text:
## How to Grow a Painting With Generative Art

What if you could plant a “seed”, and grow a painting? Surely that would save the artist a lot of trouble learning pesky things like how to draw.

As it turns out, such a thing is possible. Computer generated artwork is nothing new. Who could forget Dire Straits’ Money for Nothing?

And, of course, my own blog gallery is full of digital artwork.

But with digital painting and drawing, the computer is simply another tool, like a paintbrush or charcoal stick, used to create an image.

I’m proposing we step away from that, and let the computer take the creative reins.

It’s an application of genetic programming.

If you’ve never heard of genetic algorithms and evolutionary programming, the idea is to apply the concepts of evolution and “survival of the fittest” to solve problems in mathematics and computer science.

To solve a problem, you consider potential solutions to be “organisms.” As organisms, solutions mate and produce offspring. “Organisms” which represent better solutions are given a higher chance of mating. After several generations, you should have a fairly good set of solutions.

A classic example of an application would be the traveling librarian problem.

Suppose your local librarian had to deliver books to all five of her branches. In which order should she visit the cities so that she travels the fewest miles?

View Larger Map

In this example, the solution is represented as the permutation of five towns

Solution 1 “gene” = { Warsaw, Faison, Beulaville, Rose Hill, Magnolia}

and represents a total journey of 76.3 miles.

The permutation below represents a total journey of 90.0 miles.

Solution 2 “gene” = { Faison, Rose Hill, Beulaville, Magnolia, Warsaw }

Let’s suppose these two “organisms” are selected to mate. Just like human children inherit their parents’ eyes, hair, etc. These “children” can inherit their own parents’ properties.

To produce the children, we can apply a “reorder crossover” on the last three towns.

Solution 1 has Beulaville, Rose Hill, Magnolia as the last three towns. In solution 2, they appear in the order: Rose Hill, Beulaville, Magnolia.

To produce the first child, we take solution 1 and reorder these last three towns so that they appear in the order the mate, solution 2, has them in. The first child is:

Child 1 “gene” = { Warsaw, Faison, Rose Hill, Beulaville, Magnolia}

Solution 2 has Beulaville, Magnolia, and Warsaw as the last three towns. In solution 1, they appear in the order: Warsaw, Beulaville, Magnolia. The second child is:

Child 2 “gene” = { Faison, Rose Hill, Warsaw, Beulaville, Magnolia}

Child 1 ends up representing a distance of 82.9 miles. Child 2 ends up representing a distance of 90.2 miles.

View Larger Map
Child 2 never lived up to its parents’ expectations.
They wanted him to be a doctor. He went into ventriloquism.

Neither children were as successful as their parents. In the next generation, they’ll be less likely to mate.

This is due to what is called the fitness function. The job of the fitness function is to assign to each solution a chance of mating. A simple one for the traveling librarian problem would be to divide 100 by the distance traveled, so that shorter trips have a higher chance of surviving to the next generation.

The idea is to identify traits that yield good results, and try to create solutions that take on those traits.

In our problem, Magnolia and Rose Hill are near each other. Solutions that place these towns next to one another are more likely survive to the next generation, just like real organisms develop traits that aid in their survival in the wild.

So what about art? A popular way to create art on a computer is to apply transformations, functions which alter the image. As a quick example, consider this drawing of Agreeable Andy:

“I don’t want to be an example!”

And three functions: “Rotate by 90 degrees”, “Flip horizontally”, and “Color the left half red.”

Depending on the order we apply them to Andy, we get a different result:

“Get me off this thing!”

Look familiar? Each drawing is the product of a permutation of functions. Each permutation can be an “organism.”

The organisms are:

S1 = { Horizontal flip, Rotate 90 degrees, Left half red }

S2 = { Left half red, Rotate 90 degrees, Horizontal flip }

Now all we need is a fitness function, something to determine which qualities are desirable.

Oh, it turns out computers aren’t all that great at having these kinds of preferences.

But you are! All you need is a program that presents you with art. You select which ones you like, and the program will eventually identify which qualities you like. You get to play nature, selecting which “creatures” are most likely to survive.

Such a program exists, too. If you want something to play with, go download Evolvotron. It produced the image at the top of this post.

Evolvotron screenshot.

The screenshot above shows the program at launch. The images are fairly simple, but if you keep at it long enough, you can generate some intricate and complex ones.

I don’t see why generative art needs to live entirely in the realm of the abstract. If it’s not out there already, I don’t think it’ll be too long before someone thinks of combining Evolvotron with, say, Makehuman.

“Evolved art” raises an interesting question, though. Who is the artist? See the image at the top of the post? I didn’t really do anything to make it. I didn’t do the work involved in painting the image. I didn’t even write the program.

On the other hand, the image wouldn’t exist if it weren’t for me.

Am I the artist merely because I have an eye for a particular result? What do you think?

## Mathematics vs. Nature for People Who Hate Math

… For People Who Hate Math is the worst title for a feature on this blog. If you hate math, why are you reading this?

Maybe you were searching for pictures of cats, and came upon this page by mistake because the universe is out to get you. In that case, I hope this cheers you up.

