Algebraic intuition behind inversion of circles
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I am struggling to understand inversions of shapes (namely, circles and lines) algebraically. For one, I understand how to apply inversions to general equations of circles and lines (by replacing all instances of the variables $x$ and $y$ with $1/x$ and $1/y$ respectively), but I can't seem to go the further level by applying this to, say, 2 tangent circles with inversions at point of tangency (or another point). Apparently,
tangent circles with an inversion NOT at the point of tangency become the same tangent cirlces
tangent circles with an inversion at the point of tangency become a pair of parallel lines perpendicular to the segment joining their centres
But, I cannot seem to formulate equations for these more complex and interacting cases.
geometry euclidean-geometry circle inversive-geometry
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
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add a comment |
$begingroup$
I am struggling to understand inversions of shapes (namely, circles and lines) algebraically. For one, I understand how to apply inversions to general equations of circles and lines (by replacing all instances of the variables $x$ and $y$ with $1/x$ and $1/y$ respectively), but I can't seem to go the further level by applying this to, say, 2 tangent circles with inversions at point of tangency (or another point). Apparently,
tangent circles with an inversion NOT at the point of tangency become the same tangent cirlces
tangent circles with an inversion at the point of tangency become a pair of parallel lines perpendicular to the segment joining their centres
But, I cannot seem to formulate equations for these more complex and interacting cases.
geometry euclidean-geometry circle inversive-geometry
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
$endgroup$
$begingroup$
@Lubin yes you are right, my bad. I edited it.
$endgroup$
– helpneeded
Dec 19 '18 at 18:08
add a comment |
$begingroup$
I am struggling to understand inversions of shapes (namely, circles and lines) algebraically. For one, I understand how to apply inversions to general equations of circles and lines (by replacing all instances of the variables $x$ and $y$ with $1/x$ and $1/y$ respectively), but I can't seem to go the further level by applying this to, say, 2 tangent circles with inversions at point of tangency (or another point). Apparently,
tangent circles with an inversion NOT at the point of tangency become the same tangent cirlces
tangent circles with an inversion at the point of tangency become a pair of parallel lines perpendicular to the segment joining their centres
But, I cannot seem to formulate equations for these more complex and interacting cases.
geometry euclidean-geometry circle inversive-geometry
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
$endgroup$
I am struggling to understand inversions of shapes (namely, circles and lines) algebraically. For one, I understand how to apply inversions to general equations of circles and lines (by replacing all instances of the variables $x$ and $y$ with $1/x$ and $1/y$ respectively), but I can't seem to go the further level by applying this to, say, 2 tangent circles with inversions at point of tangency (or another point). Apparently,
tangent circles with an inversion NOT at the point of tangency become the same tangent cirlces
tangent circles with an inversion at the point of tangency become a pair of parallel lines perpendicular to the segment joining their centres
But, I cannot seem to formulate equations for these more complex and interacting cases.
geometry euclidean-geometry circle inversive-geometry
geometry euclidean-geometry circle inversive-geometry
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
edited Dec 19 '18 at 18:18
helpneeded
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
asked Dec 19 '18 at 17:46
helpneededhelpneeded
897
897
$begingroup$
@Lubin yes you are right, my bad. I edited it.
$endgroup$
– helpneeded
Dec 19 '18 at 18:08
add a comment |
$begingroup$
@Lubin yes you are right, my bad. I edited it.
$endgroup$
– helpneeded
Dec 19 '18 at 18:08
$begingroup$
@Lubin yes you are right, my bad. I edited it.
$endgroup$
– helpneeded
Dec 19 '18 at 18:08
$begingroup$
@Lubin yes you are right, my bad. I edited it.
$endgroup$
– helpneeded
Dec 19 '18 at 18:08
add a comment |
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@Lubin yes you are right, my bad. I edited it.
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– helpneeded
Dec 19 '18 at 18:08