Solved by verified expert:Please make the paper as the question list and give the clear geometric (euclidean geometry) prove of question 2,4,6,8,10,12. and all question some finished by file “final”
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Dalhousie University
Projects
Winter 2018
MATH 2051 – Problems in Geometry
Project 4: Convexity
Students: Lindsay Duggan, Zhengluhao Zhang.
Poster presentations will be on Thursday, March 22.
The written projects are due on Thursday, April 5.
The Notion of Convexity
Convexity is a notion that is used both inside and outside of mathematics.
(1) Give the definition that people use outside of mathematics and the mathematical definition for
when a subset of the plane (2-dimensional space) or 3-dimensional space is convex; describe
the relation between the two.
(2) Give some examples of convex sets and of non-convex sets.
(3) Show that the intersection of two convex sets is convex.
Planar Convex Sets
An important family of convex sets in the plane are those that are bounded by a simple closed curve.
Definition: a plane curve is the graph of a set of equations of the form x = f (t), y = g(t), where f
and g are functions defined on a closed interval [a, b] of real numbers. If the mapping is one-to-one
(when the curve doesn’t intersect itself), we call the curve simple. When the beginning and end
points match up, i.e., when f (a) = f (b) and g(a) = g(b), we call the curve closed. For any simple
closed curve we can talk about the region inside the curve.
(4) Give some examples of curves in the plane with the following properties. (you don’t need to
give the precise function description)
(4-1) Examples of curves which are closed but not simple;
(4-2) Examples of curves which are simple but not closed;
(4-3) Examples of curves which are not simple and not closed;
(4-4) Examples of curves which are simple and closed.
(5) Give some examples of simple closed curves that form the boundary of a convex set, and some
that don’t.
1
Dalhousie University
Projects
Winter 2018
Convex Hulls
When scientists collect data points through measurements, one of the first things they may consider
is the convex hull of their data set.
Definition: For any set S of points in the plane or in 3-dimensional space, the convex hull is the
smallest convex set containing S.
(6) Give some examples of both convex and non-convex sets together with their convex hulls.
(7) Show that a set is convex if and only if it is its own convex hull.
(8) Consider some examples of finite sets of points S in the plane, together with their convex
hulls. What kind of objects are their convex hulls?
(9) Show that any convex polygonal region can be described as the intersection of finitely many
half planes.
(10) Describe an algorithm that would find the convex hull of a finite set of points.
(11) Show that the convex hull of any finite set of points in the plane is a convex polygonal region.
(12) Conversely, show that any convex polygonal region is the convex hull of the set of vertices on
its boundary.
Finite point sets are easier to work with than simple closed curves. So when we have a simple
closed curve it would be nice to be able to use finite point sets to determine whether it bounds a
convex region.
(13) Show that T is the boundary of a convex set K if and only if, for every S, a finite set of points
on T , no point of S is an interior point of the convex hull K ′ of S. (Give some examples of
simple closed curves with sets of points to show how this works.)
2
1) Give the definition that people use outside of mathematics and the mathematical
definition for when a subset of the plane (2-dimensional space) or 3-dimensional
space is convex; describe the relation between the two.
Outside of math :
Having a surface that is curved or rounded outward like the exterior of a sphere or a
circle.
Mathematical definition :
Of a Set : Having the property that for each pair of points in the set the line joining the
points is wholly contained in the set.
Of a Polygon : Comprising a convex set when combined with its interior. All interior
angles are less than or equal to 180deg.
Relation :
If you take a convex object outside of mathematics, for example a convex mirror which
extrudes further as you approach the center, and consider the solid space bounded by
the convex exterior then that space would be convex by the mathematical definition as
well.
2) Give some examples of convex sets and of non-convex sets.
Convex set: Any point A and B in set, the segment AB lies in the set completely, called
convex set. (figure 2-1)
Non convex set: segment AB is not contained in the set completely, called non convex
set. (figure 2-2)
3) Show that the intersection of two convex sets is convex.