Or, maybe you don’t really hate math, but understanding what the gist of it is just proved rather elusive. In that case, this week’s post is just for you.

Mathematics is different from the other sciences. If you take a chemistry course, you can see the power of chemistry on the first day. On my first day of chemistry in high school, my teacher explained how there was enough chemical power in the storage room to level the building, should terrorists attack. Note that these were the pre-9/11 days.

I’m not sure how reducing the building to rubble before the terrorists could was a viable anti-terrorist strategy. But in Texas, we just didn’t question these things. Then she set the ceiling on fire.

Chemistry!

But what of mathematics? Chemistry is hard. From day one, though, I had at least a general idea of what we were after. We were seeking a better understanding of real, observable events. Math, on the other hand, seemed like just an arbitrary set of made-up rules.

There’s a good reason for that, too. Math is an arbitrary set of made-up rules.

This is a common question I hear: Where does ___ arise in nature? The answer, more often than not, is “it doesn’t.”

Case in point, 2 + 2 can equal 5. All you need to do is change the rules, which isn’t that hard to do since they were all made up in the first place.

There’s no natural reason 2 + 2 must equal 4. There is a practical reason. We use mathematics as a language. Just like we use the word cow to describe a large, horned animal that serves as the butt of many Far Side cartoons, we use the word two to describe the quantity of cats currently whining at me for their lunch.

The rules that people learn when they take a class like Algebra were designed for a purpose. There’s no reason they can’t change.

The difficulty in learning math is that not only is it a set of rules, it’s a very complex set of rules, all dependent on each other.

For example, let’s take a clock – and see how this simple device unravels almost everything you learned in elementary school.

We’ve all known for a while that any number, added to zero, is itself. So if you take the 4 position on a clock, advance the hour hand 12 spaces, it’s back to 4. Hence, on a clock,

$4 + 12 = 4$

But 4 + 0 = 4. This means 0 and 12 are the same numbers.

This completely messes up multiplication. Now $4 \times 3 = 0$. It also messes up division. $4 \times 5 = 20$, but since 20 = 8 + 12, and 12 is the same as 0, then 20 also equals 8 + 0.

So $4 \times 5 = 8$.

It gets worse. Try to divide both sides of that equation by 5, and you’ll see the “fraction” $\frac{8}{5}$ equals 4.

But it turns out you can’t really do that – not because it seems strange. There are stranger looking things that turn out to be valid. No, it turns out we completely obliterated the concept of division itself.

The concept of division itself is built upon rules that we just threw out. Whoops. Who needs multiplication and division, anyway?

The science of mathematics is in figuring out what happens when you change the rules, or add new ones. The catch is that, unlike chemistry or physics, mathematics is a beast mankind created. That’s right. We did this to ourselves.

The difficulty in learning math is that there isn’t much that arises out of nature to help you. The chemists who first cracked the secrets of fire had easy access to all the samples they needed. The physicists who determined the physics of falling objects had an ample supply of gravity at their disposal.

Everything in math arises out of something someone else had previously established.

Here’s another illustration. One of the most arbitrary constructions of them all – polynomials – comes out of a very small set of groundwork rules.

Polynomials don’t occur in nature, and nothing in nature seems to suggest the idea. But two simple man-made rules do.

Polynomials are members of a set of objects that adhere to two specific rules: Any two polynomials added together must be in the set, as well as any two polynomials multiplied together.

Let’s start with x. If x is in the set of polynomials, then x + x, or 2x, is in the set. This means x + 2x, or 3x, is in the set. So is 2x + 3x = 5x, 2x – 5x = -3x, etc. Following this logic, any multiple of x is in the set.

We can apply the same principle to multiplication. Since x is in the set of polynomials, the product of x and x is also in the set. So $x^2$ is also a polynomial.

But this means the product of x and $x^2$, or $x^3$, is a polynomial.

Don’t forget we can add any of these powers to themselves any number of times. Since $x^3$ is a polynomial, so is $2x^3$ and $31x^3$.

Finally, we can add all these multiples of powers. So $x + x^2 + 31x^3$ is also a polynomial. Every polynomial is constructed from multiples of powers of x.

Nothing in nature suggested we could do this. It was simply the product of saying, “let’s create a set and a couple of rules that its objects must obey, and see what happens when we run with it.”

If you’re in the crowd of “people who never saw the gist of mathematics,” try looking at the subject from the perspective of a game: learning how to play, given a set of rules printed on the back of the box cover.

When you get really good, or just bored, you can look at it from the perspective of that kid who had to change the rules so he’d always win. The changes didn’t always make sense to the observer, but that kid sure knew what he was doing.

## Author Obligations and the Millennial Generation

Like last week’s post, this week’s topic was suggested by my wife. For those not in the know: you, too, can suggest a topic. Almost anything that falls within the realm of “storytelling, art, and mathematics” is welcome – even if the fit is vague.

If you haven’t seen my “about me” page, I teach mathematics. I started teaching my senior year at The University of Tulsa. I was put in charge of what were called “quiz sections.” Some of you may have heard of “supplemental instruction,” peer-lead tutoring sessions which numerous colleges provide. “Quiz sections” were similar, although I was given the additional responsibility of administering quizzes. The novelty of grading papers lasted one day.

