a {{c1::homogeneous transformation matrix}}.
Published	10/15/2024	In a homogeneous transformation matrix, {{c1::R}} represents the rotation, and {{c2::d}} represents the translation vector.
Published	10/15/2024	The {{c1::axis-angle representation}} describes a rotation by specifying an axis of rotation and an angle of rotation.
Published	10/15/2024	{{c1::Euler angles}} use three successive rotations around different axes to describe a 3D orientation.
Published	10/15/2024	The inverse of {{c1::a rotation matrix}} is its transpose: \(R^{-1} = R^T\).
Published	10/15/2024	The {{c1::Rodrigues' formula}} computes the rotation matrix given an axis and angle: \(R=I+sinθK+(1−cosθ)K2\), where \(K\) is the skew-symmetric …
Published	10/15/2024	A 3D rotation matrix for a rotation about the x-axis is given by {{c1::\(R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & …
Published	10/15/2024	A {{c1::skew-symmetric matrix}} is used to represent cross-product operations for vectors in 3D space.
Published	10/15/2024	The {{c1::composition of transformations}} is achieved by multiplying their respective matrices: \(H_3 = H_1 H_2\).
Published	10/15/2024	The {{c1::yaw-pitch-roll}} representation of orientation uses three successive rotations around the z, y, and x axes.
Published	10/15/2024	The 3D {{c1::rotation matrix}} for a rotation about the z-axis is given by [ \(R_z(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\…
Published	10/15/2024	The {{c1::homogeneous transformation}} combines both rotational and translational components into a single transformation.
Published	10/15/2024	The {{c1::rotation group SO(3)}} includes all rotation matrices that preserve distances and angles in 3D space.
Published	10/15/2024	A {{c1::rotation matrix}} is orthogonal, meaning its transpose equals its inverse: \(R^T = R^{-1}\).
Published	10/15/2024	{{c1::Euler angles}} describe a rotation by using three angles applied in a specific sequence: yaw, pitch, and roll.
Published	10/15/2024	The {{c1::Denavit-Hartenberg parameters}} are used to describe the kinematic relationship between two robotic joints.
Published	10/15/2024	A {{c1::rigid body transformation}} can be represented as a 4x4 homogeneous matrix that combines both rotation and translation.
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