Transcript CIRCUMCENTER OF MASS Emmanuel Tsukerman University of California, Berkeley Based on joint work with Prof.

CIRCUMCENTER OF MASS
Emmanuel Tsukerman
University of California, Berkeley
Based on joint work with Prof. Sergei Tabachnikov
Based on joint work with Prof. Sergei Tabachnikov
Also A. Akopyan
Based on joint work with Prof. Sergei Tabachnikov
Also A. Akopyan
Both papers in Discr. Comp. Geom., 2014.
OUTLINE
I.
Motivation
II. Definition
III. Properties
IV. Open problem
I. MOTIVATION
Motivation comes from integrable systems
Motivation comes from integrable systems
CCM is an integral of two interesting discrete conjecturally completely integrable systems
1. POLYGON RECUTTING
1. POLYGON RECUTTING
1. POLYGON RECUTTING
V. Adler, Recuttings of polygons. Funct. Anal. Appl. 27 (1993), 141-143
V. Adler, Integrable deformations of a polygon. Phys. D 87 (1995), 52-57
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
Bicycle transformation: imagine you have the rear track of a bike whose length you know
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
Bicycle transformation: imagine you have the rear track of a bike whose length you know
You can place the bike so that its back wheel matches the track and trace out the front track
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
Bicycle transformation: imagine you have the rear track of a bike whose length you know
You can place the bike so that its back wheel matches the track and trace out the front track
Actually you can do this in two ways
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
Bicycle transformation: imagine you have the rear track of a bike whose length you know
You can place the bike so that its back wheel matches the track and trace out the front track
Actually you can do this in two ways
Related to Sherlock Holmes story
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
V1W1V2W2 is an isosceles trapezoid
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
V1W1V2W2 is an isosceles trapezoid
2. DISCRETE DARBOUX (BICYCLE) TRANSFORMATION
V1W1V2W2 is an isosceles trapezoid
U. Pinkall, B. Springborn, S. Weissmann, A new doubly discrete analogue of smoke ring flow and the real time
simulation of fluid flow. J. Phys. A 40 (2007), 12563-12576
S. Tabachnikov, E. T., On the discrete bicycle transformation. Publ. Math. Uruguay, 14 (2013), 201-220.
II. DEFINITION
Construction:
Construction:
- 𝑂 arbitrary point (off the sides)
Construction:
- 𝑂 arbitrary point (off the sides)
- Triangulate radially via 𝑂
Construction:
- 𝑂 arbitrary point (off the sides)
- Triangulate radially via 𝑂
- 𝐶𝑖 are the circumcenters with weights 𝐴𝑖 , the signed areas of triangles
Construction:
- 𝑂 arbitrary point (off the sides)
- Triangulate radially via 𝑂
- 𝐶𝑖 are the circumcenters with weights 𝐴𝑖 , the signed areas of triangles
- 𝐶𝐶𝑀 is center of mass of the 𝐶𝑖
Theorem: The circumcenter of mass
𝒏
𝑪𝑪𝑴 𝑷 =
is well-defined.
𝒊=𝟏
𝑨𝒊
𝑪
𝑨(𝑷) 𝒊
Theorem: The circumcenter of mass
𝒏
𝑪𝑪𝑴 𝑷 =
𝒊=𝟏
is well-defined.
𝑨𝒊
𝑪
𝑨(𝑷) 𝒊
Formula: If the vertices are (𝑥𝑖 , 𝑦𝑖 ), then
𝑛−1
1
𝐶𝐶𝑀 𝑃 =
(
4𝐴(𝑃)
𝑛−1
2
2
2
2
𝑦𝑖 𝑥𝑖−1
+ 𝑦𝑖−1
− 𝑥𝑖+1
− 𝑦𝑖+1
,
𝑖=0
2
2
2
2
−𝑥𝑖 (𝑥𝑖−1
+ 𝑦𝑖−1
− 𝑥𝑖+1
− 𝑦𝑖+1
))
𝑖=0
Theorem: The circumcenter of mass
𝒏
𝑪𝑪𝑴 𝑷 =
𝒊=𝟏
is well-defined.