I was the same age as the students, and lived on campus. Both of these facts resulted in more than one amusing situations.

My geology lab partner was one of my students.

Every so often I’d bump into them at a party, where we’d learn we had mutual friends.

Except for the kid who dropped the class after one of these encounters, any initial awkwardness was swiftly eradicated by the realization, Oh! We have the same friends. And we’re the same age.

We’re even in the same clubs. My last year in Tulsa, I joined APO, which is a community service fraternity. A few of my students joined with me, and we’d work at the food bank and build houses for Habitat for Humanity together.

Then I graduated, moved to North Carolina, and grew older. Sometime over the span of the next four years, I managed to complete my doctorate. In my spare time, I aged a bit more.

One day, not too long ago, I realized I was now a different person. Then I realized I was wrong. I’m pretty much the same me. The students were different.

Sometime between 2004 and 2013, the world decided it needed to be constantly plugged in. Then it decided it no longer needed wires.

My “internet empire” was recently upgraded, but my art gallery still reflects the hand-crafted-HTML internet I grew up with. What happened?

I’m not that old. I was born in 1983. So whose generation am I in?

It turns out I’m in the gray area between Generation X and the Millennial Generation. I missed out on most of the events which shaped Generation X. But the information age blast that affects so many of the Millennials didn’t really hit me until I was almost in high school.

1996, the year of Space Jam and Dolly the Sheep, ushered in the end of pay-by-hour AOL. For the majority of my childhood, “online” meant HAL-PC, Houston’s local BBS. Next year’s collegiate freshman class had just been born.

For today’s eighth-grader, “online” means near-instantaneous access to just about every form of media imaginable. In the palm of their hands. If this doesn’t blow your mind, chances are you’re a Millennial.

Let’s play a comparison game. My childhood. Theirs. I don’t think we grew up out of the same world.

It’s not just access to information that has changed. Even more so, the role of creator has changed.

First, there’s the obvious: everyone can be a content creator. How do you think this blog got here? It wasn’t because I submitted my posts to a publishing agency that decided I had something worthwhile to say.

One day I decided I’m going to author a blog. Then I did. And if anybody doesn’t like it, I can stick out my tongue and go, pfffffffffffffttttttttttt, which used to be theme song of internet connectivity.

Second, the relationship between creators and consumers has changed. There’s a lot that can be said about this. Intellectual property and copyright is a dissertation in itself.

So I’m going to address a question my wife brought up at dinner last night.

Q: What obligations do authors have to their readers?

Both of us are in fields where there’s diminishing tolerance for being set in our ways. I was reminded of this late last semester, when I received an email from a student at 1 am. I didn’t respond until I woke up the next morning, after he already asked me why I didn’t reply.

I don’t think it’s such a horrible thing to not be available 24/7. People aren’t, but information is. At least, that’s what we’re now used to.

As for authors, and content creators in general, there’s less and less room for the person who produces work behind a curtain.

In a way, we seem to be heading back to the days of the storyteller, telling a tale before a crowd and a fire. These were the days when the audience had a personal connection with the creator. They were there. They could ask questions, discuss the story, or whatever else they wished.

I’d hesitate a bit to say the internet provides “personal connections,” but how many of you have tracked down the website of your favorite author so you could ask a question or find out a bit more about them? How many of you have looked for their Facebook, Google Plus, or Twitter page, only to be vastly disappointed to find out they had none of these?

If you were born after 1995, “the internet” has been a household name for just about your entire life. “If it’s not online, it probably doesn’t matter.” If it’s not the case now, eventually that line will be a death sentence for the hopes of anyone wanting to publish without an internet presence.

This isn’t a case of traditional publishing versus electronic publishing. Whatever media is bought, in whatever form, this is the case of the consumer’s growing desire to feel connected to the producer.

Popular books aren’t just books. They’re cultures. Think of the vast library of fan fictions associated with top novels. I had never heard of “fan fiction” until I got to college. Now, fan fiction is endorsed by Amazon.

Picking up a copy of, say, Harry Potter, was doing more than just selecting a book to read. It was an invitation to participate in a sub-culture.

Going forward, I expect to see the wall dividing the producer and the consumer continue to crumble. Lurking in the shadows will be increasingly difficult. The author will take more of the role of a performer, expected to interact with their audience.

What do you think? Are we headed to a new golden age of authorship and readership?

* * *

Speaking of interaction, let me say “thank you” to everyone who has submitted a solution to A Conundrum of Eggs 1. If you’re sitting on an answer, it’s not too late to submit one yourself!

Much like its inspiration, A Tangled Tale, the answers aren’t revealed until the end of the tale. Since A Conundrum of Eggs has three installments, this rule may need to be tweaked so that nobody has to wait until October.


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# The banana problem

There is a SQUARE field and 44 BANANA trees are planted across the square field. Find the peremeter of that square field, if the distance between 2 BANANA tree is 2 metre.

Note by Jeet Dubey
7 years ago

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352 m is right i suppose considering question is right ! 44 banana across the field (assuming the line of 44 bananas parallel to one of the side) having 2 metre between each tree so length of one side comes to 88m . Since perimeter is asked we have to take addition four time because it is square, so 88*4=352.

