Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Projective Geometry and Pappus’ Theorem
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Kelly McKinnie
March 23, 2010
Pappus of Alexandria
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
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Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Pappus of Alexandria
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Pappus of Alexandria was a Greek mathematician.
Pappus of Alexandria
Projective
Geometry and
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Theorem
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McKinnie
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Theorem
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Theorem
Proof of
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Theorem
Pappus of Alexandria was a Greek mathematician.
He lived around the time of the 3rd century AD.
Pappus of Alexandria
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
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Picturing the
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plane
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geometry
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Theorem
Proof of
Pappus’
Theorem
Pappus of Alexandria was a Greek mathematician.
He lived around the time of the 3rd century AD.
Appears to have witnessed a solar eclipse in 320 AD, but
there is some confusion about this.
Pappus of Alexandria - Mathematical contributions
Projective
Geometry and
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Theorem
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Pappus of Alexandria - Mathematical contributions
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
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Theorem
Proof of
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Theorem
His major contribution is a book called Synagoge, also known
as “Mathematical Collections”.
Pappus of Alexandria - Mathematical contributions
Projective
Geometry and
Pappus’
Theorem
His major contribution is a book called Synagoge, also known
as “Mathematical Collections”.
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
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projective
plane
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geometry
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Theorem
Proof of
Pappus’
Theorem
The Mathematical Collections of Pappus in a translation of
Federico Commandino (1589).
Vatican Exhibit of Pappus’ Collections
Projective
Geometry and
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Theorem
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McKinnie
History
Pappus’
Theorem
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Theorem
Proof of
Pappus’
Theorem
Vatican Exhibit of Pappus’ Collections
Projective
Geometry and
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Theorem
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Pappus’
Theorem
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Theorem
Proof of
Pappus’
Theorem
Caption on the exhibit:
Vatican Exhibit of Pappus’ Collections
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Caption on the exhibit:
History
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Theorem
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Proof of
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Theorem
“Pappus’s “Collection,” consisting of supplements to earlier
treatises on geometry, astronomy, and mechanics, dates from
the late third century A.D. and is the last important work of
Greek mathematics. This manuscript reached the papal library
in the thirteenth century, and is the archetype of all later
copies, of which none is earlier than the sixteenth century. ”
Courtesy of http://www.ibiblio.org
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
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McKinnie
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Pappus’
Theorem
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geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
One piece of mathematics in ”Collection” is a seemingly
original theorem that has since become known as Pappus’
Theorem.
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
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McKinnie
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Pappus’
Theorem
Geometries
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plane
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Pappus’
Theorem
Proof of
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
Kelly
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Theorem
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Theorem
Proof of
Pappus’
Theorem
A
B
C
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line.
History
Pappus’
Theorem
Geometries
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plane
A
B
C
Lines in
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geometry
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Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line. Then the three pairwise intersections AB 0 ∩ A0 B
Pappus’
Theorem
Geometries
Picturing the
projective
plane
A
B
C
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line. Then the three pairwise intersections AB 0 ∩ A0 B,
AC 0 ∩ A0 C
Geometries
Picturing the
projective
plane
A
B
C
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line. Then the three pairwise intersections AB 0 ∩ A0 B,
AC 0 ∩ A0 C and BC 0 ∩ B 0 C
Geometries
Picturing the
projective
plane
A
B
C
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line. Then the three pairwise intersections AB 0 ∩ A0 B,
BC 0 ∩ C 0 B and AC 0 ∩ A0 C are incident to a third straight line.
Geometries
Picturing the
projective
plane
A
B
C
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Let three points A, B, C be incident to a single straight line
and another three points A0 , B 0 , C 0 incident to another straight
line. Then the three pairwise intersections AB 0 ∩ A0 B,
BC 0 ∩ C 0 B and AC 0 ∩ A0 C are incident to a third straight line.
Geometries
Picturing the
projective
plane
A
B
C
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Java Applet on the web at
http://www.cut-the-knot.org/pythagoras/Pappus.shtml
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
As you can see from the Java Applet, Pappus’ Theorem should
really read:
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
As you can see from the Java Applet, Pappus’ Theorem should
really read: Let three points A, B, C be incident to a single
straight line and another three points A0 , B 0 , C 0 incident to
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
As you can see from the Java Applet, Pappus’ Theorem should
really read: Let three points A, B, C be incident to a single
straight line and another three points A0 , B 0 , C 0 incident to
(generally speaking)
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
As you can see from the Java Applet, Pappus’ Theorem should
really read: Let three points A, B, C be incident to a single
straight line and another three points A0 , B 0 , C 0 incident to
(generally speaking) another straight line. Then the three
pairwise intersections AB 0 ∩ A0 B, BC 0 ∩ C 0 B and AC 0 ∩ A0 C
are incident to a third straight line.
