Translators- harmonic and rotary-harmonic motion !

Translatory-harmonic motion is represented in Cartesian coordinates by the equation 
y = f(x) = a + b cos px/q
while rotary-harmonic motion requires the polar equation 
r = f(t) = a + b cos pt/q.
In both case a, b, p and q are constants. This latter cyclic-harmonic curve has been studied kinematically by Moritz [4]. When he varied the ratio p/q he was able to describe four species of cyclo-harmonic curves; curtate, cuspitate, prolate and foliate.

FIGURE 1 shows cases

1/7, 2/7, 2/5,
2/11, 3/11, 4/11

r = 1 + tan pt/q when q is odd. Zeros, double points and intersections of axes can be counted and thus verify the theorems since the ratio p/q which is in the upper right corner. For example: the trajectory of the 2/7 case crosses the pole 4 times, and the x axis 18 times and has 24 double points (4 of which are off the screen).
Now when a tangent function replaces the cosine function in the polar equation quite different properties occur. For instance, the sum and product of the integers p and q determine how often the resulting new curve crosses itself (multiple points), how often the curve crosses the x and y axes, and how often it passes through the pole (zeros of r). These polar curves are of special interest because they represent the cross section of the surfaces (with z constant, z= b) in the class of ruled surfaces of rational order r = a + z tan pt/q which were examined by Emch [1]. Such a cross section has been used to prove that a Moebius band (in this case p/q = 1/2) self intersects in a straight line [3].
This is a study of these polar curves for the case a = b = 1. The matter is presented in theorem form followed by computer generated illustrations. The first theorem concerns the zeros of the curves, the second theorem concerns multiple points (where the trajectory crosses itself) and the last two concern points on the x and y axes. Consider the periodic trajectory of a particle moving along the polar curve 
r = f(t) = 1 + tan pt/q
where p and q are relatively prime integers and p < q . The members of this collection of curves form curious variations of the single loop (one complete trajectory) which is characteristic of the polar equation r = tan t/2. This curve passes through the pole only once and self-intersects at the point (1, pi /2). Its domain for one period is (-pi , pi ) . In the general case, however, when p and q are arbitrary and relatively prime positive integers, the domain of one period is (-q pi , q pi ) for q odd, and (-q pi/2, q pi /2) for q even. Outside each of these periods, the particle retraces a path which is composed of p or 2p loops depending on whether q is even or odd. Since the range of the tangent function sweeps through all real numbers, the variable r takes on all real values from -pi to +pi in the course of each loop. These curves self-intersect at the pole and have multiple points apart from the pole. Although p and q vary independently their product and sum can be used to classify properties of these polar curves. These properties are listed in the following four theorems.

FIGURE 2 shows cases

1/2, 1/8, 3/8,
3/4, 3/10, 5/12

r = 1 + tan pt/q when q even. For example: the trajectory of the 3/10 case crosses the pole 3 times, the y axis 13 times and has 15 double points (3 of which are off the screen).




Theorem l:

At the pole the curve r = 1 + tan pt/q has a multiple point of order p or 2p depending on whether q is even or odd.

Proof: The curve (1) passes through the pole (where r = 0) so the multiple points occur when tan pt/q = -1. If t' is a solution of this equation then t' + kq pi /p for k="0", 1, 2, 3, . . . are also solutions because the period of this function is q pi/p and so 
tan p( t' + kq/p)/q = tan (pt'/q + k ) = tan pt'/q = -1. 
Of these solutions, some have different principal values t* which are the distinct values of 0 in the interval [0, 2 ], that is for 0 < t* < 2 . These values t* occur when proper multiples of 2 (complete revolutions of the ray) are subtracted from t' + kq/p so that t' and t* are related by the equation
t' + k /p - 2N = t* (for integral values of N).
Notice that t' and t* will not be distinct solutions of (1) when ktq/p - 2N is an even integer since this would cause t' and t* to differ by an integral multiple of 2 , so would not belong to the set of principle values. If q is even, then k = p is the first value of k for which kq/p - 2N is even and t* is not essentially different from t'. If q is odd, then the first such integer is k = 2p. Therefore, the equation tan(pt/q) = -1 has p distinct solutions to obtained for k = 0, 1, 2, . . ., p - 1 when q is even or 2p distinct solutions t* obtained for k="0", I, 2, ..., 2p -1 when q is odd, and the theorem follows.

