Projective Geometry and Pappus' Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie History Projective Geometry and Pappus’ Theorem Pappus’ Theorem

Geometries

Picturing the Kelly McKinnie projective plane

Lines in projective geometry March 23, 2010 Back to Pappus’ 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

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 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

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Pappus of Alexandria was a Greek mathematician. Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Appears to have witnessed a solar eclipse in 320 AD, but there is some confusion about this.

Pappus of Alexandria

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Pappus of Alexandria was a Greek mathematician. Theorem He lived around the time of the 3rd century AD. Geometries

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’ Pappus of Alexandria was a Greek mathematician. Theorem He lived around the time of the 3rd century AD. Geometries

Picturing the Appears to have witnessed a solar eclipse in 320 AD, but projective plane there is some confusion about this.

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem His major contribution is a book called Synagoge, also known as “Mathematical Collections”.

The Mathematical Collections of Pappus in a translation of Federico Commandino (1589).

Pappus of Alexandria - Mathematical contributions

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 Mathematical Collections of Pappus in a translation of Federico Commandino (1589).

Pappus of Alexandria - Mathematical contributions

Projective His major contribution is a book called Synagoge, also known Geometry and Pappus’ as “Mathematical Collections”. 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 - Mathematical contributions

Projective His major contribution is a book called Synagoge, also known Geometry and Pappus’ as “Mathematical Collections”. Theorem

Kelly McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ 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 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’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

Vatican Exhibit of Pappus’ Collections

Projective Geometry and Pappus’ Theorem Kelly Caption on the exhibit: McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Vatican Exhibit of Pappus’ Collections

Projective Geometry and Pappus’ Theorem Kelly Caption on the exhibit: McKinnie

History

Pappus’ Theorem “Pappus’s “Collection,” consisting of supplements to earlier Geometries treatises on geometry, astronomy, and mechanics, dates from Picturing the projective the late third century A.D. and is the last important work of plane Greek mathematics. This manuscript reached the papal library Lines in projective in the thirteenth century, and is the archetype of all later geometry copies, of which none is earlier than the sixteenth century. ” Back to Pappus’ Courtesy of http://www.ibiblio.org Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem One piece of mathematics in ”Collection” is a seemingly

Geometries original theorem that has since become known as Pappus’

Picturing the Theorem. projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem C B A

Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Let three points A, B, C be incident to a single straight line 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 - Pappus’ Theorem

Projective Geometry and Pappus’ Let three points A, B, C be incident to a single straight line Theorem

Kelly McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective C plane B Lines in A projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Let three points A, B, C be incident to a single straight line 0 0 0 Kelly and another three points A , B , C incident to another straight McKinnie line. History

Pappus’ Theorem

Geometries Picturing the C projective B plane A Lines in projective geometry

Back to Pappus’ Theorem A' B' C' Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Pappus’ Theorem

Geometries C Picturing the B projective A plane

Lines in projective geometry

Back to Pappus’ Theorem A' B' C'

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Let three points A, B, C be incident to a single straight line Kelly McKinnie and another three points A0, B0, C 0 incident to another straight 0 0 History line. Then the three pairwise intersections AB ∩ A B, 0 0 Pappus’ AC ∩ A C Theorem Geometries C Picturing the B projective A plane

Lines in projective geometry

Back to Pappus’ A' B' C' Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Let three points A, B, C be incident to a single straight line Kelly McKinnie and another three points A0, B0, C 0 incident to another straight 0 0 History line. Then the three pairwise intersections AB ∩ A B, 0 0 0 0 Pappus’ AC ∩ A C and BC ∩ B C Theorem Geometries C Picturing the B projective A plane

Lines in projective geometry

Back to Pappus’ A' B' C' Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Let three points A, B, C be incident to a single straight line Kelly 0 0 0 McKinnie and another three points A , B , C incident to another straight 0 0 History line. Then the three pairwise intersections AB ∩ A B, 0 0 0 0 Pappus’ BC ∩ C B and AC ∩ A C are incident to a third straight line. Theorem Geometries C Picturing the B projective A plane

Lines in projective geometry

Back to Pappus’ A' B' C' Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Lines in projective geometry

Back to Pappus’ A' B' C' Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem Java Applet on the web at Geometries http://www.cut-the-knot.org/pythagoras/Pappus.shtml Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Let three points A, B, C be incident to a single straight line and another three points A0, B0, C 0 incident to (generally speaking) another straight line. Then the three pairwise intersections AB0 ∩ A0B, BC 0 ∩ C 0B and AC 0 ∩ A0C are incident to a third straight line.

Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History As you can see from the Java Applet, Pappus’ Theorem should Pappus’ really read: Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem (generally speaking) another straight line. Then the three pairwise intersections AB0 ∩ A0B, BC 0 ∩ C 0B and AC 0 ∩ A0C are incident to a third straight line.

Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem another straight line. Then the three pairwise intersections AB0 ∩ A0B, BC 0 ∩ C 0B and AC 0 ∩ A0C are incident to a third straight line.

Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History As you can see from the Java Applet, Pappus’ Theorem should Pappus’ really read: Let three points A, B, C be incident to a single Theorem straight line and another three points A0, B0, C 0 incident to Geometries (generally speaking) Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Lines in are incident to a third straight line. projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Lines in are incident to a third straight line. projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem C A B

A' B' C'

Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Kelly 0 McKinnie i.e., in Euclidean Geometry we shouldn’t allow the points A , B0, C 0 to be in the conﬁguration History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Pappus of Alexandria - Pappus’ Theorem

Projective Geometry and Pappus’ Theorem Kelly 0 McKinnie i.e., in Euclidean Geometry we shouldn’t allow the points A , B0, C 0 to be in the conﬁguration History

Pappus’ Theorem C A Geometries B Picturing the projective plane

Lines in projective geometry A' B' C' Back to Pappus’ Theorem

Proof of Pappus’ Theorem We need a new “geometry” in which EVERY non-equal pair of lines meets in exactly one point.

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History We would like to put Pappus’ Theorem in a setting where we Pappus’ Theorem don’t have to make lots of EXCEPTIONS for certain

Geometries conﬁgurations.

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History We would like to put Pappus’ Theorem in a setting where we Pappus’ Theorem don’t have to make lots of EXCEPTIONS for certain

Geometries conﬁgurations.

Picturing the We need a new “geometry” in which EVERY non-equal pair of projective plane lines meets in exactly one point.

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.

How do we describe a line in the Euclidean plane? They are the points that are the solutions to

y = mx + b

for ﬁxed m and b ∈ R,or solutions to

x = a

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 How do we describe a line in the Euclidean plane? They are the points that are the solutions to

y = mx + b

for ﬁxed m and b ∈ R,or solutions to

x = a

for a ﬁxed a ∈ R.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem They are the points that are the solutions to

y = mx + b

for ﬁxed m and b ∈ R,or solutions to

x = a

for a ﬁxed a ∈ R.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ How do we describe a line in the Euclidean plane? Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem y = mx + b

for ﬁxed m and b ∈ R,or solutions to

x = a

for a ﬁxed a ∈ R.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ How do we describe a line in the Euclidean plane? Theorem They are the points that are the solutions to Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem or solutions to

x = a

for a ﬁxed a ∈ R.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ How do we describe a line in the Euclidean plane? Theorem They are the points that are the solutions to Geometries

Picturing the projective y = mx + b plane Lines in for ﬁxed m and b ∈ , projective R geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem for a ﬁxed a ∈ R.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ How do we describe a line in the Euclidean plane? Theorem They are the points that are the solutions to Geometries

Picturing the projective y = mx + b plane Lines in for ﬁxed m and b ∈ ,or solutions to projective R geometry

Back to x = a Pappus’ Theorem

Proof of Pappus’ Theorem Euclidean Geometry - Review

Projective Geometry and Pappus’ Theorem Recall in 2-dimensional (plane) Euclidean geometry, every point Kelly McKinnie is given by a pair (x, y) with x and y ∈ R.

History

Pappus’ How do we describe a line in the Euclidean plane? Theorem They are the points that are the solutions to Geometries

Picturing the projective y = mx + b plane Lines in for ﬁxed m and b ∈ ,or solutions to projective R geometry

Back to x = a Pappus’ Theorem

Proof of for a ﬁxed a ∈ R. Pappus’ Theorem Two lines y = m1x + b1 and y = m2x + b2 intersect if there is a solution

m1x + b1 = m2x + b2

(m1 − m2)x = b2 − b1 b − b x = 2 1 m1 − m2

i.e., we need m1 6= m2 in order for there to exist a point of intersection.