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
As you can see from the Java Applet, Pappus’ Theorem should
really read: Let three points A, B, C be incident to a single
straight line and another three points A0 , B 0 , C 0 incident to
(generally speaking) another straight line. Then the three
pairwise intersections AB 0 ∩ A0 B, BC 0 ∩ C 0 B and AC 0 ∩ A0 C
are incident to a third straight line.
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
i.e., in Euclidean Geometry we shouldn’t allow the points A0 ,
B 0 , C 0 to be in the configuration
Pappus of Alexandria - Pappus’ Theorem
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
i.e., in Euclidean Geometry we shouldn’t allow the points A0 ,
B 0 , C 0 to be in the configuration
Pappus’
Theorem
Geometries
B
A
C
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plane
Lines in
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geometry
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Theorem
Proof of
Pappus’
Theorem
A'
B'
C'
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
We would like to put Pappus’ Theorem in a setting where we
don’t have to make lots of EXCEPTIONS for certain
configurations.
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
We would like to put Pappus’ Theorem in a setting where we
don’t have to make lots of EXCEPTIONS for certain
configurations.
We need a new “geometry” in which EVERY non-equal pair of
lines meets in exactly one point.
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
History
Pappus’
Theorem
Geometries
Picturing the
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plane
Lines in
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geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
How do we describe a line in the Euclidean plane?
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
How do we describe a line in the Euclidean plane?
They are the points that are the solutions to
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
History
Pappus’
Theorem
How do we describe a line in the Euclidean plane?
They are the points that are the solutions to
Geometries
y = mx + b
Picturing the
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Pappus’
Theorem
Proof of
Pappus’
Theorem
for fixed m and b ∈ R,
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
History
Pappus’
Theorem
How do we describe a line in the Euclidean plane?
They are the points that are the solutions to
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
y = mx + b
for fixed m and b ∈ R,or solutions to
x =a
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
Recall in 2-dimensional (plane) Euclidean geometry, every point
is given by a pair (x, y ) with x and y ∈ R.
History
Pappus’
Theorem
How do we describe a line in the Euclidean plane?
They are the points that are the solutions to
Geometries
y = mx + b
Picturing the
projective
plane
Lines in
projective
geometry
for fixed m and b ∈ R,or solutions to
x =a
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
for a fixed a ∈ R.
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Two lines
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Two lines
y = m 1 x + b1
Kelly
McKinnie
and
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Pappus’
Theorem
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Theorem
Proof of
Pappus’
Theorem
y = m 2 x + b2
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Two lines
y = m 1 x + b1
Kelly
McKinnie
and
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
y = m 2 x + b2
intersect if there is a solution
m1 x + b1 = m2 x + b2
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Two lines
y = m 1 x + b1
Kelly
McKinnie
and
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
y = m 2 x + b2
intersect if there is a solution
m1 x + b1 = m2 x + b2
(m1 − m2 )x
= b2 − b1
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Two lines
y = m 1 x + b1
Kelly
McKinnie
and
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
y = m 2 x + b2
intersect if there is a solution
m1 x + b1 = m2 x + b2
(m1 − m2 )x
x
= b2 − b1
b2 − b1
=
m1 − m2
Euclidean Geometry - Review
Projective
Geometry and
Pappus’
Theorem
Two lines
y = m 1 x + b1
Kelly
McKinnie
and
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
y = m 2 x + b2
intersect if there is a solution
m1 x + b1 = m2 x + b2
(m1 − m2 )x
x
= b2 − b1
b2 − b1
=
m1 − m2
i.e., we need m1 6= m2 in order for there to exist a point of
intersection.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
We want to fix this “flaw” of Euclidean geometry.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
We want to fix this “flaw” of Euclidean geometry. We want
every pair of distinct lines to intersect in exactly one point.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Where else have people noted a need for “extra” points?
