8/16/2023 0 Comments Rigid motion in mathematicsSince we are working with finite dimensional real spaces isometries from a metric subspace can always be extended to an isometry of the whole space (the extension may not be unique, but I don't think the choice really matters).ĭoes my definition characterize "proper isometries" (in the context of finite dimensional real spaces) ?Ĭlearly a map satisfying my definition is a proper isometry because the determinant is a continous map. My idea was that moving a subspace $S$ through the Euclidean space is just like a continous family of isometric embeddings that starts with the inclusion of $S$ into $E$. "A rigid motion in an $n$-dimensional Euclidean space $E$ is an isometry $\mathcal_E$ (and the intermediate maps of this homotopy are also isometries)." In my opinion, the most intuitive option is the following: Nevertheless I didn't find this definition satisfying so I tried to come up with my own rigorous definition of "rigid motion". I'm aware that this problem can be solved by restricting ourselves to "proper isometries" (i.e. It can describe, for example, the motion of a rigid body around a fixed point. Any rotation is a motion of a certain space that preserves at least one point. Rotation in mathematics is a concept originating in geometry. "A rigid motion of a subspace $S$ of Euclidean space is a motion of $S$ that preserves the shape of $S$."Īccording to this definition a reflection in plane geometry is not a rigid motion (but nonentheless it is an isometry). Rotation of an object in two dimensions around a point O. Yesterday I read again the (informal) definition of "rigid motion": I've always thought (until this day) that "rigid motion" and "isometry" were synonimous in Euclidean geometry.
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