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 y = m1x + b1 and y = m2x + b2 intersect if there is a solution

m1x + b1 = m2x + b2

(m1 − m2)x = b2 − b1 b − b x = 2 1 m1 − m2

i.e., we need m1 6= m2 in order for there to exist a point of intersection.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem intersect if there is a solution

m1x + b1 = m2x + b2

(m1 − m2)x = b2 − b1 b − b x = 2 1 m1 − m2

i.e., we need m1 6= m2 in order for there to exist a point of intersection.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly y = m1x + b1 McKinnie and History y = m2x + b2 Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem (m1 − m2)x = b2 − b1 b − b x = 2 1 m1 − m2

i.e., we need m1 6= m2 in order for there to exist a point of intersection.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly y = m1x + b1 McKinnie and History y = m2x + b2 Pappus’ Theorem intersect if there is a solution Geometries

Picturing the projective m1x + b1 = m2x + b2 plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem b − b x = 2 1 m1 − m2

i.e., we need m1 6= m2 in order for there to exist a point of intersection.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly y = m1x + b1 McKinnie and History y = m2x + b2 Pappus’ Theorem intersect if there is a solution Geometries

Picturing the projective m1x + b1 = m2x + b2 plane (m − m )x = b − b Lines in 1 2 2 1 projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem i.e., we need m1 6= m2 in order for there to exist a point of intersection.

Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly y = m1x + b1 McKinnie and History y = m2x + b2 Pappus’ Theorem intersect if there is a solution Geometries

Picturing the projective m1x + b1 = m2x + b2 plane (m − m )x = b − b Lines in 1 2 2 1 projective b − b geometry x = 2 1 Back to m1 − m2 Pappus’ Theorem

Proof of Pappus’ Theorem Euclidean Geometry - Review

Projective Geometry and Pappus’ Two lines Theorem

Kelly y = m1x + b1 McKinnie and History y = m2x + b2 Pappus’ Theorem intersect if there is a solution Geometries

Picturing the projective m1x + b1 = m2x + b2 plane (m − m )x = b − b Lines in 1 2 2 1 projective b − b geometry x = 2 1 Back to m1 − m2 Pappus’ Theorem i.e., we need m1 6= m2 in order for there to exist a point of Proof of Pappus’ intersection. Theorem 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 We want to ﬁx this “ﬂaw” of Euclidean geometry. 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 We want to ﬁx this “ﬂaw” of Euclidean geometry. We want Geometries every pair of distinct lines to intersect in exactly one point. Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem In perspective drawing in art.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem Where else have people noted a need for “extra” points? 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 Where else have people noted a need for “extra” points? In Geometries perspective drawing in art. 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 Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem The point on the horizon that looks like the intersection of the Geometries parallel lines is called an ideal point. 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 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. 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.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide:

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide: History What are the points of the new geometry? Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide: History What are the points of the new geometry? Pappus’ Theorem We want to incorporate all of the Euclidean points into our

Geometries projective geometry.

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem We want to incorporate our old lines into our new ones. We want all of our new lines to intersect in exactly one point.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide: History What are the points of the new geometry? Pappus’ Theorem We want to incorporate all of the Euclidean points into our

Geometries projective geometry. Picturing the What are the lines of the new geometry? projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem We want all of our new lines to intersect in exactly one point.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide: History What are the points of the new geometry? Pappus’ Theorem We want to incorporate all of the Euclidean points into our

Geometries projective geometry. Picturing the What are the lines of the new geometry? projective plane We want to incorporate our old lines into our new ones. Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie To make the new geometry we need to decide: History What are the points of the new geometry? Pappus’ Theorem We want to incorporate all of the Euclidean points into our

Geometries projective geometry. Picturing the What are the lines of the new geometry? projective plane We want to incorporate our old lines into our new ones. Lines in We want all of our new lines to intersect in exactly one projective geometry point.

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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.

Projective Geometry

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ We don’t want to add more points than we need. 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’ We don’t want to add more points than we need. What we Theorem really need is a new point for every SLOPE in the Euclidean Geometries plane. Then we can say that two parallel lines intersect at this Picturing the projective new point, the point that corresponds to their slope. 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 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 Projective Geometry

Projective Geometry and New Point! 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 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 New point!

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 How do we do this concretely? Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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].

2 The symbol for the projective plane is RP . 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!

Projective Geometry - points

Projective Geometry and Deﬁnition Pappus’ Theorem A point in the projective plane is given by a TRIPLE[ x, y, z] Kelly McKinnie with x, y, z ∈ R

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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].