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
Where else have people noted a need for “extra” points? In
perspective drawing in art.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
The point on the horizon that looks like the intersection of the
parallel lines is called an ideal point.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
Satire on false perspective by William Hogarth.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
To make the new geometry we need to decide:
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
To make the new geometry we need to decide:
What are the points of the new geometry?
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
To make the new geometry we need to decide:
What are the points of the new geometry?
We want to incorporate all of the Euclidean points into our
projective geometry.
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
To make the new geometry we need to decide:
What are the points of the new geometry?
We want to incorporate all of the Euclidean points into our
projective geometry.
What are the lines of the new geometry?
Projective Geometry
Projective
Geometry and
Pappus’
Theorem
Kelly
McKinnie
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
To make the new geometry we need to decide:
What are the points of the new geometry?
We want to incorporate all of the Euclidean points into our
projective geometry.
What are the lines of the new geometry?
We want to incorporate our old lines into our new ones.
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To make the new geometry we need to decide:
What are the points of the new geometry?
We want to incorporate all of the Euclidean points into our
projective geometry.
What are the lines of the new geometry?
We want to incorporate our old lines into our new ones.
We want all of our new lines to intersect in exactly one
point.
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We don’t want to add more points than we need.
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We don’t want to add more points than we need. What we
really need is a new point for every SLOPE in the Euclidean
plane. Then we can say that two parallel lines intersect at this
new point, the point that corresponds to their slope.
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New Point!
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New point!
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How do we do this concretely?
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
two points [x1 , y1 , z1 ] and [x2 , y2 , z2 ] are equivalent if there
is a scalar λ ∈ R − {0} such that
λ[x1 , y1 , z1 ] = [x2 , y2 , z2 ].
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
two points [x1 , y1 , z1 ] and [x2 , y2 , z2 ] are equivalent if there
is a scalar λ ∈ R − {0} such that
λ[x1 , y1 , z1 ] = [x2 , y2 , z2 ].
The symbol for the projective plane is RP2 .
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
two points [x1 , y1 , z1 ] and [x2 , y2 , z2 ] are equivalent if there
is a scalar λ ∈ R − {0} such that
λ[x1 , y1 , z1 ] = [x2 , y2 , z2 ].
The symbol for the projective plane is RP2 .
For example, in the projective plane,
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
two points [x1 , y1 , z1 ] and [x2 , y2 , z2 ] are equivalent if there
is a scalar λ ∈ R − {0} such that
λ[x1 , y1 , z1 ] = [x2 , y2 , z2 ].
The symbol for the projective plane is RP2 .
For example, in the projective plane,
[1, 2, 3] = [5, 10, 15] and [2, 2, 2] = [3, 3, 3].
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Definition
A point in the projective plane is given by a TRIPLE [x, y , z]
with x, y , z ∈ R such that
not all x = 0, y = 0 and z = 0, and
two points [x1 , y1 , z1 ] and [x2 , y2 , z2 ] are equivalent if there
is a scalar λ ∈ R − {0} such that
λ[x1 , y1 , z1 ] = [x2 , y2 , z2 ].
The symbol for the projective plane is RP2 .
For example, in the projective plane,
[1, 2, 3] = [5, 10, 15] and [2, 2, 2] = [3, 3, 3].
[0, 0, 0] is not a point in the projective plane!
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In other words, a point of the projective plane [x, y , z]
corresponds to all points on the line through the origin
containing the point (x, y , z) in Euclidean 3-space (R3 ) except
the origin.
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k(x,y,z)
The red line is all scalar multiplies of (x,y,z)
The projective plane
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y
x
The projective plane
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y
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How does the Euclidean plane “fit into” the projective plane?
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How does the Euclidean plane “fit into” the projective plane?
The Euclidean point (x, y ) is the point [x, y , 1] in RP2 .
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How does the Euclidean plane “fit into” the projective plane?
The Euclidean point (x, y ) is the point [x, y , 1] in RP2 .
Every point [x, y , z] ∈ RP2 with z 6= 0 is equivalent to
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x y
[ , , 1].
z z
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(x, y, z)
(x/z, y/z, 1)
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So the only “new points” we are adding are those with the
third coordinate z = 0.
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A line in RP2 is given by the solutions to
Ax + By + Cz = 0
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for some A, B, C ∈ R, not all zero.
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A line in RP2 is given by the solutions to
Ax + By + Cz = 0
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for some A, B, C ∈ R, not all zero.
