On the Coincidences of Pascal Lines
Jaydeep Chipalkatti
October 19, 2014
Halifax, Nova Scotia.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Pascal’s Theorem
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
/ 12
Pascal’s Theorem
The points AE ∩ BF , AD ∩ CF , BD ∩ CE are collinear.
Pascal line
Jaydeep Chipalkatti ()
A B C
F E D
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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A B C
F E D
Jaydeep Chipalkatti ()
,
A D C
F B D
,
On the Coincidences of Pascal Lines
A C
D B
E
F
.
October 19, 2014 Halifax, Nova Scotia.
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A B C
F E D
,
A D C
F B D
,
A C
D B
E
F
.
In general, we get
6!
= 60 distinct Pascal lines.
2 × 3!
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Points in Involution
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Points in Involution
Then
A B D
A E C
F B C
A B C
.
=
=
=
F E C
F B D
A E D
F E D
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Given six points
in general position
Jaydeep Chipalkatti ()
sixty distinct lines.
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Given six points
in general position
in involution
Jaydeep Chipalkatti ()
sixty distinct lines.
four of the sixty coincide.
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Given six points
in general position
in involution
sixty distinct lines.
four of the sixty coincide.
Is the converse true?
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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in involution
four of the sixty coincide.
Is the converse true?
?
Some of the Pascals coincide ⇒
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The Main Theorem:
The converse is false, but almost true.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The Main Theorem:
The converse is false, but almost true.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The Main Theorem:
The converse is false, but almost true.
Given six distinct points, assume that some of the Pascals coincide.
Then the points must be
either in involution, or
in ricochet configuration.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The ricochet configuration
Start with A, B, C , D on the conic:
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The ricochet configuration
Start with A, B, C , D on the conic:
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The ricochet configuration
Start with A, B, C , D on the conic:
Then
Jaydeep Chipalkatti ()
A B C
F E D
=
A E C
D B F
On the Coincidences of Pascal Lines
.
October 19, 2014 Halifax, Nova Scotia.
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Recapitulation
Six points on a conic
Jaydeep Chipalkatti ()
6 degrees of freedom.
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Recapitulation
Six points on a conic
6 degrees of freedom.
5 degrees of freedom.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Recapitulation
Six points on a conic
6 degrees of freedom.
5 degrees of freedom.
4 degrees of freedom.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The Proof
Think of the conic as P1 . Let
A = 0,
Jaydeep Chipalkatti ()
B = 1,
C = ∞,
D = p,
On the Coincidences of Pascal Lines
E = q,
F = r.
October 19, 2014 Halifax, Nova Scotia.
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The Proof
Think of the conic as P1 . Let
A = 0,
Then
A B
F E
B = 1,
C
D
Jaydeep Chipalkatti ()
C = ∞,
D = p,
E = q,
F = r.
= [q − r , pr − pq + p − q, rq − rp ].
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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The Proof
Think of the conic as P1 . Let
A = 0,
Then
A B
F E
B = 1,
C
D
C = ∞,
D = p,
E = q,
F = r.
= [q − r , pr − pq + p − q, rq − rp ].
For the ‘other’ line, there are 59 (but really only 9) possibilities.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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A B C
F E D
Jaydeep Chipalkatti ()
= [q − r , pr − pq + p − q, rq − rp ].
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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A B C
F E D
= [q − r , pr − pq + p − q, rq − rp ].
Example (leads nowhere)
A D F
= [p − r , r − pq, pr (q − 1)]
C E B
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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A B C
F E D
= [q − r , pr − pq + p − q, rq − rp ].
Example (leads nowhere)
A D F
= [p − r , r − pq, pr (q − 1)]
C E B
The solutions
p = r = 0, q = 1, r = 0, q = p, r = 0,
p = q = 1, q = 1, r = p, p = q = r ,
are all useless.
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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As before
A B C
F E D
Jaydeep Chipalkatti ()
= [q − r , pr − pq + p − q, rq − rp ].
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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As before
A B C
F E D
= [q − r , pr − pq + p − q, rq − rp ].
Example (leads somewhere)
A B D
After discarding the rubbish,
leads to
E C F
q=
Jaydeep Chipalkatti ()
p
,
p+1
r=
p
.
1 − p2
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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As before
A B C
F E D
= [q − r , pr − pq + p − q, rq − rp ].
Example (leads somewhere)
A B D
After discarding the rubbish,
leads to
E C F
q=
p
,
p+1
r=
p
.
1 − p2
This fits into
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Out of nine cases,
five cases
three cases
one case
Jaydeep Chipalkatti ()
nowhere
points in involution
ricochet configuration
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Out of nine cases,
five cases
three cases
one case
Jaydeep Chipalkatti ()
nowhere
points in involution
ricochet configuration
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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Out of nine cases,
five cases
three cases
one case
nowhere
points in involution
ricochet configuration
Find out the defining equations for the 4-dimensional ‘ricochet
subvariety’ in P6 .
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
/ 12
Out of nine cases,
five cases
three cases
one case
nowhere
points in involution
ricochet configuration
Find out the defining equations for the 4-dimensional ‘ricochet
subvariety’ in P6 .
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
/ 12
Out of nine cases,
five cases
three cases
one case
nowhere
points in involution
ricochet configuration
Find out the defining equations for the 4-dimensional ‘ricochet
subvariety’ in P6 .
Thank you!
Jaydeep Chipalkatti ()
On the Coincidences of Pascal Lines
October 19, 2014 Halifax, Nova Scotia.
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