Unizor - Geometry3D - Elements - Cylinders

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Consider a plane α that we will call a base plane and a circle c with a center at point O and radius R on this plane. This circle c will be used as a directrix of a cylindrical surface we will construct.
Also assume there is another plane β parallel to a plane α that we will also call a base plane. To differentiate between two base planes, we will call one of them "bottom" and another - "top".
Finally, assume that we have a straight line d that is not parallel to base planes.

Let's construct a cylindrical surface σ using circle c as a directrix and line d as a generatrix.

Circular cylinder is an object in solid geometry formed by a part of a cylindrical surface σ between base planes α and β (called its side surface) and parts of the base planes inside that cylindrical surface - circle c on the "bottom" base plane α and a corresponding circle c' with center O' and the same radius R on the "top" base plane β.
In most cases, when we use a term cylinder, we mean circular cylinder.

Cylinders have no vertices, edges or sides.
The segment connecting centers of base circles OO' is called an axis of a cylinder.
Radius of a base circle is considered as a radius of a cylinder.

It's easy to prove (and we will do it in one of the future lectures) that the top base of a cylinder is also a circle and it is congruent to the bottom circle.

If a generatrix is perpendicular to bases then a cylinder is called a right cylinder. All cylinders that do not have this characteristic, that is if a generatrix is not perpendicular to bases, are called oblique cylinders.
In most cases we will be dealing with right circular cylinders and call them just cylinders.

Unizor - Geometry3D - Elements - Prisms

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Consider a plane α that we will call a base plane and a polygon ABCDEF on this plane (we specify 6-sided polygon, but it's not important, any polygon will do). This polygon will be used as a directrix of a cylindrical surface we will construct.
Also assume there is another plane β parallel to a plane α that we will also call a base plane. To differentiate between two base planes, we will call one of them "bottom" and another - "top".
Finally, assume that we have a straight line d that is not parallel to base planes.

Let's construct a cylindrical surface σ using polygon ABCDEF as a directrix and line d as a generatrix.

Prism is an object in solid geometry formed by a part of a cylindrical surface σ between base planes α and β and parts of the base planes inside that cylindrical surface - polygon ABCDEF on the "bottom" base plane α and a corresponding polygon A'B'C'D'E'F' on the "top" base plane β.

Points A, B, A', B' and all others are called vertices of a prism.
Segments AB, BC, EE' and all others are called edges of a prism.
Polygons ABCDEF and A'B'C'D'E'F' are, as we mentioned before, called bases of a prism.
Polygons ABB'A', BCC'B' and others are called sides of a prism.
Bases and sides of a prism are generically referred to as faces. Sides are also referred to as lateral faces.

It's easy to prove (and we will do it in one of the future lectures) that all sides ABB'A', BCC'B' etc. are parallelograms and bases ABCDEF and A'B'C'D'E'F' are congruent to each other.

If a generatrix is perpendicular to bases (and, therefore, all side edges, like AA', BB' etc. are also perpendicular to bases) then a prism is called a right prism. All prisms that do not have this characteristic, that is if a generatrix is not perpendicular to bases, are called oblique prisms.

Another classification of prisms is by their bases.
If a base is a triangle, the prism is called triangular prism.
If a base is a rectangle, the prism is called rectangular prism.
If a base is a parallelogram, the prism is called parallelepiped.
If a base is a square and it's a right prism and all edges are of the same length, the prism is called cube.
If a base is a hexagon, the prism is called hexagonal prism.
If a base is a regular N-sided polygon and it's a right prism, it might be called N-prism.

Unizor - Geometry3D - Elements - Cylindrical Surface

Unizor - Creative Minds through Art of Mathematics - Math4Teens

The subject of this lecture is to introduce a concept of cylindrical surface in three-dimensional space.

Let's assume that we have a curve c in the three-dimensional space and, separately, a straight line d.
A curve does not necessarily is a "flat" one (that is, we do not require all its points to lie on the same plane).
Now imagine that through each point on this curve c we construct a straight line parallel to the given straight line d. The set of all points of all these lines we have constructed form a surface that we call cylindrical.