Since we have many representations for one point, we need to
check that this makes sense!
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
Aλx0 + Bλy0 + C λz0 =
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
Aλx0 + Bλy0 + C λz0 = λ(Ax0 + By0 + Cz0 )
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
Aλx0 + Bλy0 + C λz0 = λ(Ax0 + By0 + Cz0 )
= λ(0)
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
Aλx0 + Bλy0 + C λz0 = λ(Ax0 + By0 + Cz0 )
= λ(0)
= 0
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If the point [x0 , y0 , z0 ] satisfies Ax + By + Cz = 0, does
λ[x0 , y0 , z0 ] = [λx0 , λy0 , λz0 ] also satisfy the equation?
Aλx0 + Bλy0 + C λz0 = λ(Ax0 + By0 + Cz0 )
= λ(0)
= 0
So the equation of the line is well defined.
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You can see the Euclidean lines by setting z = 1:
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You can see the Euclidean lines by setting z = 1:
Ax + By + Cz = 0 − Projective line
Ax + By + C = 0 − Euclidean line
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You can see the Euclidean lines by setting z = 1:
Ax + By + Cz = 0 − Projective line
Ax + By + C = 0 − Euclidean line
As long as B 6= 0, the Euclidean line has slope
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You can see the Euclidean lines by setting z = 1:
Ax + By + Cz = 0 − Projective line
Ax + By + C = 0 − Euclidean line
As long as B 6= 0, the Euclidean line has slope m = −A/B.
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In projective geometry the line given by z = 0 is called the line
at infinity.
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In projective geometry the line given by z = 0 is called the line
at infinity. The points on the line at infinity are of the form
[x, y , 0] with not both x = 0 and y = 0.
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Let’s compute some intersections.
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
{[x, y , 0]} ∩ {[x, y , z] | Ax + By + Cz = 0}
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
{[x, y , 0]} ∩ {[x, y , z] | Ax + By + Cz = 0}
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= {[x, y , 0] | Ax + By = 0}
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
{[x, y , 0]} ∩ {[x, y , z] | Ax + By + Cz = 0}
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= {[x, y , 0] | Ax + By = 0}
A
= {[x, y , 0] | y = − x}
B
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
{[x, y , 0]} ∩ {[x, y , z] | Ax + By + Cz = 0}
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= {[x, y , 0] | Ax + By = 0}
A
= {[x, y , 0] | y = − x}
B
A
= {[x, − x, 0]}
B
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Let’s compute some intersections. What is
{Line at ∞} ∩ {line given by Ax +By +Cz = 0 with B 6= 0}?
{[x, y , 0]} ∩ {[x, y , z] | Ax + By + Cz = 0}
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= {[x, y , 0] | Ax + By = 0}
A
= {[x, y , 0] | y = − x}
B
A
= {[x, − x, 0]}
B
A
= [1, − , 0]
B
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The line Ax + By + Cz = 0 and the line at infinity intersect at
the point [1, m, 0] corresponding to the slope m = −A/B of
the Euclidean line Ax + By + C = 0.
Projective Geometry - lines
Projective
Geometry and
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The line Ax + By + Cz = 0 and the line at infinity intersect at
the point [1, m, 0] corresponding to the slope m = −A/B of
the Euclidean line Ax + By + C = 0.
Pappus’
Theorem
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Proof of
Pappus’
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In particular, any two distinct parallel lines with slope m
(m 6= ∞) intersect at the line at infinity at the projective point
[1, m, 0].
Projective Geometry - lines
Projective
Geometry and
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The line Ax + By + Cz = 0 and the line at infinity intersect at
the point [1, m, 0] corresponding to the slope m = −A/B of
the Euclidean line Ax + By + C = 0.
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
In particular, any two distinct parallel lines with slope m
(m 6= ∞) intersect at the line at infinity at the projective point
[1, m, 0].
You an check that all vertical lines intersect at the projective
point [0, 1, 0].
Projective Geometry - lines
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Pappus’
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Theorem
Proof of
Pappus’
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Projective Geometry - lines
Projective
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Pappus’
Theorem
Geometries
Picturing the
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Lines in
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geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
Projective Geometry - lines
Projective
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Pappus’
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Picturing the
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Lines in
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geometry
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Pappus’
Theorem
Proof of
Pappus’
Theorem
Projective Geometry - lines
Projective
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Pappus’
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Geometries
Picturing the
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Lines in
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geometry
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Proof of
Pappus’
Theorem
[1,1,0]
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Pappus’
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Proof of
Pappus’
Theorem
Let’s take another look at the configuration of points in
Pappus’ Theorem that didn’t work.