- 6 years, 11 months ago

4 Banana trees are planted ONE at each corner of a square. and 10 trees are planted at each side between two corners of a side so Perimeter of the square field=4+(10*4)=44 meter

- 6 years, 11 months ago

88

- 4 years, 9 months ago

- 7 years ago

44 is not a square number so the field can't be square

- 7 years ago

I agree that this question doesn't make much sense, and that 144 (likely a typo) would be a much better interpretation.

However, you're making the assumption that the banana trees must be planted parallel to the sides of the field, which is not necessary. Hence, an interpretation of this problem (which likely isn't the one originally intended) could be to find the minimum area of a field that contains exactly 44 banana trees.

Staff - 7 years ago

If Jeet D intended to type 144,then the answer would be the same as Luca B's answer.Otherwise it should be $6 \times 2m \times 4=48m$ because 49 is the smallest square number larger than 44 and $\sqrt{49}=7$

- 7 years ago

Not quite. This is actually much more interesting. You are still making the same assumption that the banana trees must be planted parallel to the sides of the field.

For example, if we only wanted 2 trees, then your approach will require a square field of side length 2, which has 4 banana trees on the corners. Whereas, if you take the 'diamond', you can see that a square field of side length $\sqrt{2}$ is sufficient, with 2 banana trees on the opposite corners. Can you find an argument to show that this is the minimum length possible?

What happens in the case for 3 banana trees? Can we also do it with a square field of side length $\sqrt{2}$? If not, what is the minimum length?

What happens in the case for 4 banana trees? Is the minimum length 2? Can we not do any better? Why, or why not?

What happens in the case for 44 banana trees?

Staff - 7 years ago

Double interpretation with same result LOL

- 7 years ago

Perimeter is 88 m. 12 banana trees per side, 22 metre per side.

- 7 years ago

Why are there 12 banana trees per side?

Staff - 7 years ago

I thought the 44 trees are in the perimeter...12 per side with 4 in commons in the corners. This is my only way to interpret the problem so that it's logical...

- 7 years ago

That's certainly a valid interpretation, given the ambiguity in the question. Thanks for explaining!

Staff - 7 years ago

No problem :)

- 7 years ago

format: plain


text:
# How to do imbalanced classification in deep learning (tensorflow, RNN)?

I am trying to do binary classification of News Articles (Sports/Non-Sports) using recurrent neural net in tensorflow. The training data is highly skewed [Sports:Non-Sports::1:9].

I am using cross-entropy as my cost function, which treats both classes equally.

What are the ways by which user can penalise one class? Or is there any other cost function suitable for this purpose?

The suggestion in both cases was to add class weights to the loss function, by multiplying logits:

loss(x, class) = weights[class] * (-x[class] + log(\sum_j exp(x[j])))

For example, in tensorflow you could do:

ratio = 31.0 / (500.0 + 31.0)
class_weight = tf.constant([ratio, 1.0 - ratio])
logits = ... # shape [batch_size, 2]
weighted_logits = tf.mul(logits, class_weight) # shape [batch_size, 2]
xent = tf.nn.softmax_cross_entropy_with_logits(
weighted_logits, labels, name="xent_raw")

• I don't think you really want to multiply logits by weights but cross entropy by weights. See this answer for a correct implementation: stackoverflow.com/questions/35155655/… Mar 7 '17 at 8:47


text:
### Optimal Pairings

F. Vercauteren

##### Abstract

In this paper we introduce the concept of an \emph{optimal pairing}, which by definition can be computed using only $\log_2 r/ \varphi(k)$ basic Miller iterations, with $r$ the order of the groups involved and $k$ the embedding degree. We describe an algorithm to construct optimal ate pairings on all parametrized families of pairing friendly elliptic curves. Finally, we conjecture that any non-degenerate pairing on an elliptic curve without efficiently computable endomorphisms different from powers of Frobenius requires at least $\log_2 r/ \varphi(k)$ basic Miller iterations.

Note: Corrected statement of theorem 2

Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Keywords
Tate pairingate pairingelliptic curvesfinite fields
Contact author(s)
frederik vercauteren @ esat kuleuven be
History
2008-03-07: last of 5 revisions
See all versions
Short URL
https://ia.cr/2008/096

CC BY

BibTeX

@misc{cryptoeprint:2008/096,
author = {F.  Vercauteren},
title = {Optimal Pairings},
howpublished = {Cryptology ePrint Archive, Paper 2008/096},
year = {2008},
note = {\url{https://eprint.iacr.org/2008/096}},
url = {https://eprint.iacr.org/2008/096}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
Home » Verbal Ability » Ordering of Sentences » Question

#### Ordering of Sentences

Direction: In each of the following questions, choose the most appropriate sequence of sentences from the given choices.