Let's introduce some terminology related to cylindrical surfaces.

A curve c determines the points through which we draw lines parallel to d. This curve c is called directrix because it directs the position of each line we draw.

Those parallel lines that we draw form the cylindrical surface and are sometimes called rulings.

The straight line d, to which all rulings must be parallel, is called generatrix because it helps to generate the surface.

Let's exemplify this construction without any rigorous proof of characteristics of constructed objects.

If a directrix c is a straight line not lying on the same plane with a generatrix d, the result of our construction will be some plane parallel to a generatrix d.

If a directrix c is a circle lying in a plane perpendicular to a generatrix d, the result of our construction will be infinitely long right circular cylinder with an axis parallel to a generatrix d and perpendicular to a plane where a directrix c lies.

If, instead of a circle we will choose a polygon, the resulting surface would be an infinitely long prism with edges parallel to a generatrix d and perpendicular to a plane where a directrix c lies.

Notice, we did not define yet concepts or perpendicularity and parallelism between planes and straight lines. In this introductory lecture we just assume that students have intuitive understanding of these concept.


Interesting property of any cylindrical surface is that, if it's made of paper, it can be flattened on a flat plane without stretching or cuts (which is not the case with a spherical surface that we will introduce in subsequent lectures).

Another viewpoint to a cylindrical surface is that it can be considered as a surface formed by a line moving parallel to a straight line called generatrix along a curve called directrix.

Unizor - Geometry3D - Elements - Angles and Planes

The concept of an angle in general is related to a process of rotation. Thus, if two rays (half-lines) on a plane have a common beginning points, the angle between them can be viewed as an angle needed to rotate one ray around the common point (strictly speaking, counterclockwise is a positive direction and clockwise is negative) to coincide with another.

Provided a unit of measurement of such a rotation (for instance, rotation around a full circle can be set as equal to 360 degree or 2π radian), any angle can be measured in such units.

Talking about an angle between two intersecting lines on a plane, we usually consider the positive smallest angle between half-lines and call it an angle between given lines.

With angles between intersecting planes we will deal analogously. Consider we have two planes α and β that intersect along a line d, which, according to axiom A2 above, is a straight line. We can rotate plane α around line of intersection d until it coincides with plane β. Similarly to the measurement of angles between the lines, we can set the full circle of rotation to 360 degree or 2π radian, thus establishing a measuring unit for angles.

After we introduce other concepts in the corresponding lectures, we will see how measurement of angles between certain lines on the planes corresponds to angles between these planes. Jumping forward, let's just mention without rigorous explanation that, if we construct the third plane γ perpendicular (not yet properly defined) to a line d of intersection of two given planes α and β, the angle between α and β is equal in measurement (degrees or radians) to an angle between two lines lying in the plane γ - a line of intersection of γ with α and a line of intersection of γ with β.

Let's introduce a concept of an angle between a line and a plane. Assume that line d intersects plane α at one point X. Let's draw a line x within plane α through this point of intersection. Depending on the position of line x, the angle between lines d and x will be different. Let's choose the position of line x that gives the smallest value of this angle. This, by definition, is the angle between line d and plane α.
Again, without rigorous explanation that we defer to a future lecture, the line x that produces a smallest angle between line d and plane α is a projection of line d onto plane α that can be constructed by dropping a perpendicular (admittedly, not yet properly defined) from any point on line d outside a plane onto plane α.

Unizor - Geometry3D - Elements - Points, Lines, Planes

Geometrical points, lines and planes are abstractions, as most other objects of mathematics, like numbers, functions etc.
It is appropriate to present definitions of these concepts given by Euclid around 300BC:
"A point is that which has no part."
"A line is breadthless length."
"A straight line is a line which lies evenly with the points on itself."
"A surface is that which has length and breadth only."
"A plane surface is a surface which lies evenly with the straight lines on itself."