2 The symbol for the projective plane is RP . 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!

Projective Geometry - points

Projective Geometry and Deﬁnition Pappus’ Theorem A point in the projective plane is given by a TRIPLE[ x, y, z] Kelly McKinnie with x, y, z ∈ R such that

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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].

2 The symbol for the projective plane is RP . 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!

Projective Geometry - points

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 2 The symbol for the projective plane is RP . 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!

Projective Geometry - points

Projective Geometry and Deﬁnition Pappus’ Theorem A point in the projective plane is given by a TRIPLE[ x, y, z] Kelly McKinnie with x, y, z ∈ R such that not all x = 0, y = 0 and z = 0, and History Pappus’ two points [x1, y1, z1] and [x2, y2, z2] are equivalent if there Theorem is a scalar λ ∈ R − {0} such that Geometries

Picturing the projective λ[x1, y1, z1] = [x2, y2, z2]. plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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!

Projective Geometry - points

Picturing the projective λ[x1, y1, z1] = [x2, y2, z2]. plane

Lines in 2 projective The symbol for the projective plane is RP . geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem [1, 2, 3] = [5, 10, 15] and [2, 2, 2] = [3, 3, 3].

[0, 0, 0] is not a point in the projective plane!

Projective Geometry - points

Picturing the projective λ[x1, y1, z1] = [x2, y2, z2]. plane

Lines in 2 projective The symbol for the projective plane is RP . geometry

Back to For example, in the projective plane, Pappus’ Theorem

Proof of Pappus’ Theorem [0, 0, 0] is not a point in the projective plane!

Projective Geometry - points

Picturing the projective λ[x1, y1, z1] = [x2, y2, z2]. plane

Lines in 2 projective The symbol for the projective plane is RP . geometry

Back to For example, in the projective plane, Pappus’ Theorem [1, 2, 3] = [5, 10, 15] and [2, 2, 2] = [3, 3, 3]. Proof of Pappus’ Theorem Projective Geometry - points

Picturing the projective λ[x1, y1, z1] = [x2, y2, z2]. plane

Lines in 2 projective The symbol for the projective plane is RP . geometry

Back to For example, in the projective plane, Pappus’ Theorem [1, 2, 3] = [5, 10, 15] and [2, 2, 2] = [3, 3, 3]. Proof of Pappus’ Theorem [0, 0, 0] is not a point in the projective plane! Projective Geometry - points

Projective Geometry and Pappus’ Theorem In other words, a point of the projective plane [x, y, z]

Kelly corresponds to all points on the line through the origin McKinnie 3 containing the point (x, y, z) in Euclidean 3-space (R ) except History the origin. Pappus’ Theorem

Geometries

Picturing the (x, y, z) projective plane

Lines in projective geometry

Back to Pappus’ k(x,y,z) Theorem

Proof of The red line is all scalar multiplies of (x,y,z) Pappus’ Theorem The projective plane

Projective Geometry and Pappus’ Theorem

Kelly McKinnie z

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective y geometry x Back to Pappus’ Theorem

Proof of Pappus’ Theorem The projective plane

Projective Geometry and Pappus’ Theorem

Kelly McKinnie z

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective y geometry x Back to Pappus’ Theorem

Proof of Pappus’ Theorem The projective plane

Projective Geometry and Pappus’ Theorem

Kelly McKinnie z

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective y geometry x Back to Pappus’ Theorem

Proof of Pappus’ Theorem The projective plane

Projective Geometry and Pappus’ Theorem

Kelly McKinnie z

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective y geometry x Back to Pappus’ Theorem

Proof of Pappus’ Theorem 2 The Euclidean point (x, y) is the point [x, y, 1] in RP . 2 Every point [x, y, z] ∈ RP with z 6= 0 is equivalent to x y [ , , 1]. z z

Projective Geometry - points

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History How does the Euclidean plane “ﬁt into” the projective plane?