1. (i) He went to his house and had himself announced.
(ii) This time, however, his greeting, though still polite, was colder than usual.
(iii) Since Monsieur de La Tremouille belonged to no faction, he ordinarily had no prejudices in his social relations.
(iv) The two men greeted each other politely, for while there was no friendship between them, there was at least respect.
(v) They were both honourable, great-hearted men.
1. (v) (iv) (i) (ii) (iii)
2. (iii) (iv) (ii) (i) (v)
3. (i) (iv) (v) (iii) (ii)
4. (i) (ii) (iii) (v) (iv)
##### Correct Option: C

Sentences (iii) -(ii) and (iv)-(v) form pairs.


text:
# [Announcement] Statistical Software Award

#### TheEcologist

##### Global Moderator
John M Chambers, or the man behind S, provides this award to grads and undergrads who have created their own statistical software applications.

Go for it ! And remember that SAS and SPSS and many others all originated as graduate student work!

Code:
John M. Chambers Statistical Software Award – 2011
Statistical Computing Section
American Statistical Association

The Statistical Computing Section of the American Statistical
Association announces the competition for the John M.  Chambers
Statistical Software Award. In 1998 the Association for Computing
Machinery presented its Software System Award to John Chambers for the
design and development of S. Dr. Chambers generously donated his award
to the Statistical Computing Section to endow an annual prize for
The prize carries with it a cash award of \$1000, plus a substantial
allowance for travel to the annual Joint Statistical Meetings where
the award will be presented.

Teams of up to 3 people can participate in the competition, with the
cash award being split among team members. The travel allowance will
be given to just one individual in the team, who will be presented the
award at JSM.  To be eligible, the team must have designed and
implemented a piece of statistical software. The individual within
the team indicated to receive the travel allowance must have begun the
development while a student, and must either currently be a student,
or have completed all requirements for her/his last degree after
January 1, 2009.  To apply for the award, teams must provide the
following materials:

Current CV’s of all team members.

A letter from a faculty mentor at the academic institution of the
individual indicated to receive the travel award.  The letter
should confirm that the individual had substantial participation in
the development of the software, certify her/his student status
when the software began to be developed (and either the current
student status or the date of degree completion), and briefly
discuss the importance of the software to statistical practice.

A brief, one to two page description of the software, summarizing
what it does, how it does it, and why it is an important
contribution.  If the team member competing for the travel
allowance has continued developing the software after finishing
her/his studies, the description should indicate what was developed
when the individual was a student and what has been added since.

An installable software package with its source code for use by the
award committee. It should be accompanied by enough information to allow
the judges to effectively use and evaluate the software (including
its design considerations.)  This information can be provided in a
variety of ways, including but not limited to a user manual (paper
or electronic), a paper, a URL, and online help to the system.

All materials must be in English.  We prefer that electronic text be
submitted in Postscript or PDF.  The entries will be judged on a
variety of dimensions, including the importance and relevance for
statistical practice of the tasks performed by the software, ease of
use, clarity of description, elegance and availability for use by the
statistical community. Preference will be given to those entries that
are grounded in software design rather than calculation.  The decision
of the award committee is final.

All application materials must be received by 5:00pm EST, Monday,
February 21, 2011 at the address below.  The winner will be announced
in May and the award will be given at the 2011 Joint Statistical
Meetings.

Information on the competition can also be accessed on the website of
the Statistical Computing Section (www.statcomputing.org or see the
ASA website, www.amstat.org for a pointer), including the names and
contributions of previous winners.  Inquiries and application
materials should be emailed or mailed to:

Chambers Software Award
c/o Fei Chen
Avaya Labs
233 Mt Airy Rd.
feic@avaya.com
Past winners:
http://stat-computing.org/awards/jmc/winners.html