4) Give some examples of curves in the plane with the following properties. (you don’t
need to give the precise function description)
(4-1) Examples of curves which are closed but not simple;
(4-2) Examples of curves which are simple but not closed;
(4-3) Examples of curves which are not simple and not closed;
(4-4) Examples of curves which are simple and closed.
Simple curve: curve does not cross itself
Not simple curve: curve cross it self
5) Give some examples of simple closed curves that form the boundary of a convex set,
and some
that don’t.
Form a Boundary of a Convex Set.
Don`t form a Boundary of a Convex Set.
6) Give some examples of both convex and non-convex sets together with their convex
hulls.
7) Show that a set is convex if and only if it is its own convex hull.
A convex hull M of a set L is, by definition, the smallest convex set which contains L.
1) Case 1: If M and L are equal in size, but vary in shape or location, then M does
not contain L, contradicts definition.
2) Case 2: If M is larger than L, it is not the smallest convex set which contains L,
contradicts definition
3) Case 3: If M is smaller than L, it does not contain L, which contradicts definition
8) Consider some examples of finite sets of points S in the plane, together with their
convex hulls. What kind of objects are their convex hulls?
&
11) Show that the convex hull of any finite set of points in the plane is a convex
polygonal region.
9) Show that any convex polygonal region can be described as the intersection of
finitely many half planes.
It is possible to create a set containing any two lines and their contents within an angle
of 180deg or less. By definition of a convex polygonal region, it can be defined by the
intersection of a finite amount of half planes.
10) Describe an algorithm that would find the convex hull of a finite set of points.
Graham Scan algorithm
12) Conversely, show that any convex polygonal region is the convex hull of the set of
vertices on its boundary.
A convex polygon, by definition, contains the convex set interior of and including its
vertices, by question 7, this convex set is equal to its own convex hull
13) Show that T is the boundary of a convex set K if and only if, for every S, a finite set
of points on T, no point of S is an interior point of the convex hull K. of S. (Give some
examples of simple closed curves with sets of points to show how this works.)
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and
Non-Euclidean Geometries by Marvin Jay Greenberg (2009-03-26)
Logic Rule 0 No unstated assumptions may be used in a proof.
Logic Rule 1 Allowable justifications.
1. “By hypothesis . . . ”.
2. “By axiom . . . ”.
3. “By theorem . . . ” (previously proved).
4. “By definition . . . ”.
5. “By step . . . ” (a previous step in the argument).
6. “By rule . . . ” of logic.
Logic Rule 2 Proof by contradiction (RAA argument).
Logic Rule 3 The tautology ∼ (∼ S) ⇐⇒ S
Logic Rule 4 The tautology ∼ (H =⇒ C) ⇐⇒ H ∧ (∼ C).
Logic Rule 5 The tautology ∼ (S1 ∧ S2 ) ⇐⇒ (∼ S1 ∨ ∼ S2 ).
Logic Rule 6 The statement ∼ (∀xS(x)) means the same as ∃x(∼ S(x)).
Logic Rule 7 The statement ∼ (∃xS(x)) means the same as ∀x(∼ S(x)).
Logic Rule 8 The tautology ((P =⇒ Q) ∧ P ) =⇒ Q.
Logic Rule 9 The tautologies
1. ((P =⇒ Q) ∧ (Q =⇒ R) =⇒ (P =⇒ R).
2. (P ∧ Q) =⇒ P and (P ∧ Q) =⇒ Q.
3. (∼ Q =⇒∼ P ) =⇒ ((P =⇒ Q).
Logic Rule 10 The tautology P =⇒ (P ∨ ∼ P ).
Logic Rule 11 (Proof by Cases) If C can be deduced from each of S1 , S2 , · · · , Sn individually, then (S1 ∨S2 ∨· · · Sn ) =⇒
C is a tautology.
Logic Rule 12 Euclid’s “Common Notions”
1. ∀X (X = X)
2. ∀X ∀Y (X = Y ⇐⇒ Y = X)
3. ∀X ∀Y ∀X ((X = Y ∧ Y = Z) =⇒ X = Z)
4. If X = Y and S(X) is a statement about X, then S(X) ⇐⇒ S(Y )
Undefined Terms: Point, Line, Incident, Between, Congruent.