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 2 Every point [x, y, z] ∈ RP with z 6= 0 is equivalent to x y [ , , 1]. z z

Projective Geometry - points

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History How does the Euclidean plane “ﬁt into” the projective plane? 2 Pappus’ The Euclidean point (x, y) is the point [x, y, 1] in RP . Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - points

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History How does the Euclidean plane “ﬁt into” the projective plane? 2 Pappus’ The Euclidean point (x, y) is the point [x, y, 1] in RP . Theorem 2 Every point [x, y, z] ∈ RP with z 6= 0 is equivalent to Geometries

Picturing the x y projective [ , , 1]. plane z z

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - points

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History (x, y, z)

Pappus’ Theorem (x/z, y/z, 1) Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - points

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem So the only “new points” we are adding are those with the Geometries third coordinate z = 0. Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Since we have many representations for one point, we need to check that this makes sense!

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie 2 History A line in RP is given by the solutions to Pappus’ Theorem Ax + By + Cz = 0 Geometries Picturing the for some A, B, C ∈ , not all zero. projective R plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie 2 History A line in RP is given by the solutions to Pappus’ Theorem Ax + By + Cz = 0 Geometries Picturing the for some A, B, C ∈ , not all zero. projective R plane Since we have many representations for one point, we need to Lines in check that this makes sense! projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Aλx0 + Bλy0 + Cλz0 = λ(Ax0 + By0 + Cz0) = λ(0) = 0

So the equation of the line is well deﬁned.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie If the point [x0, y0, z0] satisﬁes Ax + By + Cz = 0, does History λ[x0, y0, z0] = [λx0, λy0, λz0] also satisfy the equation? Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem λ(Ax0 + By0 + Cz0) = λ(0) = 0

So the equation of the line is well deﬁned.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie If the point [x0, y0, z0] satisﬁes Ax + By + Cz = 0, does History λ[x0, y0, z0] = [λx0, λy0, λz0] also satisfy the equation? Pappus’ Theorem Aλx0 + Bλy0 + Cλz0 = Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem = λ(0) = 0

So the equation of the line is well deﬁned.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie If the point [x0, y0, z0] satisﬁes Ax + By + Cz = 0, does History λ[x0, y0, z0] = [λx0, λy0, λz0] also satisfy the equation? Pappus’ Theorem Aλx0 + Bλy0 + Cλz0 = λ(Ax0 + By0 + Cz0) Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem = 0

So the equation of the line is well deﬁned.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Picturing the = λ(0) projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem So the equation of the line is well deﬁned.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Picturing the = λ(0) projective plane = 0 Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Picturing the = λ(0) projective plane = 0 Lines in projective geometry So the equation of the line is well deﬁned.

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History You can see the Euclidean lines by setting z = 1: Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem As long as B 6= 0, the Euclidean line has slope m = −A/B.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History You can see the Euclidean lines by setting z = 1: Pappus’ Theorem Ax + By + Cz = 0 − Projective line Geometries Picturing the Ax + By + C = 0 − Euclidean line projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem m = −A/B.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History You can see the Euclidean lines by setting z = 1: Pappus’ Theorem Ax + By + Cz = 0 − Projective line Geometries Picturing the Ax + By + C = 0 − Euclidean line projective plane As long as B 6= 0, the Euclidean line has slope Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Back to Pappus’ Theorem

Proof of Pappus’ Theorem The points on the line at inﬁnity are of the form [x, y, 0] with not both x = 0 and y = 0.

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem In projective geometry the line given by z = 0 is called the line

Geometries at inﬁnity.

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem In projective geometry the line given by z = 0 is called the line

Geometries at inﬁnity. The points on the line at inﬁnity are of the form

Picturing the [x, y, 0] with not both x = 0 and y = 0. projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

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 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}

= {[x, y, 0] | Ax + By = 0} A = {[x, y, 0] | y = − x} B A = {[x, − x, 0]} B A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. 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 {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0}

= {[x, y, 0] | Ax + By = 0} A = {[x, y, 0] | y = − x} B A = {[x, − x, 0]} B A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem = {[x, y, 0] | Ax + By = 0} A = {[x, y, 0] | y = − x} B A = {[x, − x, 0]} B A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0} Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem A = {[x, y, 0] | y = − x} B A = {[x, − x, 0]} B A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0} Pappus’ Theorem

Geometries

Picturing the projective = {[x, y, 0] | Ax + By = 0} plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem A = {[x, − x, 0]} B A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0} Pappus’ Theorem

Geometries

Picturing the projective = {[x, y, 0] | Ax + By = 0} plane A Lines in = {[x, y, 0] | y = − x} projective geometry B