These trajectories self-intersect (have double points) at other points in the plane besides the pole and the frequency of their occurrence again depends on the values of the constants p and q. The rays along which double points occur are seen in the case p/q = 1/2, Fig. 3a. Each ray t = t_i cuts the curve in two distinct points (r_i, t_i) and ( r_j, t_j), where the arguments differ by that is t_j= +t_i. This occurs for all values of 0 except along the ray t = t_i where non zero values of r_i = - r_j.
This is the only double point for p/q = 1/2, because of the fact that for proper multiples of pi

r= 1 + tan [(t_i +k pi)/2] =

1 + tan [t_i/2] = r_i for k even

1 + tan [(t_i+k pi )/2] = r_j for k odd

since the period of the tangent funct_ion is pi, and the equation tan [t_i/2] = tan[(t_i+pi )/2] has only one solution resulting in r_i =- r_j. 

QED

This principle is now used to prove the second theorem concerning the general p/q case.

Theorem 2:

Apart from the pole, the curve r = 1 + tan p t/q has

I. pq/2 double points if q is even

II. 2p(q-1) double points if q is odd.

Proof of Theorem 2 part I: for q even:
A similar process is used now to find the distinct double points for larger values of q, for instance q = 8 in Fig. 3b. We examine one loop of the trajectory and count the double points. These points occur along the rays t = t_k + k when k is an odd integer: 1, 3, 5, ...., q-1. These are all distinct values of r_j or -r_j because the greater values q+1, q+3, ... q+k duplicate either the values tan pt_k/q or tan p(t_k+ )/q. This is true since
tan p[t_i + (q+k) ]/q = tan [p(ti+k )/q + p ] = tan pt_i/q for p even or tan (pt + pi )/q for p odd
since the period of the tangent function is . So along each loop there are an odd number of double points depending on the size of q. That is k can be 1, 3, 5, . . . , q-1 which amounts to a total of q/2 double points (r_k, t_k). From the reasoning in Theorem I, we know there are p loops when q is even so there are pq/2 distinct double points when q is even. 

QED

Proof of theorem 2 Part II. q is odd:
A similar procedure proves the second part of Theorem 2. Examine one of the loops, Fig. 3c to count the double points (r_k, t_k). These occur along the rays t = t_k where k = 1, 2, 3, 4, . . . q - 1. As above the values of k > q-1 duplicate tan pt_k/q or tan p(t_k+k )/q already obtained but since q is odd, even values do not duplicate. So in the case q odd, each loop has q-1 double points. Now from Theorem 1, we recall that there are 2p loops for q odd. So altogether we have 2p(q-1) double points apart from the pole.

QED

Theorems 3 and 4 can be proven somewhat like theorem I and 2 by counting the number of times a loop intersects an arbitrary ray from the origin.




Theorem 3:

The curve r = 1 + tan pt/q intersects the x axis in 2(p + q)points if q is odd.

Proof: Since y = r sin t the curve (1) may be represented by the equation y = sin t (1 + tan pt/q)( 2 ) Now sin t = 0 when for k = 0,1, 2, 3,...(2q - 1) as is clear from the proof of Theorem 1. Therefore, sin t accounts for 2q distinct roots. On the other hand, the other factor in (2) (I + tan pt/q) has 2 p zeros when q is odd according to Theorem 1. Altogether, there are 2p + 2q roots for the equation y = sin t ( 1 + tan pt/q) = r sin o and the theorem follows, so the curve r = 1 + tan pt/q intersects the x-axis in 2(p + q) points. 

QED

FIGURE 3 shows cases

1/2, 3/8, 1/3

Double points (apart from the pole) when r_i = -r_j along the ray t_k:
3a Special case p/q = 1/2: only one double point (r_k, t_k ) along t_k = /4
3b. Case of q even: pq/2 = 3(8)/2 = 12 double points for q even (3 are off screen)
3c. Case of q odd: 2p(q-1 )= 2(1)(3-1 ) = 4 double points for q odd

Theorem 4:

The curve r = 1 + tan pt/q intersects the y-axis in (p + q) points if q is even.

