Isometry closely connected

A Isométrie refines is a bijective transformation of a Espace refines Euclidean in another which preserves the distances. One generalizes this concept with the bijective tranformations of a metric Espace in another which preserve the distances.

If this isometry preserves also the directed angles, then they acts of a Déplacement. If it reverses the directed angles, it is about a Antidéplacement.

Isométries plane remarkable

One indicates by $\backslash \; mathcal\; \{P\}$ the plan (i.e., more precisely, a plan refines real Euclidean directed). The following applications are isométries of $\backslash \; mathcal\; \{P\}$:

• being given a vector $\backslash \; vec\; \{U\}$ the application which, with any point $A$, associates the point $A\text{'}$ such as $\backslash \; vec\; \{AA\text{'}\}\; =\; \backslash \; vec\; \{U\}$: it is the translation of vector $\backslash \; vec\; \{U\}$. Its reciprocal is the translation of vector $-\; \backslash \; vec\; \{U\}$. It does not have any fixed point, except if $\backslash \; vec\; \{U\}\; =\; \backslash \; vec\; \{0\}$, in which case it is the identity. The translations are displacements.
• being given a right $\backslash \; Delta$ the application which, with any point $A$, associates the point $A\text{'}$ such as $\backslash \; vec\; \{AA\text{'}\}\; =2\; \backslash \; vec\; \{AH\}$, where $H$ the is projected orthogonal one of $A$ on $\backslash \; Delta$: it is the reflection of axis $\backslash \; Delta$. One can define it differently: $A\text{'}=A$ if $A\; \backslash \; in\; \backslash \; Delta$ and, if $A\; \backslash \; notin\; \backslash \; Delta$, $A\text{'}$ is such as $\backslash \; Delta$ is the mediator of $$. The reflections are involutive and are antidéplacements.
• being given a point $A$ $\backslash \; mathcal\; \{P\}$ and a reality $\backslash \; theta$ the application which fixes $A$ and, with a point $B$ distinct from $A$, associates the single point $B\text{'}$ such as $AB=AB\text{'}$ and a measurement of the angle directed $(\backslash \; vec\; \{AB\},\; \backslash \; vec\; \{AB\text{'}\})$ is $\backslash \; theta$: it is the Rotation of center $A$ and angle $\backslash \; theta$. The reciprocal one of the rotation of center $A$ and angle $\backslash \; theta$ is the rotation of center $A$ and angle $-\; \backslash \; theta$. Lastly, rotations are displacements.

Classification of the plane isométries having a fixed point

• a isometry of the plan having three fixed points not aligned is the identity.

• a isometry of the plan other than the identity having at least two fixed points has and B is the reflection compared to the right (AB).
• a isometry of the plan having a single fixed point has is a rotation of center A.

In unspecified dimension

To study the isométries closely connected in unspecified dimension, one is interested in the orthogonal Automorphisme $\backslash \; phi$ definite associate of the kind: if is a isometry closely connected of $\backslash \; mathcal\; \{E\}$, then its associated orthogonal automorphism is Consequently the study of the fixed points of $f$ and $\backslash \; phi$ makes it possible to conclude on nature from $f$.

• If $f$ admits fixed points then:

if $\backslash \; phi$ is a vectorial rotation then $f$ will be a rotation closely connected.

if $\backslash \; phi$ is the vectorial identity then $f$ will be a translation.

• If $f$ does not admit fixed points then breaks up in a single way as made up of a isometry closely connected with fixed points (one thus returns to the preceding case) and of a translation of vectors in the direction of the fixed points of the preceding isometry.

Random links:

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