Mixing
Show that these 3 projective lines intersect on the same point
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I'm stuck with the following problem of projective geometry from an assignment:
Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that any 3 of them are not collinear. Define
$Q_1=overline{P_1Q}capoverline{P_2P_3}$, $Q_2=overline{P_2Q}capoverline{P_1P_3}$ and
$Q_3=overline{P_3Q}capoverline{P_1P_2}$. Now, let $R$ be another point not contained on the lines $overline{P_1P_2}$, $overline{P_1P_3}$ and $overline{P_2P_3}$ and define $R_1$, $R_2$ and $R_3$ the same way as above.
Let $alpha_1$ be the only automorphism of the line $overline{P_2P_3}$ such that $alpha_1(Q_1)=Q_1$, $alpha_1(P_2)=P_3$ and $alpha_1(P_3)=P_2$. Define $S_1=alpha_1(R_1)$. Similarly, define $S_2$ and $S_3$. Show that the lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$ intersect in the same point.
Since the statement is a bit messy, I tried to make a figure to help understanding it (which is also a bit messy):
I'm trying to use cross-ratio to show that the points $overline{P_1S_1}capoverline{P_2S_2}$ and $overline{P_2S_2}capoverline{P_3S_3}$ are equal: I would express the cross-ratio between one of this points and 3 other (fixed) points of the line $overline{P_2S_2}$ and, using the invariance properties of the cross-ratio, express it in terms of the other know points mentioned above, to show that the cross-ratio between the other point and the 3 fixed points are the same. But none of my attempts of finding adequate central projections helped me with that, the cross-ratios obtained would always involve "strange points" for which I don't know anything (i.e. points other than those mentioned above).
Could someone please help me with a hint (not a full solution, since it's an assignment)? Thanks!
algebraic-curves projective-geometry
asked Jan 23 at 2:10
Edu W.Edu W.
155
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add a comment |
$begingroup$
I'm stuck with the following problem of projective geometry from an assignment:
Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that any 3 of them are not collinear. Define
$Q_1=overline{P_1Q}capoverline{P_2P_3}$, $Q_2=overline{P_2Q}capoverline{P_1P_3}$ and
$Q_3=overline{P_3Q}capoverline{P_1P_2}$. Now, let $R$ be another point not contained on the lines $overline{P_1P_2}$, $overline{P_1P_3}$ and $overline{P_2P_3}$ and define $R_1$, $R_2$ and $R_3$ the same way as above.
Let $alpha_1$ be the only automorphism of the line $overline{P_2P_3}$ such that $alpha_1(Q_1)=Q_1$, $alpha_1(P_2)=P_3$ and $alpha_1(P_3)=P_2$. Define $S_1=alpha_1(R_1)$. Similarly, define $S_2$ and $S_3$. Show that the lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$ intersect in the same point.
Since the statement is a bit messy, I tried to make a figure to help understanding it (which is also a bit messy):
I'm trying to use cross-ratio to show that the points $overline{P_1S_1}capoverline{P_2S_2}$ and $overline{P_2S_2}capoverline{P_3S_3}$ are equal: I would express the cross-ratio between one of this points and 3 other (fixed) points of the line $overline{P_2S_2}$ and, using the invariance properties of the cross-ratio, express it in terms of the other know points mentioned above, to show that the cross-ratio between the other point and the 3 fixed points are the same. But none of my attempts of finding adequate central projections helped me with that, the cross-ratios obtained would always involve "strange points" for which I don't know anything (i.e. points other than those mentioned above).
Could someone please help me with a hint (not a full solution, since it's an assignment)? Thanks!
algebraic-curves projective-geometry
asked Jan 23 at 2:10
Edu W.Edu W.
155
$endgroup$
add a comment |
$begingroup$
I'm stuck with the following problem of projective geometry from an assignment:
Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that any 3 of them are not collinear. Define
$Q_1=overline{P_1Q}capoverline{P_2P_3}$, $Q_2=overline{P_2Q}capoverline{P_1P_3}$ and
$Q_3=overline{P_3Q}capoverline{P_1P_2}$. Now, let $R$ be another point not contained on the lines $overline{P_1P_2}$, $overline{P_1P_3}$ and $overline{P_2P_3}$ and define $R_1$, $R_2$ and $R_3$ the same way as above.