𝑨𝒊
𝑪
𝑨(𝑷) 𝒊
Formula: If the vertices are (𝑥𝑖 , 𝑦𝑖 ), then
𝑛−1
1
𝐶𝐶𝑀 𝑃 =
(
4𝐴(𝑃)
2
2
2
2
𝑦𝑖 𝑥𝑖−1
+ 𝑦𝑖−1
− 𝑥𝑖+1
− 𝑦𝑖+1
,
𝑖=0
𝑛−1
1
(
6𝐴(𝑃)
2
2
2
2
−𝑥𝑖 (𝑥𝑖−1
+ 𝑦𝑖−1
− 𝑥𝑖+1
− 𝑦𝑖+1
))
𝑖=0
Cf. the center of mass (of the lamina):
𝐶𝑀 𝑃 =
𝑛−1
𝑛−1
(𝑥𝑖 +𝑥𝑖+1 )(𝑥𝑖 𝑦𝑖+1 − 𝑥𝑖+1 𝑦𝑖 ),
𝑖=0
(𝑦𝑖 + 𝑦𝑖+1 )(𝑥𝑖 𝑦𝑖+1 − 𝑥𝑖+1 𝑦𝑖 )
𝑖=0
The 𝐶𝐶𝑀 can be defined also for simplicial polytopes and in spherical and hyperbolic
geometries
The 𝐶𝐶𝑀 can be defined also for simplicial polytopes and in spherical and hyperbolic
geometries
My current understanding is that the “proper” setting for 𝐶𝐶𝑀 is geometric simplicial
complexes
III. PROPERTIES
ARCHIMEDES’ LEMMA
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀 (assuming two simplicial complexes)
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀 (assuming two simplicial complexes)
valuation
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀 (assuming two simplicial complexes)
valuation (isometry covariant)
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀 (assuming two simplicial complexes)
valuation (isometry covariant)
Consequence: 𝐶𝐶𝑀 can be
computed via any
nondegenerate triangulation
ARCHIMEDES’ LEMMA
Classically: if an object is divided two smaller objects, then the center of mass of the compound object is the weighted sum of
the centers of mass of the two smaller objects.
Applies also to 𝐶𝐶𝑀 (assuming two simplicial complexes)
valuation (isometry covariant)
Consequence: 𝐶𝐶𝑀 can be
computed via any
nondegenerate triangulation
Consequence: 𝐶𝐶𝑀 of an
inscribed polygon is its circumcenter
I will now discuss subtleties involving triangulations
“DANGEROUS” TRIANGLES
Flattening a triangle so its area goes to zero does not prevent the circumcenter from escaping to infinity
“DANGEROUS” TRIANGLES
Flattening a triangle so its area goes to zero does not prevent the circumcenter from escaping to infinity
“DANGEROUS” TRIANGLES
Flattening a triangle so its area goes to zero does not prevent the circumcenter from escaping to infinity
The left is “dangerous”, the right is “safe”
PARADOX?
Compute the CCM of a right triangle in two different ways:
PARADOX?
Compute the CCM of a right triangle in two different ways: counterexample?
PARADOX?
Compute the CCM of a right triangle in two different ways: counterexample?
No, because on the left we have introduced a vertex.
PARADOX?
Compute the CCM of a right triangle in two different ways: counterexample?
No, because on the left we have introduced a vertex.
This gives us the 𝐶𝐶𝑀 of a degenerate quadrilateral:
Another example where we must be careful
Another example where we must be careful is when we try to compute the CCM of a nonsimplicial polytope
Another example where we must be careful is when we try to compute the CCM of a nonsimplicial polytope
Another example where we must be careful is when we try to compute the CCM of a nonsimplicial polytope
Different triangulations will yield different “CCM”s
Another example where we must be careful is when we try to compute the CCM of a nonsimplicial polytope
Different triangulations will yield different “CCM”s
Back to properties of CCM
Proposition: If 𝑃 is an equilateral polygon, then
𝑪𝑪𝑴 𝑷 = 𝑪𝑴 𝑷 .
