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See searches and references containing RIGID TRANSFORMATION!RIGID TRANSFORMATION
Mathematical transformation that preserves distances
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space
Rigid_transformation
Distinction between meanings of Euclidean space transformations
instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in
Active and passive transformation
Active_and_passive_transformation
Topics referred to by the same term
functions) Rigid body, in physics, a simplification of the concept of an object to allow for modelling Rigid transformation, in mathematics, a rigid transformation
Rigidity
Physical object which does not deform when forces or moments are exerted on it
In classical mechanics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure
Rigid_body
Function that applies a set to itself
Infinitesimal transformation Linear transformation List of transforms Rigid transformation Transformation geometry Transformation semigroup Transformation group
Transformation_(function)
Bijection of a set using properties of shapes in space
Coordinate transformation Erlangen program Symmetry (geometry) Motion Reflection Rigid transformation Rotation Topology Transformation matrix Usiskin
Geometric_transformation
Fundamental space of geometry
every rigid transformation that is not a rigid motion is the product of r and a rigid motion. A glide reflection is an example of a rigid transformation that
Euclidean_space
Central object in linear algebra; mapping vectors to vectors
(computer vision) Rigid transformation Transformation (function) Transformation geometry Gentle, James E. (2007). "Matrix Transformations and Factorizations"
Transformation_matrix
Eight-dimensional algebra over the real numbers
to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees
Dual_quaternion
Constraint equations of a mechanical system
sequence of rigid transformations along links and around joints in a mechanical system. The principle that the sequence of transformations around a loop
Kinematics_equations
Polyhedron with 8 triangles and 6 squares
stretch and its long edges are rigid, and in the rigid-edge transformation its long edges compress and its short edges are rigid. Everything in the descriptions
Cuboctahedron
Transformation of a body from a reference configuration to a current configuration
non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation)
Deformation_(physics)
Branch of physics describing the motion of objects without considering forces
undergo rigid motion. Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between
Kinematics
Algorithm
aligning three dimensional models given an initial guess of the rigid transformation required. The ICP algorithm was first introduced by Chen and Medioni
Iterative_closest_point
Process of finding a spatial transformation that aligns two point clouds
non-rigid registration yields a non-rigid transformation which maps one point set to the other. Non-rigid transformations include affine transformations such
Point-set_registration
Geometric transformation that preserves lines but not angles nor the origin
affine transformations: those where the determinant of A {\displaystyle A} is positive. In the last case this is in 3D the group of rigid transformations (proper
Affine_transformation
Study of the effects of forces on undeformable bodies
mechanics Lagrangian mechanics Lagrangian Hamiltonian mechanics Rigid body Rigid transformation Rigid rotor Soft-body dynamics Multibody system Polhode Herpolhode
Rigid_body_dynamics
Computing a robot's end-effector position from joint values and kinematic equations
obtained using a rigid transformation [Z] to characterize the relative movement allowed at each joint and separate rigid transformation [X] to define the
Forward_kinematics
Physical quantity that changes sign with improper rotation
rigid transformations such as rotations or translations, but which does not transform like a vector under certain discontinuous rigid transformations
Pseudovector
Type of matrix
realization, if it exists, is unique up to rigid transformations, i.e. distance-preserving transformations of Euclidean space (rotations, reflections
Euclidean_distance_matrix
Form of an object
a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set of points
Shape
Mathematical model for a mechanical system
obtained using rigid transformations [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the
Kinematic_chain
an orthogonal transformation (an origin-preserving rigid transformation) with a uniform scaling (dilation). All similarity transformations (which globally
Conformal linear transformation
Conformal_linear_transformation
Application of Clifford algebra
application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving
Plane-based_geometric_algebra
Linear algebra operation
)\end{bmatrix}}} Geometric transformation Improper rotation Linear transformation Orthogonal matrix Rigid transformation Unitary transformation Rowland, Todd. "Orthogonal
Orthogonal_transformation
Distance-preserving mathematical transformation
mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective
Isometry
Property of a mathematical matrix
A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space R k {\displaystyle \mathbb {R} ^{k}}
Definite_matrix
Transformation of coordinates through an angle
dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation. Coordinate systems are essential for studying the equations of curves
Rotation of axes in two dimensions
Rotation_of_axes_in_two_dimensions
Database of handwritten digits
Yann; Denker, John (1992). "Efficient Pattern Recognition Using a New Transformation Distance". Advances in Neural Information Processing Systems. 5. Morgan-Kaufmann
MNIST_database
Isometry of the Euclidean plane
rotations together form the rigid motions or rigid displacements. This set forms a group under composition, the group of rigid motions, a subgroup of the
Euclidean_plane_isometry
In mathematics, invertible homomorphism
automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations. Category theory, which
Isomorphism
Subgroup invariant under conjugation
dimension. This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single
Normal_subgroup
Rational function of the form (az + b)/(cz + d)
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f ( z ) = a z + b c z + d {\displaystyle
Möbius_transformation
(SimilarityTransform) expands the rigid transformation by introducing isotropic scaling Affine (AffineTransform) expands the rigid transformation allowing both scaling
Elastix_(image_registration)
Symmetry group of a configuration in space
The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. Usually the term is applied
Space_group
(software). It is used in the image registration in both rigid and non-rigid transformation. Sederberg, Thomas W.; Parry, Scott R. (1986). "Free-form
Free-form_deformation
Figure formed by knights moves on a grid
extended to polyknights: free polyknights are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another
Polyknight
a transformation of an object in space and time. If this transformation preserves size and shape of the object it is known as a Rigid Transformation. Rigid
Rigid_motion_segmentation
Method of data interpolation and smoothing
parameters included in the affine transformation are not penalized. TPS has been widely used as the non-rigid transformation model in image alignment and shape
Thin_plate_spline
Number of independent parameters needed to define the state of a mechanical system
and other fields. The position of an n-dimensional rigid body is defined by the rigid transformation, [T] = [A, d], where d is an n-dimensional translation
Degrees of freedom (mechanics)
Degrees_of_freedom_(mechanics)
Research topic in computational geometry
characteristic. Editing may involve denoising, deforming, or performing rigid transformations. At the final stage of the shape's "life," it is consumed. This
Geometry_processing
Limiting form of small transformation
infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional
Infinitesimal_transformation
Concepts from linear algebra
reversed) by a given linear transformation. More precisely, an eigenvector v {\displaystyle \mathbf {v} } of a linear transformation T {\displaystyle T} is
Eigenvalues_and_eigenvectors
Point, line, or plane about which a molecule or crystal is symmetric
element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the object unchanged.