Basic Definitions
1. Three or more points are collinear if there exists a line incident with all of them.
2. Three or more lines are concurrent if there is a point incident with all of the them.
3. Two lines are parallel if they are distinct and no point is incident with both of them.
←→
←→
4. {AB} is the set of points incident with AB.
Incidence Axioms:
IA1: For every two distinct points there exists a unique line incident on them.
IA2: For every line there exist at least two points incident on it.
IA3: There exist three distinct points such that no line is incident on all three.
1
Incidence Propositions:
P2.1: If l and m are distinct lines that are not parallel, then l and m have a unique point in common.
P2.2: There exist three distinct lines that are not concurrent.
P2.3: For every line there is at least one point not lying on it.
P2.4: For every point there is at least one line not passing through it.
P2.5: For every point there exist at least two distinct lines that pass through it.
P2.6: For every point P there are at least two distinct points neither of which is P .
P2.7: For every linen l there are at least two distinct lines neither of which is l.
P2.8: If l is a line and P is a point not incident with l then there is a one-to-one correspondence between the set of
points incident with l and the set of lines through P that meet l.
P2.9: Let P be a point. Denote the set of points {X : X is on a line passing through P } by S. Then every point is
in S.
P2.10: Let l be a point. Denote the set of points {m : m is incident with a point that lies on l or m is parallel to l}
by S. Then every point is in S.
Betweenness Axioms and Notation:
Notation: A ∗ B ∗ C means “point B is between point A and point C.”
B1: If A ∗ B ∗ C, then A, B, and C are three distinct points all lying on the same line, and C ∗ B ∗ A.
←→
B2: Given any two distinct points B and D, there exist points A, C, and E lying on BD such that A∗B ∗D, B ∗C ∗D,
and B ∗ D ∗ E.
B3: If A, B, and C are three distinct points lying on the same line, then one and only one of them is between the
other two.
B4: For every line l and for any three points A, B, and C not lying on l:
1. If A and B are on the same side of l, and B and C are on the same side of l, then A and C are on the same
side of l.
2. If A and B are on opposite sides of l, and B and C are on opposite sides of l, then A and C are on the same
side of l.
Corollary If A and B are on opposite sides of l, and B and C are on the same side of l, then A and C are on opposite
sides of l.
Betweenness Definitions:
Segment AB: Point A, point B, and all points P such that A ∗ P ∗ B.
−−→
Ray AB: Segment AB and all points C such that A ∗ B ∗ C.
Same/Opposite Side: Let l be any line, A and B any points that do not lie on l. If A = B or if segment AB
contains no point lying on l, we say A and B are on the same side of l, whereas if A 6= B and segment AB does
intersect l, we say that A and B are on opposite sides of l. The law of excluded middle tells us that A and B are
either on the same side or on opposite sides of l.
Betweenness Propositions:
P3.1 (does not use BA-4): For any two points A and B:
−−→ −−→
1. AB ∩ BA = AB Proof is in text, and
−−→ −−→ ←→
2. AB ∪ BA = AB.
P3.2 Proof is in text: Every line bounds exactly two half-planes and these half-planes have no point in common.
←→
←→
Same Side Lemma: Given A ∗ B ∗ C and l any line other than line AB meeting line AB at point A, then B and C
are on the same side of line l.
←→
←→
Opposite Side Lemma: Given A ∗ B ∗ C and l any line other than line AB meeting line AB at point B, then A and
C are on opposite sides of line l.
P3.3 Proof is in text: Given A ∗ B ∗ C and A ∗ C ∗ D. Then B ∗ C ∗ D and A ∗ B ∗ D.
2
Corollary to P3.3: Given A ∗ B ∗ C and B ∗ C ∗ D. Then A ∗ B ∗ D and A ∗ C ∗ D.
P3.4 Proof is in text: If C ∗ A ∗ B and l is the line through A, B, and C, then for every point P lying on l, P either
−−→
−→
lies on ray AB or on the opposite ray AC.