As definitions, from a modern viewpoint, these do not serve the purpose because they use other concepts like "part", "length" etc., which are not defined. So, consider them not as definitions, but rather explanations of mathematical concepts of a point, a line and a plane.
The contemporary set of axioms of geometry that includes those related to solid geometry in three-dimensional space were developed by David Hilbert in the beginning of the 20th century. They are presented on the Web site that hosts this and all other lectures - UNIZOR.COM - in notes for this lecture.
We will use the following short list of main propositions related specifically to planes, that we take as axioms, and that are based on the more formal list of axioms above:

A1. If two points of a straight line belong to a plane, every point of this line belongs to this plane.

A2. If two planes have a common point, they intersect along a straight line passing through this point.

A3. For any three points not lying on the same straight line there is one and only one plane that contains them.

From the above axioms we can derive the following simple theorems:

T1: For any straight line a and a point X outside of this line there is one and only one plane α that contains both of them, line a and point X.
Proof
We have to prove two things: the existence of a plane that contains a given straight line and a point outside it and the uniqueness of this plane, that is that there is only one such plane.
Choose any two points A and B on a given line a. Using axiom A3 above, there is one and only one plane α that contains these two points and the one outside of a line - point X.
Now, using axiom A1, we can say that an entire straight line a belongs to this plane.
So, plane α satisfies the condition of existence. But is it the only one?
Assume that with two different points on line a, C and D and the same point X we construct a different plane β that contains them. Since points C and D lie on line a, by axiom A1 an entire line a belongs to plane β as well as to plane α. Hence, three points X, C and D not lying on the same line belong to both plane α and plane β. Based on axiom A3, there is one and only one plane that contains these three points. Therefore, planes α and β are identical.

T2: For any pair of intersecting straight lines a and b there is one and only one plane α that contains them both.
Proof
Let X be a point of intersection of lines a and b. Choose a point A on line a and a point B on line b.
Using axiom A3, construct a plane α that contains three points X, A and B.
This plane α contains an entire line a because it contains two points on it - X and A (axiom A1). Plane α also contains an entire line b because it contains two points on it - X and B. Therefore, the condition of existence of a plane α
is satisfied. But is it unique?
Yes, because any other plane β that contains both lines a and b would inevitably be identical to α since both planes must contain the three points mentioned above - X, A and B.

T3: For any pair of parallel lines there is one and only one plane that contains them both.
Proof
The existence of such a plane follows from the definition of the parallel lines. The uniqueness of this plane follows from axiom A3, if we choose any three points - one on one line and two on another.

T4: For any given straight line in space there are infinite number of planes that contain it.
Proof
Choose any point outside the line and construct a plane that contains both this point and an original line. Now you can choose any point outside this plane and construct another plane that contains this new point and an original line. Choose yet another point outside both previously constructed planes and construct a third plane that contains this new point and an original line. This process can continue infinitely.
All these planes are different and all contain an original straight line. They can be considered and a set of positions of one plane that rotates around a straight line - an axis or rotation.

Unizor - Probability - Advanced Problems 6

Problem A
Knowing unconditional probabilities of events A and B and a conditional probability of an event B under condition of the occurrence of an event A, determine the conditional probability of an event A under condition that B has occurred.

Example
Many good mathematicians are also good chess players.
Assume that 1% of the people you might meet are good mathematician.
Assume further that 90% of them are good chess player.
At the same time only 10% of the people you might meet are good chess player.
What is the probability that a good chess player you've met is a good mathematician?
In terms of this problem, event A occurs when you meet a good mathematician; an event B occurs when you meet a good chess player.
P(A)=0.01
P(B)=0.1
P(B|A)=0.9
What is P(A|B)?