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B
A
C
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A'
B'
C'
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B
A
C
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Proof of
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A'
B'
C'
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B
A
C
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Proof of
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A'
B'
C'
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B
A
C
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A'
B'
C'
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B
A
C
Pappus’
Theorem
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Proof of
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A'
B'
C'
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[1,m,0]
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B
A
C
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Theorem
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Proof of
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A'
B'
C'
Proof of Pappus’ Theorem
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Pappus’
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Proof of
Pappus’
Theorem
Since we now know that parallel lines really do intersect, we
can proceed to prove Pappus’ Theorem under the assumption
that all the lines intersect.
Proof of Pappus’ Theorem
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Proof of
Pappus’
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Let Lij denote the line through points Pi and Pj , and let a, b, c
denote the points of intersection between the pairs of lines
[L15 , L24 ], [L16 , L34 ], and [L35 , L26 ] respectively. Pappus’
Theorem asserts that the points a, b, c lie on a straight line.
Proof of Pappus’ Theorem
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Pappus’
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Theorem
Proof of
Pappus’
Theorem
One proof proceeds by explicitly determining the coordinates of
the points a, b, c, it is straightforward, but less trivial than one
might think.
Proof of Pappus’ Theorem
Projective
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Pappus’
Theorem
Geometries
Picturing the
projective
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Lines in
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geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
One proof proceeds by explicitly determining the coordinates of
the points a, b, c, it is straightforward, but less trivial than one
might think. Let Pi = (xi , yi ).
Proof of Pappus’ Theorem
Projective
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Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
One proof proceeds by explicitly determining the coordinates of
the points a, b, c, it is straightforward, but less trivial than one
might think. Let Pi = (xi , yi ). We can assume that
y1 , y2 , y3 = 0 by rotating and sliding the picture so that P1 ,
P2 , and P3 lie on the x-axis and O lies at the origin.
Proof of Pappus’ Theorem
Projective
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Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
One proof proceeds by explicitly determining the coordinates of
the points a, b, c, it is straightforward, but less trivial than one
might think. Let Pi = (xi , yi ). We can assume that
y1 , y2 , y3 = 0 by rotating and sliding the picture so that P1 ,
P2 , and P3 lie on the x-axis and O lies at the origin. Then for
i = 4, 5, 6, yi = kxi where k is the slope of the line OP6 .
Proof of Pappus’ Theorem
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Pappus’
Theorem
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Picturing the
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Theorem
Proof of
Pappus’
Theorem
You can now determine the coordinates of a, b and c explicitly
by writing down the equations for the lines L15 , L24 , L16 , L34 ,
L35 and L26 .
Proof of Pappus’ Theorem
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You can now determine the coordinates of a, b and c explicitly
by writing down the equations for the lines L15 , L24 , L16 , L34 ,
L35 and L26 .
History
Pappus’
Theorem
Geometries
Picturing the
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Pappus’
Theorem
Proof of
Pappus’
Theorem
You can then calculate the equation of the line through a and c
and prove that b lies on that line.
Proof of Pappus’ Theorem
Projective
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Theorem
Kelly
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You can now determine the coordinates of a, b and c explicitly
by writing down the equations for the lines L15 , L24 , L16 , L34 ,
L35 and L26 .
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
You can then calculate the equation of the line through a and c
and prove that b lies on that line.
There is a lot to keep track of!!
Proof of Pappus’ Theorem
Projective
Geometry and
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Theorem
Kelly
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You can now determine the coordinates of a, b and c explicitly
by writing down the equations for the lines L15 , L24 , L16 , L34 ,
L35 and L26 .
History
Pappus’
Theorem
Geometries
Picturing the
projective
plane
Lines in
projective
geometry
Back to
Pappus’
Theorem
Proof of
Pappus’
Theorem
You can then calculate the equation of the line through a and c
and prove that b lies on that line.
There is a lot to keep track of!!
This is the most “straightforward” proof of Pappus’ Theorem.
There are other theorems that make extensive use of the theory
of projective geometry.