EULER LINE
EULER LINE
It is a classical fact that the circumcenter, centroid, orthocenter, and many other notable points of a triangle are collinear
EULER LINE
It is a classical fact that the circumcenter, centroid, orthocenter, and many other notable points of a triangle are collinear
This line is called the Euler line
EULER LINE
It is a classical fact that the circumcenter, centroid, orthocenter, and many other notable points of a triangle are collinear
This line is called the Euler line
We can now extend the notion to polygons (more generally, simplicial polytopes) by taking affine combinations of 𝐶𝐶𝑀
and 𝐶𝑀: 𝐶𝑡 = 𝑡𝐶𝑀 + 1 − 𝑡 𝐶𝐶𝑀
EULER LINE
It is a classical fact that the circumcenter, centroid, orthocenter, and many other notable points of a triangle are collinear
This line is called the Euler line
We can now extend the notion to polygons (more generally, simplicial polytopes) by taking affine combinations of 𝐶𝐶𝑀
and 𝐶𝑀: 𝐶𝑡 = 𝑡𝐶𝑀 + 1 − 𝑡 𝐶𝐶𝑀
(orthocenter: 𝑡 = 3)
EULER LINE - EXAMPLE
EULER LINE - EXAMPLE
For a simplex, it is the line through the centroid and circumcenter
EULER LINE - EXAMPLE
For a simplex, it is the line through the centroid and circumcenter
This line passes through the Monge point
EULER LINE - EXAMPLE
For a simplex, it is the line through the centroid and circumcenter
This line passes through the Monge point (the six planes through the midpoints of the edges of a tetrahedron and
perpendicular to the opposite edges concur in a point known as the Monge point – there is also a higher dimensional
analog)
EULER LINE - EXAMPLE
For a simplex, it is the line through the centroid and circumcenter
This line passes through the Monge point (the six planes through the midpoints of the edges of a tetrahedron and
perpendicular to the opposite edges concur in a point known as the Monge point – there is also a higher dimensional
analog)
The Monge point is 𝐶𝑛+1 and can now be defined for simplicial polytopes
𝑛−1
EULER LINE - PROPERTIES
EULER LINE - PROPERTIES
Theorem: Suppose that a center:
- depends analytically on the polygon
- commutes with dilations
- satisfies Archimedes’ lemma
Then it is an affine combination of 𝐶𝑀 and 𝐶𝐶𝑀
EULER LINE - PROPERTIES
Theorem: Suppose that a center:
- depends analytically on the polygon
- commutes with dilations
- satisfies Archimedes’ lemma
Then it is an affine combination of 𝐶𝑀 and 𝐶𝐶𝑀
Proposition: assume that the 𝑛-gon is almost equilateral: the sides satisfy 𝑆1 = 𝑆2 = ⋯ = 𝑆𝑛−1 . Then the Euler line is
orthogonal to side 𝑆𝑛 .
Proof: reflect in side 𝑆𝑛 to obtain an equilateral polygon and use the fact that 𝐶𝐶𝑀 = 𝐶𝑀 for equilateral polygons
EULER LINE – FURTHER PROPERTIES
If 𝑃 has a line of reflection symmetry 𝐿, then 𝐸 is either 𝐿 or a point on 𝐿
EULER LINE – FURTHER PROPERTIES
If 𝑃 has a line of reflection symmetry 𝐿, then 𝐸 is either 𝐿 or a point on 𝐿
If 𝑃 has a center of rotational symmetry 𝐶 and none of the (extended) sides passes through 𝐶, then 𝐸 = 𝐶
CONTINUOUS LIMIT
Let 𝛾 be a curve enclosing a star-shaped domain.
CONTINUOUS LIMIT
Let 𝛾 be a curve enclosing a star-shaped domain. Define 𝐶𝐶𝑀 𝛾 =
𝐶 𝑡 𝑑𝐴
𝑑𝐴
CONTINUOUS LIMIT
Let 𝛾 be a curve enclosing a star-shaped domain. Define 𝐶𝐶𝑀 𝛾 =
Then 𝐶𝐶𝑀 𝛾 = 𝐶𝑀 𝛾 .
𝐶 𝑡 𝑑𝐴
𝑑𝐴
CONTINUOUS LIMIT
Let 𝛾 be a curve enclosing a star-shaped domain. Define 𝐶𝐶𝑀 𝛾 =
Then 𝐶𝐶𝑀 𝛾 = 𝐶𝑀 𝛾 .
In general, should be an intrinsic moment.
𝐶 𝑡 𝑑𝐴
𝑑𝐴
RELATION TO OTHER VALUATIONS
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood.
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
Proof: (in dim 2, general argument similar)
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
Proof: (in dim 2, general argument similar) Consider a “dangerous” isosceles triangle whose base is of length 2, aligned with the
x-axis, and which is symmetric about the y-axis.
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
Proof: (in dim 2, general argument similar) Consider a “dangerous” isosceles triangle whose base is of length 2, aligned with the
x-axis, and which is symmetric about the y-axis. Flattening gives a 0-moment vector.
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
Proof: (in dim 2, general argument similar) Consider a “dangerous” isosceles triangle whose base is of length 2, aligned with the
x-axis, and which is symmetric about the y-axis. Flattening gives a 0-moment vector. However, a calculation shows that 𝜑=(0,-1/2).
RELATION TO OTHER VALUATIONS
ℝ𝑛 -valued continuous (wrt Hausdorff metric) isometry-covariant valuations on convex compact sets are classified as linear
combinations of intrinsic moments
Namely, given convex 𝐾, let 𝐾𝜖 be its 𝜖-neighborhood. The moment vector
𝑥 𝑑𝑥
𝐾𝜖
Is a polynomial in 𝜖 and its coefficients span the space of continuous isometry-covariant valuations.
The map 𝜑: 𝑃 ⟼ 𝑉𝑜𝑙 𝑃 𝐶𝐶𝑀 𝑃 is not an element of this space
Proof: (in dim 2, general argument similar) Consider a “dangerous” isosceles triangle whose base is of length 2, aligned with the
x-axis, and which is symmetric about the y-axis. Flattening gives a 0-moment vector. However, a calculation shows that 𝜑=(0,-1/2).
