Re: isometries in E^3-

You are right. Orientation preserving isometry in E^3 with unique fixed
point does not exist, so I should drop that from my list.

"JEMebius" <jemebius@xxxxxxxxx> wrote in message
news:43921822.9060204@xxxxxxxxxxxx
>
> TCL wrote:
>
>>In E^2, the Euclidean plane, every isometry is one of the following types:
>>identity, (line) reflection, rotation, translation, glide reflection (i.e.
>>a line reflection followed by a translation in a direction parallel to the
>>line).
>>
>>Now, in E^3, I think every isometry is one of these:
>>identity, plane reflection (reflection across a fixed plane), rotation
>>about an axis, rotation about a point (with the point as the unique fixed
>>point), plane reflection followed by a rotation about an axis normal to
>>the plane, tranlation,
>>plane reflection followed by a translation parallel to the plane, rotation
>>about an axis followed by a tranlation parallel to the axis.
>>
>>Did I miss anything? Or did I list too many?
>>
>>
>>
> You missed Euler's theorem on 3D rotational displacements, which says that
> every displacement of E3 which lets a certain point O fixed also lets an
> entire straight line thru O pointwise fixed.
>
> Please note that this theorem is about displacements and not about
> motions.
>
> A displacement is just an orientation-preserving isometry.
> A displacement is the net result of a motion, acutally of an infinite
> number of motions. This is in a nutshell a main difference between
> kinematics and geometry.
>
> A rotational =motion= in 3D about a fixed point O has in general O as its
> unique fixed point; but the isometry that results from a motion with a
> beginning and an end can always be obtained by a rotational motion around
> a fixed axis thru O.
>
> BTW, Who can tell the newsgroup who gave the first geometrical proof of
> the theorem that in E3 each orientation-reversing isometry with a fixed
> point O can be obtained as a rotational displacement around an axis A
> followed by (or preceded by) a reflection across the plane thru O
> perpendicular to A?
>
> This theorem is the rotation-reflection counterpart of Euler's rotation
> theorem. Did Euler perhaps also prove this theorem?
>
> Johan E. Mebius


.

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