http://stat-computing.org/awards/jmc/index.html

#### Dragan

##### Super Moderator
Great!...Thanks, Marco. I will pass this along to the grad students here at my university.


text:
# Trigonometry In The Workplace

 Hints of Trigonometry on a 3,700-Year-Old Babylonian Tablet Image An ancient Babylonian tablet known as Plimpton 322 consists of a table of 60 numbers organized into 15 rows and four columns. How to solve trigonometry problems or questions? Step 1: If no diagram is given, draw one yourself. Students explore the concept of similar right triangles and how they apply to trigonometric ratios. Guild Of Light - Tranquility Music. The calculator will generate a step by step explanations. Here's a diagram to awaken your memory. by - Blackie And Son Limited , 1919. A right-angled triangle is a triangle in which one of the angles is a right-angle. COMING SOON – More on Trigonometry. Fix a pipe of aluminum at centre of protractor in such a way that the pipe is free to rotate. Advance Algebra and Trigonometry is a high-school course that allows students to further strengthen their algebra skills and study trigonometry before enrolling in a precalculus course. Alternating current is described by sine waves, so an understanding of trigonometric functions is indispensable. Fuck trigonometry. trigonometry noun definition in Mathematical terminology topic from the Oxford Advanced Learner's Dictionary. Let's work here on our understanding of "invention". If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. Labelling the sides. Some examples:. Mechanical Engineers use trigonometry to inspect every component made to make sure that they are to a satisfactory standard. Question #37967. The relation between the sides and angles of a right triangle is the basis for trigonometry. They work mostly in engineering services, research and development, manufacturing industries, and the federal government. Today we looked at examples of "real world" applications of trigonometry. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own. Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Home - Shape and Space - Triangles - Trigonometry. In the sexagesimal system, multiplication or division by 120 (twice 60) is analogous to multiplication or division by 20 (twice 10) in the decimal system. We are confronted with a smile, the workplace is no real information. You will see a line that says “Radian” and “Degree”. A summary of Vectors in 's Trigonometry: Angles. The subject has numerous elegant and unexpected theorems. trigonometric formulas for all:) This App has all the important formulas and concepts -Basic definitions -Trigonometric functions -Inverse trig functions -Unit circle -Law of sine and cosine -Negative angle formulas -Periodic formulas -Double & triple angle formulas -Power-reduction formulas -Degree to radians formulas -Sum and difference formulas -Product to sum formulas -Sum to product formulas -Reduction formulas -Compositions of trig functions -Trigonometric equations -Euler's formula. 2000 Simcoe Street North Oshawa, Ontario L1G 0C5 Canada. Use these practical worksheets to ground students in the Law of Sines, the Law of Cosines, tangents, trigonometric functions, and much more!. In the second set, students will solve problems with a trigonometry emphasis. So trigonometry is a part of pre-calculus, and is related to other pre-calculus topics, such as exponential and logarithmic functions, and complex numbers. Forum posts are owned by the individual posters. How to Use Right Angled Trigonometry. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own. QA Registration Information: If you are a matriculated student, you must register for credit classes. In today's post I focus on ideas and resources for teaching trigonometry. Most Plus articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. Give angles in degrees and minutes. The answer should be written in. Questions 1 and 2 require Sine, questions 3 and 4 require Cosine, question 5 and 6 require Tangent. My teacher has asked us to write a paper on how trigonometry affects my life and careers. Trigonometry is a mathematical method used to define relations between elements of a triangle. 13º y = 180º - 135º = 45º (interior angles) x = 360º - 53. Trigonometry & Calculus - powered by WebMath. Use black ink or ball-point pen. Problem 1: A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: 30°, 45°, and 60°. Bartholomaeus Pitiscus is best known for his book called Trigonometria which was first published in Heidelburg in 1595. This Trigonometry Handbook was developed primarily through work with a number of High School and College Trigonometry classes. The basic unit vectors are i = (1, 0) and j = (0, 1) which are of length 1 and have directions along the positive x-axis and y-axis respectively. It is more convenient to write such expressions using one single term. Algebra 2 Trigonometry Work Answer Key. Recall that sr T (angle in radians) and 1 2 2 ArT. It is alos used to find the lengths of such components of a. He produced a trigonometric table using sine. To solve problems involving trigonometry in 3D shapes students are taught to identify 2D right-angled triangles and apply the Sine, Cosine and Tangent ratios. Questions are organized in Practice Tests, which draw from various topics taught in Trigonometry; questions are also organized by concept. Following is a more detailed description of the sections within each chapter. Spherical trigonometry is the study of curved triangles, triangles drawn on the surface of a sphere. A right triangle is a triangle in which one angle is a right angle. The endeavor has been to make the student feel that he is in actual touch with real things. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. The way I've defined it so far, this will only work in right triangles. Definitely, it will be the latter but at an affordable price. Unit 7 - Trigonometric Functions Trigonometry is essentially the study of how lengths vary compared to the rotations or angles that create the length. Trigonometry Academic Credit. This trigonometry textbook is different than other trigonometry books in that it is free to download, and the reader is expected to do more than read the book and is expected to study the material in the book by working out examples rather than just reading about them. The following guide is all about how to memorize the definitions of the trig functions in the easiest way possible. Forum posts are owned by the individual posters. Find the lengths of all sides of the right triangle below if its area is 400. Problem : A student stands on the ground at point, which is 10 m away from the foot of a pole. All of the above applies, including solving for multiple triangles. For example, trigonometric functions can be used to work out the heights of mountains, the quantity of water in a lake, or the square footage of a piece of land that is an unusual shape. Invention of a science like trigonometry was niether a one man's job nor it was one day job. Alternating current is described by sine waves, so an understanding of trigonometric functions is indispensable. Questions 1 and 2 require Sine, questions 3 and 4 require Cosine, question 