Proof: A proof is used which is similar to that of theorem 3. Substituting x = r cos t in (1) we write (1) as x = cos t(1 + tan pt/q) ( 3 )
A proof is used which is similar to the proof for theorem 3. Substituting x = r cos t in (1) we write (1) as x = cos t (1 + Tan pt/q). As in the proof for theorem 3, the zeros for the factors of (3) can be added together. Since cos t = O whenever t = (2k-1) /2 for k = 1, 2, 3,. . . q then cos t has q zeros. The factor (1 + tan pt/q ), however, has p zeros according to Theorem I. So the total number of roots for (3) is (p + q) and the theorem is proven: that is r = 1 + tan pt/q intersects the y-axis in (p + q) points.

QED

Figures I and 2 illustrate members of this collection of curves for the varying values of p and q. The ratio p/q for each graph is stated in the upper right hand corner. Because of the fact that the graphs extend only to six units along the x and y axes, some points of intersection cannot be seen, only extrapolated.
This collection of polar curves reveals properties not easily found in Cartesian curves. Although the integers p and q are independent, except for the obvious fact that p and q are relatively prime, this study illustrates these relations.
1. The number of times the curves passes through the pole depends on whether q is odd or even and is a simple function of p. 
2. The number of double points depends on the product of p and q 
3. The frequency in passing through the axes depends on the sum of p and q . The points of these trajectories in the polar plane can be calculated using a computer for any ratio p/q and then by using computer graphics these four theorems can be demonstrated.

References

[1] A. Emch, "On a certain class of rational ruled surfaces",
American Journal of Mathematics, 42 (1920) 189-210

[2] W. C. Graustein, Introduction to Higher Geometry, Macmillan, New York 1940

[3] J. MacDonnell, S.J., "A ruled Moebius band which self-intersects in a straight line",
American Mathematical Monthly, 91 (1984) 125-127

[4] R. E. Moritz, "On the construction of certain curves given in polar coordinates", American Mathematical Monthly, 26 (1917) 213-220.


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10.7 graph of polar equation

Today we learnt the graph of polar equations 

The graph is either a circle, a limacon or rose curve, or dumbbell shape. 

First we have to test if it is symmetrical 

Then analyze the graph by finding its max value and zero

Finally let's analyze a rose curve and other special shapes

10.6 polar coordinate

This is an image of the polar coordinate system. 

     Ex 1 plotting points in polar coordinate system 

Ex 2 multiple rep of points 

Conic section

In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with aplane. In analytic geometry, a conic may be defined as a plane algebraic curveof degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.

Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.

The conic sections were named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Latus Rectum is the line through the focus and parallel to the directrix.

The length of the Latus Rectum is 2b²/a.

10.2 hyperbola

Hyperbola is the set of all points (x,y) the difference from. 2 foci is constant.

Here artwo kinds of hyperboles:

This is a clearer picture of the hyperbole

The following is the formulas we need to know

10.2 eclipse

After parabola, let's continue to review algebra 2 materials: eclipse

Eclipse is a set of all points (x,y) the sum of whose distances from 2 fixed point is constant.

Example:

1. Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
2. 
   
   

Father of parabola

Menaechmus is remembered by mathematicians for his discovery of the conic sections and his solution to the problem of doubling the cube.^[1] Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola, and the hyperbola, as a by-product of his search for the solution to the Delian problem.^[2] Menaechmus knew that in a parabola y² = lx, where l is a constant called the latus rectum, although he was not aware of the fact that any equation in two unknowns determines a curve.^[3] He apparently derived these properties of conic sections and others as well. Using this information it was now possible to find a solution to the problem of theduplication of the cube by solving for the points at which two parabolas intersect, a solution equivalent to solving a cubic equation.^[3]

There are few direct sources for Menaechmus' work; his work on conic sections is known primarily from an epigram by Eratosthenes, and the accomplishment of his brother (of devising a method to create a square equal in area to a given circle using the quadratrix),Dinostratus, is known solely from the writings of Proclus. Proclus also mentions that Menaechmus was taught by Eudoxus. There is a curious statement by Plutarch to the effect that Plato disapproved of Menaechmus achieving his doubled cube solution with the use of mechanical devices; the proof currently known appears to be solely algebraic.

Menaechmus was said to have been the tutor of Alexander the Great; this belief derives from the following anecdote: supposedly, once, when Alexander asked him for a shortcut to understanding geometry, he replied "O King, for traveling over the country, there are royal road and roads for common citizens, but in geometry there is one road for all" (Beckmann 1989, p. 34). However, this quote is first attributed to Stobaeus, about 500 AD, and so whether Menaechmus really taught Alexander is uncertain.

