![]() reflections, which have a fixed line (the "mirror"), and are orientation-reversing.rotations, which have one fixed point (the "centre"), and are orientations-preserving.translations, which have no fixed points, and are orientation-preserving.In the Euclidean plane, we may classify rigid motions as: The group of all rigid motions is generated by the reflections: for example, a translation is a product of reflections in two parallel mirrors, and a rotation about a point is a product of two reflections in mirrors which meet at that point. This process exhibits the Euclidean group as the semi-direct product. The orientation preserving maps are those of determinant +1, forming the special orthogonal group SO( n). Hence they may be represented by the orthogonal matrix group O( n). If we fix a particular point O in the Euclidean space and consider the rigid motions which fix this point, we find that these must be linear maps of the underlying vector space which preserve distance. ![]() All translations are orientation-preserving. If we regard Euclidean space of n dimensions as an affine space built on a real vector space R n then the translations are the maps of the formįor a particular a in R n. It is a matter of convention whether the orientation-reversing maps such as reflections are considered "proper" rigid motions.Īn important subclass of rigid motions are the translations or displacements. Rigid motions are invertible functions, whose inverse functions are also rigid motions, and hence form a group, the Euclidean group.Īn important distinction is between those motions which preserve orientation or "handedness" and those which do not (for example, those three-dimensional motions which would transform a right-handed into a left-handed glove). Since Euclidean properties may be defined in terms of distance, the rigid motions are the distance-preserving mappings or isometries. With clarity and precision, describe a sequence of rigid motions that would map figure ABC onto figure A'B'C'.In Euclidean geometry, a rigid motion is a transformation which preserves the geometrical properties of the Euclidean space. Let two figures ABC and A'B'C' be given so that the length of curved segment AC = the length of curved segment A'C', |∠ B| = |∠ B'| = 80 ° and |AB| = |A'B'| = 5. Which basic rigid motion, or sequence of, would map one triangle onto the other?ĥ. Transparency and mapped onto triangle A'B'C'. In each pair, triangle ABC can be traced onto a In the following picture, we have two pairs of triangles. Which basic rigid motion, or sequence of, would map one triangle onto the other?Ĥ. In the following picture, triangle ABC can be traced onto a transparency and mapped onto triangle A'B'C'. So now we let E 3 be the image of E after the Translation 1(E)įollowed by the Rotation 2(E) followed by the Rotation 3(E)ġ - 3.
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