Solution
From the definition of a conditional probability, the probability of the occurrence of both events A and B (that is, the probability of an event A AND B or A∩B) can be represented in two ways:
(1) probability of the occurrence of A multiplied by a conditional probability of the occurrence of B under condition of the occurrence of A;
(2) probability of the occurrence of B multiplied by a conditional probability of the occurrence of A under condition of the occurrence of B.
In both case we should have the same result:
P(A∩B) = P(A)·P(B|A)
P(B∩A) = P(B)·P(A|B)
Of course,
P(A∩B) = P(B∩A)
Therefore,
P(A)·P(B|A) = P(B)·P(A|B),
from which we can derive the resulting formula:
P(A|B) = P(A)·P(B|A) / P(B)

Applied to the example above,
P(A|B)=0.01·0.9/0.1=0.09
So, there is 9% chance that a good chess player you've met is also a good mathematician. The knowledge about a person to be a good chess player increases the chances that he is also a good mathematician in 9 times compared to a random selection.

Problem B
This is a similar problem and it leads to a Bayes formula.
Assume there are two mutually exclusive events, A and B, that cover the whole sample space.
That is, A∩B=∅ (an empty set) and A∪B=Ω (an entire sample space).
Now we have another event X, the probability of which depends on whether A or B has occurred.
Knowing probability of event A (and, therefore, knowing the probability of B as a complementary event) and conditional probability of event X under condition of occurrence of events A and B, determine the conditional probability of event A under condition of occurrence of event X.

Example
For instance, A is an event describing a case of a person that is sick with some specific illness and B is an event that he has no such illness.
Now we run some diagnostic medical test X that is supposed to identify a specific illness defined by an event A.
Usually (but not always) the test performed on a sick person gives a positive result (event X occurred) and, if performed on a person who has no such type of illness, usually (but not always) gives a negative result (event X did not occur).
In medicine this is a typical situation when tests not always correctly identify the illness for a sick person and sometimes give false positive results on a non-sick person.
Assume that only 1% of people are sick with the illness in question, so we know the probability of the occurrence of event A:
P(A)=0.01
The rest 99% of the people are not sick, so the complementary event B has probability; P(B)=0.99
If the diagnostic test is performed on a sick person, it gives positive results in 95% of the cases, that is
P(X|A)=0.95
If the diagnostic test is performed on a non-sick person, it gives negative results in 95% of the cases, but in 5% of the cases gives false positive result.
P(X|B)=0.05
When the person comes to a doctor for a routine check-up, a doctor decides to run this diagnostic test and the result is positive.
What is the probability that this person is indeed sick with an illness in question?

Solution
P(A|X) = P(A)·P(X|A) / [P(A)·P(X|A)+P(B)·P(X|B)]

Applied to an example above,
P(A)=0.01
P(B)=0.99
P(X|A)=0.95
P(X|B)=0.05
Therefore, the probability of getting a positive result from the test is
P(X) = 0.01·0.95+0.99·0.05 =
= 0.059
Hence, the conditional probability of a person to be really sick if his test shows positive result is
P(A|X) = 0.01·0.95 / 0.059 ≅
≅ 0.161
As you see, relatively reliable test (95% reliability for sick people) delivers only 16% of reliability for randomly chosen person because of 5% of false positive results it delivers for healthy people. Really low reliability and insufficient to properly diagnose the person with the illness in question.

Unizor - Probability - Advanced Problems 5

Problem A
Imagine you are a member of a tourist group in Bangkok, Thailand in front of the Emerald Buddha Temple. There are N people in your group and all of them have the same kind of shoes of the same size. To enter, all of you have to take off your shoes. You do, enter the temple and, when you got out, each of you looks for the shoes at the entrance and sees that they are all the same (left and right shoes differ, but all the left and all the right are the same).
What is the probability that each of you will put on his or her own pair of shoes?

Answer: (1/N!)^2

Problem B
A city is supplied with electricity from the power station through two substations connected sequentially. The probability of an accident on each substation during a period of one year when the electricity is interrupted is P.
(a) What is the probability of uninterrupted flow of electricity to a city during a period of one year?
(b) What should be the value of P if you would like the probability of uninterrupted supply of electricity during a period of one year to be 0.99?
(c) What is the probability of uninterrupted flow of electricity to a city during a period of one year if it's sequentially connected through N substations?