Pasch’s Theorem Proof is in text: If A, B, and C are distinct noncollinear points and l is any line intersecting
AB in a point between A and B, then l also intersects either AC, or BC. If C does not lie on l, then l does not
intersect both AC and BC.
P3.5: Given A ∗ B ∗ C. Then AC = AB ∪ BC and B is the only point common to segments AB and BC.
−−→
−−→
−−→ −→
P3.6: Given A ∗ B ∗ C. Then B is the only point common to rays BA and BC, and AB = AC.
Angle Definitions:
←→
Interior: Given an angle ] CAB, define a point D to be in the interior of ] CAB if D is on the same side of AC as B
←→
and if D is also on the same side of AB as C. Thus, the interior of an angle is the intersection of two half-planes.
(Note: the interior does not include the angle itself, and points not on the angle and not in the interior are on
the exterior).
−−→
−→
−−→
−−→
−→
Ray Betweenness: Ray AD is between rays AC and AB provided AB and AC are not opposite rays and D is interior
to ] CAB.
Triangle: The union of the three segments formed by three non-collinear points.
Interior of a Triangle: The interior of a triangle is the intersection of the interiors of its thee angles. Define a point
to be exterior to the triangle if it in not in the interior and does not lie on any side of the triangle.
Angle Propositions:
←→
P3.7: Given an angle ] CAB and point D lying on line BC. Then D is in the interior of ] CAB iff B ∗ D ∗ C.
−−→
“Problem 9”: Given a line l, a point A on l and a point B not on l. Then every point of the ray AB (except A) is
on the same side of l as B.
P3.8: If D is in the interior of ] CAB, then:
−−→
1. so is every other point on ray AD except A,
−−→
2. no point on the opposite ray to AD is in the interior of ] CAB, and
3. if C ∗ A ∗ E, then B is in the interior of ] DAE.
−−→
−→
−−→
−−→
Crossbar Theorem: If AD is between AC and AB, then AD intersects segment BC.
P3.9:
1. If a ray r emanating from an exterior point of 4ABC intersects side AB in a point between A and B, then
r also intersects side AC or BC.
2. If a ray emanates from an interior point of 4ABC, then it intersects one of the sides, and if it does not pass
through a vertex, then it intersects only one side.
Congruence Axioms:
C1: If A and B are distinct points and if A0 is any point, then for each ray r emanating from A0 there is a unique
point B 0 on r such that B 0 6= A0 and AB ∼
= A0 B 0 .
C2: If AB ∼
= CD and AB ∼
= EF , then CD ∼
= EF . Moreover, every segment is congruent to itself.
C3: If A ∗ B ∗ C, and A0 ∗ B 0 ∗ C 0 , AB ∼
= A0 B 0 , and BC ∼
= B 0 C 0 , then AC ∼
= A0 C 0 .
−−→
−→
−→
C4: Given any ] BAC (where by definition of angle, AB is not opposite to AC and is distinct from AC), and given
←−→
−−−→
−−→
any ray A0 B 0 emanating from a point A0 , then there is a unique ray A0 C 0 on a given side of line A0 B 0 such that
] B 0 A0 C 0 ∼
= ] BAC.
∼
C5: If ] A = ] B and ] A ∼
= ] C, then ] B ∼
= ] C. Moreover, every angle is congruent to itself.
C6 (SAS): If two sides and the included angle of one triangle are congruent, respectively, to two sides and the included
angle of another triangle, then the two triangles are congruent.
3
Congruence Propositions:
Corollary to SAS: Proof is in text Given 4ABC and segment DE ∼
= AB, there is a unique point F on a given
←→
∼
side of line DE such that 4ABC = 4DEF .
P3.10 Proof is in text: If in 4ABC we have AB ∼
= AC, then ] B ∼
= ] C.
P3.11: (Segment Subtraction) If A ∗ B ∗ C, D ∗ E ∗ F , AB ∼
= DE, and AC ∼
= DF , then BC ∼
= EF .
