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51:. In particular, the lead refers correctly to transformations of Euclidean spaces, while the sections describe only the case of Euclidean vector spaces or of spaces of coordinate vectors. The "formal definition" section does not specify which kind of objects are represented by the variables, call them vaguely as "vectors", suggests implicitly that a basis and a
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1062:{\displaystyle d(\mathbf {v} +\mathbf {d} ,\mathbf {w} +\mathbf {d} )^{2}=(\mathbf {v} +\mathbf {d} -\mathbf {w} -\mathbf {d} )\cdot (\mathbf {v} +\mathbf {d} -\mathbf {w} -\mathbf {d} )=(\mathbf {v} -\mathbf {w} )\cdot (\mathbf {v} -\mathbf {w} )=d(\mathbf {v} ,\mathbf {w} )^{2}.}
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which shows that the matrix can have a determinant of either +1 or −1. Orthogonal matrices with determinant −1 are reflections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can be viewed as consisting of two manifolds in
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625:{\displaystyle d\left(\mathbf {X} ,\mathbf {Y} \right)^{2}=\left(X_{1}-Y_{1}\right)^{2}+\left(X_{2}-Y_{2}\right)^{2}+\dots +\left(X_{n}-Y_{n}\right)^{2}=\left(\mathbf {X} -\mathbf {Y} \right)\cdot \left(\mathbf {X} -\mathbf {Y} \right).}
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of objects in the
Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a
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1549:{\displaystyle d(\mathbf {v} ,\mathbf {w} )^{2}=(\mathbf {v} -\mathbf {w} )\cdot (\mathbf {v} -\mathbf {w} )=((\mathbf {v} -\mathbf {w} ))\cdot ((\mathbf {v} -\mathbf {w} )).}
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136:, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the
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It is easy to show that this is a rigid transformation by showing that the distance between translated vectors equal the distance between the original vectors:
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1710:{\displaystyle d(\mathbf {v} ,\mathbf {w} )^{2}=(\mathbf {v} -\mathbf {w} )^{\mathsf {T}}^{\mathsf {T}}(\mathbf {v} -\mathbf {w} ).}
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227:-dimensional Euclidean spaces. The set of rigid motions is called the special Euclidean group, and denoted
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1191:{\displaystyle L(\mathbf {V} )=L(a\mathbf {v} +b\mathbf {w} )=aL(\mathbf {v} )+bL(\mathbf {w} ).}
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A rigid transformation is formally defined as a transformation that, when acting on any vector
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This condition actually requires the columns of these matrices to be orthogonal unit vectors.
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In dimension at most three, any improper rigid transformation can be decomposed into an
805:{\displaystyle d(g(\mathbf {X} ),g(\mathbf {Y} ))^{2}=d(\mathbf {X} ,\mathbf {Y} )^{2}.}
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1328:{\displaystyle d(\mathbf {v} ,\mathbf {w} )^{2}=d(\mathbf {v} ,\mathbf {w} )^{2},}
350:(an orientation-preserving orthogonal transformation). Indeed, when an orthogonal
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A linear transformation is a rigid transformation if it satisfies the condition,
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where is the identity matrix. Matrices that satisfy this condition are called
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as the sum of the squares of the distances along the coordinate axes, that is
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to every vector in the space, which means it is the transformation
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164:. In dimension three, every rigid motion can be decomposed as the
1909:{\displaystyle \det \left(^{\mathsf {T}}\right)=\det^{2}=\det=1,}
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of a rotation and a translation, and is thus sometimes called a
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380:. The formula gives the distance squared between two points
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Matrices that satisfy this condition form a mathematical
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does not produce a reflection, and hence it represents a
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under the operation of matrix multiplication called the
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Now use the fact that the scalar product of two vectors
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Using this distance formula, a rigid transformation
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Mathematical transformation that preserves distances
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172:. In dimension three, all rigid motions are also
2020:Galarza, Ana Irene Ramírez; Seade, José (2007),
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1809:Compute the determinant of the condition for an
331:A proper rigid transformation has, in addition,
1721:is rigid if its matrix satisfies the condition
354:produces a reflection, its determinant is −1.
198:and size after a proper rigid transformation.
1204:can be represented by a matrix, which means
156:In dimension two, a rigid motion is either a
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1942:because it has the structure of a manifold.
61:. There might be a discussion about this on
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815:Translations and linear transformations
2006:Introduction to Theoretical Kinematics
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2022:Introduction to classical geometries
124:The rigid transformations include
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1976:O. Bottema & B. Roth (1990).
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1773:{\displaystyle ^{\mathsf {T}}=,}
1717:Thus, the linear transformation
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1982:. Dover Publications. reface.
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1198:A linear transformation
111:geometric transformation
103:Euclidean transformation
2003:J. M. McCarthy (2013).
1952:Deformation (mechanics)
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1979:Theoretical Kinematics
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2009:. MDA Press. reface.
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43:This article
41:
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2024:, Birkhauser
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1978:
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1933:
1930:and denoted
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1200:
1089:, preserves
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388:
382:
372:
370:formula for
361:
343:
341:
330:
324:
310:
306:
303:
298:
294:
291:
287:
283:
279:of the form
274:
270:
264:
261:
252:screw motion
244:rigid bodies
237:
230:
218:
210:
200:
193:
182:
169:
155:
150:
146:
143:rigid motion
142:
130:translations
123:
106:
102:
98:
92:
77:
68:
57:Please help
44:
821:translation
209:called the
189:reflections
166:composition
158:translation
134:reflections
95:mathematics
71:August 2021
53:dot product
2047:Kinematics
2036:Categories
1963:References
1813:to obtain
240:kinematics
215:, denoted
138:handedness
49:to readers
1940:Lie group
1694:−
1639:−
1530:−
1507:⋅
1493:−
1450:−
1430:⋅
1410:−
1014:−
1003:⋅
992:−
970:−
962:−
943:⋅
932:−
924:−
607:−
594:⋅
581:−
544:−
522:⋯
495:−
452:−
176:(this is
126:rotations
1946:See also
1335:that is
1233:matrix.
1211: :
1080: :
704: :
348:rotation
162:rotation
681:, ...,
651:, ...,
338:(R) = 1
322:), and
314:(i.e.,
149:, or a
109:) is a
45:may be
1986:
632:where
364:metric
318:is an
304:where
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196:shape
160:or a
113:of a
1984:ISBN
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662:and
386:and
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223:for
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