Symmetry_element
Transformation of coordinates that moves the origin
defined similarly. A translation of axes is a rigid transformation, but not a linear map. (See Affine transformation.) Coordinate systems are essential for studying
Translation_of_axes
Geometric shapes formed from squares
polykings for enumeration: free polykings are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another
Pseudo-polyomino
Geometric space with six dimensions
crystallography of icosahedral quasicrystals. In three dimensional space a rigid transformation has six degrees of freedom, three translations along the three coordinate
Six-dimensional_space
Property of objects which are scaled or mirrored versions of each other
similarity coefficient. When r = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under
Similarity_(geometry)
Medical condition
(seconds) for HbS polymerization and the subsequent flexible-to-rigid transformation. If the transit time of RBC through the microvasculature is longer
Sickle_cell_disease
Ability to initiate and control actions
very young human observers are sensitive to self-propulsion, non-rigid transformation of the object's surface, irregular path movement, causation at a
Agency_(psychology)
Technique in digital image processing
sought-after object is partly hidden in an image; detection of non-rigid transformations, when an object is distorted or imaged from different angles; sensitivity
Template_matching
On lattices and sphere packing in Euclidean space
usually in three dimensions, with its symmetry operation as the rigid transformations of the pattern that leave it unchanged. This question was answered
Hilbert's_eighteenth_problem
Property that is not changed by mathematical transformations
operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated
Invariant_(mathematics)
Matrix of inner products of vectors
j {\displaystyle w_{i}\cdot w_{j}} are equal if and only if some rigid transformation of R k {\displaystyle \mathbb {R} ^{k}} transforms the vectors v
Gram_matrix
Geometric shape formed from squares
polyominoes for enumeration: free polyominoes are distinct when none is a rigid transformation (translation, rotation, reflection, or glide reflection) of another
Polyomino
Open-source algorithm library
just a rigid transformation of another. Normal Distributions Transform (NDT) is a registration algorithm that can be used to determine a rigid transformation
Point_Cloud_Library
Powered mechanical device
assembly of rigid components allows rotational and translational movement to be modeled mathematically as Euclidean, or rigid, transformations. This allows
Machine
Capacity to understand 3D relationships
is a non-rigid spatial transformation ability which means features of the manipulated object end up changing unlike mental rotation. In rigid manipulations
Spatial_ability
Coordinate transformation that preserves the form of Hamilton's equations
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations
Canonical_transformation
Matrix representing a Euclidean rotation
Rotation operator (vector space) Transformation matrix Yaw-pitch-roll system Kabsch algorithm Isometry Rigid transformation Rotations in 4-dimensional Euclidean
Rotation_matrix
Four-dimensional algebra over the real numbers
algebra over the real numbers. Their primary application is in representing rigid body motions in 2D space. In this article, certain applications of the dual
Applications of dual quaternions to 2D geometry
Applications_of_dual_quaternions_to_2D_geometry
Shape with four equal sides and angles
The square is the most symmetrical of the quadrilaterals. Eight rigid transformations of the plane take the square to itself: For an axis-parallel square
Square
Change in the shape or size of an object
rotations (rigid transformations). Deformation are changes in the relative position between internals points on the object, excluding rigid transformations, causing
Deformation_(engineering)
Method for specifying point positions
(linear) position of points and the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred
Coordinate_system
Computer vision concept
These trajectories are constrained globally at each frame by the rigid transformation which the shape is undergoing, i.e., trajectory of every point will
Tomasi–Kanade_factorization
Non-commutative group with 6 elements
the collection of all rigid transformations made by reflections, rotations, and combinations of these both. Six transformations are the results. Three
Dihedral_group_of_order_6
American mathematician (1947–2023)
Zimmer was greatly influenced by the work of Mikhail Gromov on rigid transformation groups and he extended and connected Gromov's theory to the Zimmer
Robert_Zimmer
Concept in 16th-century Aztec/Mexica cosmology
directions, although not being limited to these directions in a static or rigid way. Scholar Gabriel S. Estrada states that "as cosmic movement, ollin is
Nahui_Ollin
Various molecular structures formed only by rotation about single bonds
the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. Protein side chains exhibit rotamers, whose distribution
Rotamer
Group of symmetries of an n-dimensional hypercube
(\pm 1,\ldots ,\pm 1)} . The hyperoctahedral group consists of all rigid transformations w of R n {\displaystyle \mathbb {R} ^{n}} that send H {\displaystyle
Hyperoctahedral_group
Concept in special relativity, governing a body's dynamics at high speeds
answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics. The concept was introduced