Let $alpha_1$ be the only automorphism of the line $overline{P_2P_3}$ such that $alpha_1(Q_1)=Q_1$, $alpha_1(P_2)=P_3$ and $alpha_1(P_3)=P_2$. Define $S_1=alpha_1(R_1)$. Similarly, define $S_2$ and $S_3$. Show that the lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$ intersect in the same point.
Since the statement is a bit messy, I tried to make a figure to help understanding it (which is also a bit messy):
I'm trying to use cross-ratio to show that the points $overline{P_1S_1}capoverline{P_2S_2}$ and $overline{P_2S_2}capoverline{P_3S_3}$ are equal: I would express the cross-ratio between one of this points and 3 other (fixed) points of the line $overline{P_2S_2}$ and, using the invariance properties of the cross-ratio, express it in terms of the other know points mentioned above, to show that the cross-ratio between the other point and the 3 fixed points are the same. But none of my attempts of finding adequate central projections helped me with that, the cross-ratios obtained would always involve "strange points" for which I don't know anything (i.e. points other than those mentioned above).
Could someone please help me with a hint (not a full solution, since it's an assignment)? Thanks!
algebraic-curves projective-geometry
asked Jan 23 at 2:10
Edu W.Edu W.
155
$endgroup$
I'm stuck with the following problem of projective geometry from an assignment:
Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that any 3 of them are not collinear. Define
$Q_1=overline{P_1Q}capoverline{P_2P_3}$, $Q_2=overline{P_2Q}capoverline{P_1P_3}$ and
$Q_3=overline{P_3Q}capoverline{P_1P_2}$. Now, let $R$ be another point not contained on the lines $overline{P_1P_2}$, $overline{P_1P_3}$ and $overline{P_2P_3}$ and define $R_1$, $R_2$ and $R_3$ the same way as above.
Let $alpha_1$ be the only automorphism of the line $overline{P_2P_3}$ such that $alpha_1(Q_1)=Q_1$, $alpha_1(P_2)=P_3$ and $alpha_1(P_3)=P_2$. Define $S_1=alpha_1(R_1)$. Similarly, define $S_2$ and $S_3$. Show that the lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$ intersect in the same point.
Since the statement is a bit messy, I tried to make a figure to help understanding it (which is also a bit messy):
I'm trying to use cross-ratio to show that the points $overline{P_1S_1}capoverline{P_2S_2}$ and $overline{P_2S_2}capoverline{P_3S_3}$ are equal: I would express the cross-ratio between one of this points and 3 other (fixed) points of the line $overline{P_2S_2}$ and, using the invariance properties of the cross-ratio, express it in terms of the other know points mentioned above, to show that the cross-ratio between the other point and the 3 fixed points are the same. But none of my attempts of finding adequate central projections helped me with that, the cross-ratios obtained would always involve "strange points" for which I don't know anything (i.e. points other than those mentioned above).
Could someone please help me with a hint (not a full solution, since it's an assignment)? Thanks!
algebraic-curves projective-geometry
algebraic-curves projective-geometry
asked Jan 23 at 2:10
Edu W.Edu W.
155
asked Jan 23 at 2:10
Edu W.Edu W.
155
asked Jan 23 at 2:10
Edu W.Edu W.
155
asked Jan 23 at 2:10
Edu W.Edu W.
155
asked Jan 23 at 2:10
Edu W.Edu W.
155
155
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add a comment |
1 Answer
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The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:
One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.
With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$.
answered Jan 26 at 12:35
Edu W.Edu W.
155
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1 Answer
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The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:
One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.
With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$.
answered Jan 26 at 12:35
Edu W.Edu W.
155
$endgroup$
add a comment |
$begingroup$
The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:
One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.
With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$.
answered Jan 26 at 12:35
Edu W.Edu W.
155
$endgroup$
add a comment |
$begingroup$
The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:
One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.
With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$.
answered Jan 26 at 12:35
Edu W.Edu W.
155
$endgroup$
The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:
One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.
With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $overline{P_1S_1}$, $overline{P_2S_2}$ and $overline{P_3S_3}$.
answered Jan 26 at 12:35
Edu W.Edu W.
155
answered Jan 26 at 12:35
Edu W.Edu W.
155
answered Jan 26 at 12:35
Edu W.Edu W.
155
answered Jan 26 at 12:35
Edu W.Edu W.
155
155
add a comment |
add a comment |
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