5 and 6 require Tangent. Homework helpers trigonometry with who can help write reaserch. Written in stone: the world's first trigonometry revealed in an ancient Babylonian tablet August 24, 2017 3. Menu Geometry / Right triangles and trigonometry / The converse of the Pythagorean theorem and special triangles If we know the sides of a triangle - we can always use the Pythagorean Theorem backwards in order to determine if we have a right triangle, this is called the converse of the Pythagorean Theorem. Each unit introduces the topic, provides space for practice, but more importantly, provides opportunities for students to reflect on the work in order to deepen their. This Teacher Resource Guide has been developed to provide supporting materials to help educators successfully implement the Indiana Academic Standards for Trigonometry. All you do is cover what you want to calculate and what's left is the formula to calculate it. So in high school math, and probably college, we are taught how to use trig functions, what they do, and what kinds of problems they solve. A standard vector is a vector in standard position, which means a vector with initial point at the origin in the Cartesian coordinate system. It is strongly recommended that students complete Math 1050 (College Algebra) before taking Math 1060. Watch the best videos and ask and answer questions in 40 topics and 6 chapters in Trigonometry. “It is a fascinating mathematical work that demonstrates undoubted genius. These involve ratios of the lengths of the sides in a right triangle. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective. Trigonometry. This is done in order to maintain your confidentiality, and so that you may purchase with piece of mind. The NRICH Project aims to enrich the mathematical experiences of all learners. This right triangle calculator helps you to calculate angle and sides of a triangle with the other known values. Find x the length of BC. Trigonometry. Includes practice exercises and answers. Thus, this unit begins with careful development of essential angle and rotation terminology, including reference angles, quadrant work, and radian angle measurement. Procedure: Step 1. However, some of the classic works by great mathematicians of Greece and Rome (and, of course, the Indian and Chinese ones) have endured over two millennia. trigonometry definition: Trigonometry is defined as the branch of math that deals with calculations related to the sides and angles of triangles. Finding the Height of an Object Using Trigonometry, Example 2 Finding the Height of an Object Using Trigonometry, Example 3 Trigonometry Word Problem, Finding The Height of a Building, Example 1. This course examines trigonometric concepts in terms of the Cartesian coordinate plane and the rectangular and polar coordinate systems. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own. Trigonometry is where you apply your knowledge of triangles from geometry and use the resulting formulas to help you solve problems. Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! Right-Angled Triangle. The following problems require the use of these six basic trigonometry derivatives :. Differential equations have a remarkable ability to predict the world around us. It is possible that some chapter exercises will appear on the test that are not on the pre-test. 1 The trigonometric circle Take an x-axis and an y-axis (orthonormal) and let O be the origin. What the CSS — Implementing Trigonometry in SCSS, and lessons learnt CSS, at its core, is a style sheet language. This model is a working model and you should read this if you are searching for a quick maths model to make. The versatility that this calculator brings makes it perfect for everything from college work, homework, testing and answer verification. You can also search a book's contents by keyword. A discussion of international electrical quality standards and why basic. Electrical Engineers. See more ideas about Trigonometry, Teaching math and Precalculus. The word “Trigonometry” comes from the Greek words: ‘ Trigonon ’ meaning ‘triangle’ and ‘ metron ’ meaning a ‘measure’. They may seem complicated at first glance, but most of them boil down to a few simple concepts. Guild Of Light - Tranquility Music. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions. Trigonometry involves calculating angles and sides in triangles. Building a Town and Building Silos are practical based activities in which students realise a three dimensional structure based upon two dimensional planning. Unit 7 – Trigonometric Functions Trigonometry is essentially the study of how lengths vary compared to the rotations or angles that create the length. 4 of the Elements is the angle-side-angle congruence theorem which states that a triangle is determined by any two angles and. Materials required for examination Items included with question papers. Real life applications of trigonometry Trigonometry simply means calculations with triangles (that’s where the tri comes from). Trigonometry is the mathematical basis for describing rotations and periodic phenomena of all kinds, which are found in astronomy, mechanical/electrical engineering, music, biology, and much more. Lesson plans for math teachers. Trigonometry Review with the Unit Circle: All the trig. Students explore the concept of similar right triangles and how they apply to trigonometric ratios. The three fundamental trig functions are sine, cosine, and tangent. Pythagoras is used when you have a right angled triangle and you need to work out one of the missing side lengths. Use this lesson as a refresher of what trig ratios are and how they work. Tangent and cotangent are similarly related. He said, "Well, what angle is the ramp?" The first thing that popped to my mind was, "Holy crap, that's basic trig. Students will be able to apply principles of angles, fundamental identities and identity verifications of trigonometry, and solving and graphing trigonometric functions. Problem 1: A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Ancient societies used trigonometry as a navigation method in sailing. Your email address will not be published. The field was first studied in the third century B. The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. PreCalculus, Trigonometry: Proving Trig Identities Cut & Paste ActivityEnjoy a small sample of a cut & paste activity for your PreCalculus or Trigonometry students. 10 Real Life Application in Trigonometry. ALGEBRA 2/TRIGONOMETRY The possession or use of any communications device is strictly prohibited when taking this examination. in a right-angled triangle. A right angle has a value of 90 degrees ($90^\circ$). Trying on Trigonometry At first glance, it may not seem relevant to study the relationships between the sides and angles of right triangles, but trigonometry is essential to electrical engineering. Sine and Cosine Rules. Help With Your Math Homework. Indeed, what may be most surprising is that only. Graphs of Trigonometric Functions, which are really helpful for understanding what is going on in trigonometry. 