P3.12 Proof is in text: Given AC ∼
= DF , then for any point B between A and C, there is a unique point E between
D and F such that AB ∼
DE.
=
P3.13: (Segment Ordering)
1.
2.
3.
4.
Exactly one of the following holds: AB < CD, AB ∼
= CD, or AB > CD.
∼
If AB < CD and CD = EF , then AB < EF .
If AB > CD and CD ∼
= EF , then AB > EF .
If AB < CD and CD < EF , then AB < EF .
P3.14: Supplements of congruent angles are congruent.
P3.15: 1. Vertical angles are congruent to each other.
2. An angle congruent to a right angle is a right angle.
P3.16 Proof is in text: For every line l and every point P there exists a line through P perpendicular to l.
∼ ] D, ] C ∼
P3.17 (ASA): Given 4ABC and 4DEF with ] A =
= ] F , and AC ∼
= DF , then 4ABC ∼
= 4DEF .
∼ ] C, then AB =
∼ AC and 4ABC is isosceles.
P3.18: If in 4ABC we have ] B =
−−→
−−→
−−→ −−→
−−→
−−→
P3.19 Proof is in text: (Angle Addition) Given BG between BA and BC, EH between ED and EF , ] CBG ∼
=
] F EH and ] GBA ∼
= ] HED. Then ] ABC ∼
= ] DEF .
−−→
−−→
−−→ −−→
−−→
−−→
∼ ] F EH and
P3.20: (Angle Subtraction) Given BG between BA and BC, EH between ED and EF , ] CBG =
∼ ] DEF . Then ] GBA =
∼ ] HED.
] ABC =
P3.21: (Ordering of Angles)
Exactly one of the following holds: ] P < ] Q, ] P ∼
= ] Q, or ] P > ] Q.
∼
If ] P < ] Q and ] Q = ] R, then ] P < ] R.
If ] P > ] Q and ] Q ∼
= ] R, then ] P > ] R.
If ] P < ] Q and ] Q < ] R, then ] P < ] R.
P3.22 (SSS): Given 4ABC and 4DEF . If AB ∼
= 4DEF .
= DF , then 4ABC ∼
= EF , and AC ∼
= DE, BC ∼
1.
2.
3.
4.
P3.23 Proof is in text: (Euclid’s Fourth Postulate) All right angles are congruent to each other.
Corollary (not numbered in text) If P lies on l then the perpendicular to l through P is unique.
Definitions:
Segment Inequality: AB < CD (or CD > AB) means that there exists a point E between C and D such that
AB ∼
= CE.
−−→
−−→
−−→
Angle Inequality: ] ABC < ] DEF means there is a ray EG between ED and EF such that ] ABC ∼
= ] GEF .
Right Angle: An angle ] ABC is a right angle if has a supplementary angle to which it is congruent.
Parallel: Two lines l and m are parallel if they do not intersect, i.e., if no point lies on both of them.
−−→
Perpendicular: Two lines l and m are perpendicular if they intersect at a point A and if there is a ray AB that is a
−→
part of l and a ray AC that is a part of m such that ] BAC is a right angle.
Triangle Congruence and Similarity: Two triangles are congruent if a one-to-one correspondence can be set up
between their vertices so that corresponding sides are congruent and corresponding angles are congruent. Similar
triangles have this one-to-one correspondence only with their angles.
Circle (with center O and radius OA): The set of all points P such that OP is congruent to OA.
Triangle: The set of three distinct segments defined by three non-collinear points.
Acute, Obtuse Angles An angle is acute if it is less than a right angle, obtuse if it is greater than a right angle.
Hilbert Plane A model of our incidence, betweenness, and congruence axioms is called a Hilbert Plane.
4
Continuity Axioms and Principles:
Circle-Circle Continuity Principle If a circle γ has one point inside and one point outside another circle γ 0 , then
the two circles intersect in two points.
Line-Circle Continuity Principle If a line passes through a point inside a circle, then the line intersects the circle
in two points.
Segment-Circle Continuity Principle In one endpoint of a segment is inside a circle and the other outside, then
the segment ...
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