Effect of plane isometry on punctured disc
$begingroup$
I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other isometric transformation(like reflection, rotation, glide reflection) can just be viewed as the effect of a translation. Am I correct? The same will hold for the punctured disc?
EDIT: I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as the translation only because the effect of other three isometries on punctured disk can be viewed as the special cases of translation, due to the 'nice' symmetry of punctured disc? Hope this clarifies my question.
real-analysis geometry differential-geometry euclidean-geometry analytic-geometry
asked Jan 6 at 17:59
ershersh
294112
$endgroup$
add a comment |
$begingroup$
I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other isometric transformation(like reflection, rotation, glide reflection) can just be viewed as the effect of a translation. Am I correct? The same will hold for the punctured disc?
EDIT: I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as the translation only because the effect of other three isometries on punctured disk can be viewed as the special cases of translation, due to the 'nice' symmetry of punctured disc? Hope this clarifies my question.
real-analysis geometry differential-geometry euclidean-geometry analytic-geometry
asked Jan 6 at 17:59
ershersh
294112
$endgroup$
$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42
add a comment |
$begingroup$
I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other isometric transformation(like reflection, rotation, glide reflection) can just be viewed as the effect of a translation. Am I correct? The same will hold for the punctured disc?
EDIT: I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as the translation only because the effect of other three isometries on punctured disk can be viewed as the special cases of translation, due to the 'nice' symmetry of punctured disc? Hope this clarifies my question.
real-analysis geometry differential-geometry euclidean-geometry analytic-geometry
asked Jan 6 at 17:59
ershersh
294112
$endgroup$
I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other isometric transformation(like reflection, rotation, glide reflection) can just be viewed as the effect of a translation. Am I correct? The same will hold for the punctured disc?
EDIT: I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as the translation only because the effect of other three isometries on punctured disk can be viewed as the special cases of translation, due to the 'nice' symmetry of punctured disc? Hope this clarifies my question.
real-analysis geometry differential-geometry euclidean-geometry analytic-geometry
real-analysis geometry differential-geometry euclidean-geometry analytic-geometry
asked Jan 6 at 17:59
ershersh
294112
asked Jan 6 at 17:59
ershersh
294112
edited Jan 7 at 21:13
ersh
asked Jan 6 at 17:59
ershersh
294112
asked Jan 6 at 17:59
ershersh
294112
asked Jan 6 at 17:59
ershersh
294112
294112
$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42
add a comment |
$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42
$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42
add a comment |
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$begingroup$
There are no translations on $D$ in Euclidean geometry. Are you perhaps referring to hyperbolic geometry using the Poincaré metric? And if you are, then I still do not understand your question: What do you mean by the assertion that "every other isometric transformation... can just be viewed as the effect of a translation"? What kind of "effect" did you have in mind?
$endgroup$
– Lee Mosher
Jan 6 at 18:41
$begingroup$
We have four isometries of plane: I am asking whether we can view Rotation, Reflection as a Translation in the following way?1)Rotating disc ≡ No translation. 2) Reflection of disc about some line ≡ Translation by some l(say?
$endgroup$
– ersh
Jan 6 at 20:09
$begingroup$
No, you cannot view them that way. Every (non-identity) isometry of the plane falls into exactly one of those four classes, there are no overlaps between them as you suggest. In particular, a reflection about some line is not a translation in any way. Nonetheless, I fear that I still misunderstand your question, because a particular open unit disc $D$ in the plane has no special relation to isometries of the plane.
$endgroup$
– Lee Mosher
Jan 6 at 22:46
$begingroup$
@LeeMosher I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as translation only because other the effect of other three isometries on punctured disk can be viewed as the special cases of translation?
$endgroup$
– ersh
Jan 7 at 19:42