Back to Pappus’ Theorem

Proof of Pappus’ Theorem A = [1, − , 0] B

Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0} Pappus’ Theorem

Geometries

Picturing the projective = {[x, y, 0] | Ax + By = 0} plane A Lines in = {[x, y, 0] | y = − x} projective geometry B A Back to = {[x, − x, 0]} Pappus’ Theorem B

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Let’s compute some intersections. What is Geometry and Pappus’ Theorem {Line at ∞} ∩ {line given by Ax+By +Cz = 0 with B 6= 0}? Kelly McKinnie

History {[x, y, 0]} ∩ {[x, y, z] | Ax + By + Cz = 0} Pappus’ Theorem

Geometries

Picturing the projective = {[x, y, 0] | Ax + By = 0} plane A Lines in = {[x, y, 0] | y = − x} projective geometry B A Back to = {[x, − x, 0]} Pappus’ Theorem B A Proof of = [1, − , 0] Pappus’ B Theorem 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.

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

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 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

Projective Geometry and Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem You an check that all vertical lines intersect at the projective point [0, 1, 0].

Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Kelly The line Ax + By + Cz = 0 and the line at inﬁnity intersect at McKinnie the point [1, m, 0] corresponding to the slope m = −A/B of History the Euclidean line Ax + By + C = 0. Pappus’ Theorem Geometries In particular, any two distinct parallel lines with slope m Picturing the (m 6= ∞) intersect at the line at inﬁnity at the projective point projective plane [1, m, 0]. Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

Projective Geometry and Pappus’ Theorem

Proof of Pappus’ Theorem Projective Geometry - lines

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 - lines

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 - lines

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 - lines

Projective Geometry and Pappus’ Theorem Kelly [1,1,0] McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem Let’s take another look at the conﬁguration of points in Geometries Pappus’ Theorem that didn’t work. Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Back to Pappus’ Theorem

Projective Geometry and Pappus’ Theorem [1,m,0] Kelly McKinnie C A B History

Pappus’ Theorem

Geometries

Picturing the projective A' B' C' plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Proof of Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem Since we now know that parallel lines really do intersect, we

Geometries can proceed to prove Pappus’ Theorem under the assumption

Picturing the that all the lines intersect. projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Proof of Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry Back to Let Lij denote the line through points Pi and Pj , and let a, b, c Pappus’ Theorem denote the points of intersection between the pairs of lines Proof of [L15, L24], [L16, L34], and [L35, L26] respectively. Pappus’ Pappus’ Theorem Theorem asserts that the points a, b, c lie on a straight line. 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

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History One proof proceeds by explicitly determining the coordinates of Pappus’ the points a, b, c, it is straightforward, but less trivial than one Theorem might think. Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Then for i = 4, 5, 6, yi = kxi where k is the slope of the line OP6.

Proof of Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

History One proof proceeds by explicitly determining the coordinates of Pappus’ the points a, b, c, it is straightforward, but less trivial than one Theorem might think. Let Pi = (xi , yi ). We can assume that Geometries y , y , y = 0 by rotating and sliding the picture so that P , Picturing the 1 2 3 1 projective P , and P lie on the x-axis and O lies at the origin. plane 2 3

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Proof of Pappus’ Theorem

Projective Geometry and Pappus’ Theorem

Kelly McKinnie

Lines in i = 4, 5, 6, yi = kxi where k is the slope of the line OP6. 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.

Proof of Pappus’ Theorem

Projective Geometry and Pappus’ Theorem You can now determine the coordinates of a, b and c explicitly Kelly by writing down the equations for the lines L15, L24, L16, L34, McKinnie L35 and L26. History

Pappus’ Theorem

Geometries

Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem 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.

Proof of Pappus’ Theorem

Pappus’ Theorem You can then calculate the equation of the line through a and c Geometries and prove that b lies on that line. Picturing the projective plane

Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem This is the most “straightforward” proof of Pappus’ Theorem. There are other theorems that make extensive use of the theory of projective geometry.

Proof of Pappus’ Theorem

Pappus’ Theorem You can then calculate the equation of the line through a and c Geometries and prove that b lies on that line. Picturing the projective plane There is a lot to keep track of!! Lines in projective geometry

Back to Pappus’ Theorem

Proof of Pappus’ Theorem Proof of Pappus’ Theorem

Back to This is the most “straightforward” proof of Pappus’ Theorem. Pappus’ Theorem There are other theorems that make extensive use of the theory Proof of of projective geometry. Pappus’ Theorem