10.1 parabola

Today in class we reviewed parabola 

Parabola: the set of all points (x,y) that are equidistant from a fixed line and a fixed point not on the line.

(X-h)^2=4p(y-k) vertical y= k-p

If p is more than 0, the parabola opens upward. If less than 0, it opens downward.

Focus (h,k+p)

Directrix: y= k-p

Axis of symmetry: x= h

(Y-k)^2=4p (x-h) horizontal 

Focus( k+p, h) 

Directrix: x= h-p

Axis of symmetry: y=k

Now let's see one problem: 

Find th standard form of a parabola with vertex (2,1) and focus (2,4) 

(X-2)^2= 4p (y-1)

P=3

So (x-2)^2=12(y-1)

AP Stats Intro - how data are collected

To derive conclusions from data, we need to know how the data were collected; that is, we need to know the method(s) of data collection.

Methods of Data Collection

There are four main methods of data collection.

• Census. A census is a study that obtains data from every member of a population. In most studies, a census is not practical, because of the cost and/or time required.
  
  
• Sample survey. A sample survey is a study that obtains data from a subset of a population, in order to estimate population attributes.
  
  
• Experiment. An experiment is a controlled study in which the researcher attempts to understand cause-and-effect relationships. The study is "controlled" in the sense that the researcher controls (1) how subjects are assigned to groups and (2) which treatments each group receives.
  
  In the analysis phase, the researcher compares group scores on some dependent variable. Based on the analysis, the researcher draws a conclusion about whether the treatment (independent variable) had a causal effect on the dependent variable.
  
  
• Observational study. Like experiments, observational studies attempt to understand cause-and-effect relationships. However, unlike experiments, the researcher is not able to control (1) how subjects are assigned to groups and/or (2) which treatments each group receives.

Data Collection Methods: Pros and Cons

Each method of data collection has advantages and disadvantages.

• Resources. When the population is large, a sample survey has a big resource advantage over a census. A well-designed sample survey can provide very precise estimates of population parameters - quicker, cheaper, and with less manpower than a census.
  
  
• Generalizability. Generalizability refers to the appropriateness of applying findings from a study to a larger population. Generalizability requires random selection. If participants in a study are randomly selected from a larger population, it is appropriate to generalize study results to the larger population; if not, it is not appropriate to generalize. 
  
  Observational studies do not feature random selection; so generalizing from the results of an observational study to a larger population can be a problem.
  
  
• Causal inference. Cause-and-effect relationships can be teased out when subjects are randomly assigned to groups. Therefore, experiments, which allow the researcher to control assignment of subjects to treatment groups, are the best method for investigating causal relationships.
• 
  
  

9.8/9.9 Data collection

Today in class we touched upon stuffs from AP Stats!!! Excited?

First let's review the notion of mean, median and mode.

Mean : average number of a number set

Median: the middle number of a number set

Mode: the most repeated number 

Then the novel stuff begins!

Measures of dispersion (with mean m)

Variance = ((x1-m)^2 +(x2-m)^2+...+ (xn-m)^2)/n

Standard deviation= variance ^(1/2)

Quartile:( with median m)

Lower: median of the numbers that occur before m

Upper: median of the numbers that occur after m

Example: 

42,62,40,29,32,70

Mean =(42+62+40+29+32+70)/6=45.8

Median= 41

Mode= N/A

Variance= (42-45.8)^2+ (62-45.8)^2+...+ (70-45.8)^2/6= 228

Standard deviation= 15.104

Is that hard for you???

9.7 probability

Heyyyy today lets talk about probability!

First lets be clear about some vocabulary:

 Independent event: the occurrence of one event has no event has no effect on the second event    Complement of an event: probability that the event does not occur

Experiment: the result is uncertain

Outcome: possible results

Probability= n(E)/ p(E) must be between 0 and 1 

If p (E) = 0, impossible event 

If p(E)= 1, certain event 

Mutually exclusive event: P( A U B)= P(A) + P (B) - P(A n B) 

Independent event P (A and B) = P(A)X P(B)

Complement: P(A')= 1-P(A)

Example : if there are 3 red marbles, 2 black marbles and 5 yellow, what is the probability of picking out one red marble?

 P = 3/10