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anonymous
 one year ago
Which statement about rigid transformations is true?
anonymous
 one year ago
Which statement about rigid transformations is true?

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donnie1999
 one year ago
Best ResponseYou've already chosen the best response.0dw:1434552933830:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points. Rigid transformations of the plane \(R^2\), space \(R^3\), or real ndimensional space \(R^n\) are termed a Euclidean transformation because they form the basis of Euclidean geometry. The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures 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, this smaller class of transformations is known as proper rigid transformations (informally, also known as rototranslations). In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections). Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a group called the Euclidean group, denoted E(n) for ndimensional Euclidean spaces. The set of proper rigid transformation is called special Euclidean group, denoted SE(n). In kinematics, proper rigid transformations in a 3dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement. A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form \[T(v) = R v + t \] where RT = R−1 (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin. A proper rigid transformation has, in addition, \[ \det(R) = 1\] which means that R does not produce a reflection, and hence it represents a rotation (an orientationpreserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is –1. I hope I was helpful! \(\Huge\ddot\smile\) Source: https://en.wikipedia.org/wiki/Rigid_transformation
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