Answer:
(a) (1−P)^2
(b) P = 1−√0.99 ≅ 0.0050
(c) (1−P)^N

Problem C
The same problem as B above, but the substations are connected in parallel.

Answer:
(a) 1−P^2
(b) P = √(1−0.99) = 0.1
(c) 1−P^N

Problem D
A city needs a solution to some specific engineering problem. There are N companies specializing in this field, each with its own rate of success.
Let's set the probability of a company #i to be able to solve the problem during a specified time to be Pi.
The city executive invites these companies in sequence, from company #1 to company #N.
If a particular company cannot solve this problem in a specified time, the city executive asks the next company to try. If all N companies failed to provide a solution, each is given another period of time to think about it, also in sequence. This process continues until a solution is found or until each company tried and failed T times. In case of the latter, the city gives up on a project as unsolvable.
What is the probability that the company #K will be the one that solves the problem?

Answer:
ProbK = Σj∈[1,T](A^(j−1)·BK)
where
A = Πn∈[1,N](1−Pn)
BK = PK·Πn∈[1,K−1](1−Pn)

Unizor - Probability - Advanced Problems 4

The following problems are about conditional probability.
It is useful to represent events as certain closed sets of points on a plane (like a circle or a square or any other closed area of any, even irregular, shape).
Then a statement "event A has occurred" would be represented as "a point randomly thrown on a plane falls inside area A". The relationship "the occurrence of event B follows from the occurrence of event A" can be represented as an area A lying completely inside of area B. If a randomly thrown point falls inside A (that is, "event A has occurred"), it is automatically follows that this point is located inside B and event B can be considered as occurred.
A reminder about conditional probability of occurrence of event B if event A has occurred:
P(B|A) = P(B∩A)/P(A)
Graphically it can be represented as a big area Ω representing an entire sample space, areas A and B are inside it and intersect each other. In this case independence between A and B would be represented if the ratio of the area of their intersection A∩B towards the area of A equals to the ratio of the area of B to the area of Ω.

Problem A

Event X follows from event Y. True or false?
(a) event X follows from event NOT Y;
(b) event NOT X follows from event Y;
(c) event NOT X follows from event NOT Y;
(d) event Y follows from event X;
(e) event Y follows from event NOT X;
(f) event NOT Y follows from event X;
(g) event NOT Y follows from event NOT X;

Answer:
(a) false; (b) false; (c) false;
(d) false; (e) false; (f) false;
(g) true.

Problem B

Event X follows from event Y and also follows from event Z. True or false?
(a) event X follows from event Y AND Z;
(b) event X follows from event Y OR Z;
(c) event NOT Y follows from event (NOT X) OR (NOT Z);
(d) event NOT Y follows from event NOT (X OR Z);
(e) event NOT (Y AND Z) follows from event NOT X;
(f) event NOT (Y OR Z) follows from event NOT X;

Answer:
(a) true; (b) true; (c) false;
(d) true; (e) true; (f) true;

Problem C

Events X and Y are mutually exclusive.
(a) Are events X and Y independent?
(b) Are events X and NOT Y independent?
(c) Are events NOT X and NOT Y independent?

Answer:
(a) No; (b) No; (c) No.

Problem D

Given a standard deck of 52 cards ranking from 2 to 10, Jack, Queen, King and Ace in four different suits Spades, Hearts, Diamonds and Clubs.
A random experiment consists of pulling one card out of this deck.
Consider the following events:
E1: A card that is a Queen is pulled;
E2: A card that belongs to a suit of Spades is pulled;
E3: A card that belongs to a suit of Spades or Hearts is pulled;
E4: A Queen of Hearts is pulled;
(a) Are events E1 and E2 independent?
(b) Are events E1 and E3 independent?
(c) Are events E1 and E4 independent?

Answer:
(a) Yes; (b) Yes; (c) No;