We can also see that it has a contribution from the “paradox”
HISTORICAL NOTE
Once we had understood CCM well-enough, we started asking other researchers if they had seen
this object (CCM) before
HISTORICAL NOTE
Once we had understood CCM well-enough, we started asking other researchers if they had seen
this object (CCM) before
•It seems that Giusto Bellavitis was the first who noted the existence of the circumcenter of mass
of a planar polygon in 1834
HISTORICAL NOTE
Once we had understood CCM well-enough, we started asking other researchers if they had seen
this object (CCM) before
•It seems that Giusto Bellavitis was the first who noted the existence of the circumcenter of mass
of a planar polygon in 1834
•In 1993, it was independently noticed by Adler for the case of triangulation of planar polygon
by diagonals
HISTORICAL NOTE
Once we had understood CCM well-enough, we started asking other researchers if they had seen
this object (CCM) before
•It seems that Giusto Bellavitis was the first who noted the existence of the circumcenter of mass
of a planar polygon in 1834
•In 1993, it was independently noticed by Adler for the case of triangulation of planar polygon
by diagonals
•in the private correspondence of G.C. Shephard and B. Grünbaum. They also noted that the
circumcenter could be replaced by any point on the Euler line, that is, by a fixed affine
combination of the centroid and the Circumcenter.
HISTORICAL NOTE
Once we had understood CCM well-enough, we started asking other researchers if they had seen
this object (CCM) before
•It seems that Giusto Bellavitis was the first who noted the existence of the circumcenter of mass
of a planar polygon in 1834
•In 1993, it was independently noticed by Adler for the case of triangulation of planar polygon
by diagonals
•in the private correspondence of G.C. Shephard and B. Grünbaum. They also noted that the
circumcenter could be replaced by any point on the Euler line, that is, by a fixed affine
combination of the centroid and the Circumcenter.
•Myakishev proved the existence of Euler line for a quadrilateral.
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
- The map ∆⟼ 𝐶 ∆ commutes with similarities
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
- The map ∆⟼ 𝐶 ∆ commutes with similarities
- The map ∆⟼ 𝐶(∆) is invariant under permutations of the vertices of ∆
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
- The map ∆⟼ 𝐶 ∆ commutes with similarities
- The map ∆⟼ 𝐶(∆) is invariant under permutations of the vertices of ∆
- The map ∆⟼ 𝑉𝑜𝑙(∆)𝐶(∆) is a polynomial in the coordinates of the vertices of the simplex ∆
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
- The map ∆⟼ 𝐶 ∆ commutes with similarities
- The map ∆⟼ 𝐶(∆) is invariant under permutations of the vertices of ∆
- The map ∆⟼ 𝑉𝑜𝑙(∆)𝐶(∆) is a polynomial in the coordinates of the vertices of the simplex ∆
Theorem: 𝐶(∆) is an affine combination of 𝐶𝑀 and 𝐶𝐶𝑀
QUESTION
Assign to every nondegenerate simplex ∆ a center 𝐶(∆) so that
- The map ∆⟼ 𝐶 ∆ commutes with similarities
- The map ∆⟼ 𝐶(∆) is invariant under permutations of the vertices of ∆
- The map ∆⟼ 𝑉𝑜𝑙(∆)𝐶(∆) is a polynomial in the coordinates of the vertices of the simplex ∆
Theorem: 𝐶(∆) is an affine combination of 𝐶𝑀 and 𝐶𝐶𝑀
Question: what are the right axioms for 𝐶𝐶𝑀?
Summary
Summary
•
CCM is a cousin of the CM
Summary
•
•
CCM is a cousin of the CM
Arises in discrete integrable systems
Summary
•
•
•
CCM is a cousin of the CM
Arises in discrete integrable systems
Satisfies Archimedes’ Lemma
Summary
•
•
•
•
CCM is a cousin of the CM
Arises in discrete integrable systems
Satisfies Archimedes’ Lemma
Allows to define an Euler line
Summary
•
•
•
•
•
CCM is a cousin of the CM
Arises in discrete integrable systems
Satisfies Archimedes’ Lemma
Allows to define an Euler line
Have several results showing Euler line centers are the only centers satisfying Archimedes’
Lemma
Summary
•
•
•
•
•
•
CCM is a cousin of the CM
Arises in discrete integrable systems
Satisfies Archimedes’ Lemma
Allows to define an Euler line
Have several results showing Euler line centers are the only centers satisfying Archimedes’
Lemma
What are the right axioms?
Thank you
Questions?