Born_rigidity
Direction and rate of rotation
velocity. A rigid body rotating about a fixed axis has each point of the body having the same orbital angular velocity. Hence such a rigid body can be
Angular_velocity
Algebraic object with geometric applications
combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse
Tensor
Ammunition type designed to penetrate armour
armour-piercing cap. This lowers the initial shock of impact to prevent the rigid projectile from shattering, as well as aiding the contact between the target
Armour-piercing_ammunition
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
kinematics and dynamics of rigid bodies. Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where
Screw_theory
velocity tensor of a rigid body (in its rest frame) is a linear transformation that maps positions to velocities (within the rigid body), it can be regarded
Angular_velocity_tensor
Description of the orientation of a rigid body
three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent
Euler_angles
Landmark textbook in classical mechanics by E. T. Whittaker
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund
Analytical Dynamics of Particles and Rigid Bodies
Analytical_Dynamics_of_Particles_and_Rigid_Bodies
Nonlinear partial differential equation
the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations. There is a theorem, sometimes called the fundamental theorem of
Sine-Gordon_equation
Feature detection algorithm in computer vision
results except under wide illumination variations and under non-rigid transformations. We begin by detecting points of interest, which are termed keypoints
Scale-invariant feature transform
Scale-invariant_feature_transform
Set of principles for modeling solid geometry
ordered binary trees where non-terminal nodes represent either rigid transformations (orientation preserving isometries) or regularized set operations
Solid_modeling
Long-term change in dominant economic activity
way. Transformational “losses” cannot be recovered or regained by definition. Not understanding that is at the core of wasteful spending of rigidly hopeless
Transformation_in_economics
Ways to represent 3D rotations
formulations to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Planar movement within a Euclidean space without rotation
Transformation matrix Translational symmetry LIMA, 2001 Edmund Taylor Whittaker (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
Translation_(geometry)
Physical theory with fields invariant under the action of local "gauge" Lie groups
transformation on the fiber over that point. A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation
Gauge_theory
solutions, or frameworks, in some metric space. A framework of a GCS is rigid in d {\displaystyle d} -dimensions, for a given d {\displaystyle d} if it
Geometric_rigidity
registration with different transformation models, such as rigid-body, affine and nonlinear warping. For example, for affine transformation the registration from
AIR_(program)
Motion of a certain space that preserves at least one point
preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an
Rotation_(mathematics)
Inorganic, nonmetallic solid prepared by the action of heat
metals. These materials do show plastic deformation. However, because of the rigid structure of crystalline material, there are very few available slip systems
Ceramic
algorithms have focused on recognizing rigid objects consisting of a single part, that is, objects whose spatial transformation is a Euclidean motion. Two general
3D_object_recognition
descriptors is for them to be invariant under certain transformations. For rigid transformations, commonly used feature descriptors include shape context
Heat_kernel_signature
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form ∑ i p i δ q i = ∑ i P i δ Q i {\displaystyle
Mathieu_transformation
Symmetry of spatially mirrored systems
In physics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also
Parity_(physics)
Type of constraints for mechanical systems
weight and L {\displaystyle L} is length of the string. The particles of a rigid body obey the holonomic constraint ( r i − r j ) 2 − L i j 2 = 0 , {\displaystyle
Holonomic_constraints
Paradox in special relativity
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity. In its original 1909 formulation as presented by Paul Ehrenfest
Ehrenfest_paradox
Type of motion
rotation around a fixed axis of a rigid body are mathematically much simpler than those for free rotation of a rigid body; they are entirely analogous
Rotation_around_a_fixed_axis
Theatrical convention of metamorphosis
rigid separation implied by the transformation, leading to the 19th century view of pantomime. In 1881, Percy Fitzgerald described the transformation
Transformation_scene
Physical components of a computer
by hardware. Hardware derived its name from the fact that it is hard or rigid with respect to changes, whereas software is soft because it is easy to
Computer_hardware
Method of analysis for systems of interacting components
particular in areas like scalability and predictability, because of its rigidness and lack of dynamism. Promise theory's point of departure from obligation
Promise_theory
Mathematical model combining space and time
universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity. In 1908, Hermann Minkowski presented
Spacetime
travel, tourism, insurance
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