1) 6 cm 8 cm 87° 24 cm² 2) 5 in 6 in 140° 9. Of course they can always be used if wished. We can find an unknown angle in a right-angled triangle, as long as we know the lengths of two of its sides. We give a few below. Tracing paper may be used. PQR is a right-angled triangle. , using dynamic geometry software, concrete materials), the properties of similar triangles (e. It is a useful tool for engineers, scientists, and surveyors and is applied even in seismology and navigation. CHAPTER 7: SIMILAR TRIANGLES AND TRIGONOMETRY Specific Expectations Addressed in the Chapter • Verify, through investigation (e. The sheets present concepts in the order they are taught and give examples of their use. How to memorize the values of trigonometric functions. Chapter 1 – The Trigonometric Functions Section 1. In the world of calculus, pre-calculus, and trigonometry, trigonometric ratios are a powerful and commonly used tool. This section looks at Sin, Cos and Tan within the field of trigonometry. Arnold, 1906. Use the fundamental, quotient, Pythagorean, co-function, and even/odd identities to verify trigonometric identities. The direction to a point is stated as the number of degrees east or west of north or south. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective. The teacher teaches that the tangent or the tan of an angle is the ratio of the opposite side to the adjacent side of a right angled triangle. Work through the entire course from start to finish and gain a firm understanding of the fundamentals. We were once students just like you. Cengage Unlimited from $119. A summary of Vectors in 's Trigonometry: Angles. Integrals Involving the Inverse Sine Function. Trigonometry is the branch of mathematics that deals with the study of: angles triangles (tri means three, tri-angles means$3$angles) lateral and angular relationships of planar and three-dimensional figures. Trigonometric Function Solver and Grapher v1. The Babylonians discovered a strange form of trigonometry The Middle Eastern civilization created a trig table 1,000 years before the Greeks. Trigonometry, as the name might suggest, is all about triangles. It is alos used to find the lengths of such components of a. Trigonometry According to Ptolemy. Early astronomers noted fixed relationships between the sides and angles of right triangles. Sine, cosine and tangent are the principle trigonometric functions. Forum posts are owned by the individual posters. Tips for teaching maths skills to our future chemists, by Paul Yates of Keele University Given that chemistry takes place in three dimensions, it is of no surprise that trigonometry plays an important role in a mathematical description of the subject. You must attribute the work in the manner speci ed by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Includes score reports and progress tracking. 100–170 ce). In calculus, sin −1 x, tan −1 x, and cos −1 x are the most important inverse trigonometric functions. SOH stands for Sine equals Opposite over Hypotenuse. Ricci left for China in 1581 and arrived in Macao in 1582. While tangent will find the ratio of the two sides of a right triangle when given an angle, arctangent can find the angle given the ratio. Trigonometry can be used to measure the height of a building or mountains: if you know the distance from where you observe the building and the angle of elevation you can easily find the height of the building. Ontario Tech University is the brand name used to refer to the University of Ontario Institute of Technology. Your background You should already be familiar with algebra and geometry before learning trigonometry. We have our function already solved for y, and we need to just plug in x=2 to evaluate the function at that point. Special Right Triangles and Right Triangle Trigonometry Reporting Category Triangles Topic Investigating special right triangles and right triangle trigonometry Primary SOL G. Trigonometry, Solving a Triangle I know enough from High School and some College that if you know certain elements such as length of two sides and the angle between them, then you can determine the two other angles and the length that closes the triangle. Use the sum and difference, double angle, half angle formulas to evaluate the exact values of trigonometric expressions. Commons, that the work may be used for free by any party so long as attribution is given to the author(s), that the work and its derivatives are used in the spirit of "share and share alike," and that no party other than the author(s) may sell this work or any of its derivatives for proﬁt. Visit Cosmeo for explanations and help with your homework problems! Home. COURSE OVERVIEW: The objective of this course is to enable the student to: a. It's a terribly unmotivated subject, and as a student you are expected to memorize double angle formulas with no proofs. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. The ACT also has significantly more questions dealing with geometry and trigonometry than the SAT does. It's a great gift for students and teachers. Brahe (second half of the 16th century), F. (8716 views) Practical Trigonometry by H. Because of this fact, there are two special right triangles which are useful to us as we begin our study of Trigonometry. Show all work and / or support your answer. I was an English teacher in Chicago Public Schools with certification in special education, and when my school was facing a shortage of certified special education teachers, I was pulled in mid-year to co-teach a junior-level trigonometry class with the math teacher. Learn exactly what happened in this chapter, scene, or section of Trigonometry: Angles and what it means. 'right triangle' trigonometry. Trigonometry manages to use everything to the utmost level and its benefits lack nowhere to convince peers to keep trigonometry at all levels. You can also search a book's contents by keyword. Give angles in degrees and minutes. That is the method to use when solving an isosceles right triangle or a 30°-60°-90° triangle. To sum up, knowledge of trigonometry can be used to create the unique projects and give things a glitter to shine. Trigonometry involves calculating angles and sides in triangles. ,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Labelling the sides. We also offer noncredit training, ESL, GED and other services. Get to know some special rules for angles and various other important functions, definitions, and translations. Trigonometry - Trigonometry - India and the Islamic world: The next major contribution to trigonometry came from India. Every angle in a triangle is adjacent to two sides, but in trig ratios opposite and adjacent always refer to LEGS of the right triangle, and never the hypotenuse. A right triangle is a triangle in which one angle is a right angle. Trigonometry is the branch of mathematics that deals with the study of: angles triangles (tri means three, tri-angles means$3\$ angles) lateral and angular relationships of planar and three-dimensional figures. The Lesson: y = sin(x) and y = cos(x) are periodic functions because all possible y values repeat in the same sequence over a given set of x values. Definition of Trigonometry. If we wish to find the resultant of two vectors, we can use either the law of cosines, or alternatively can add corresponding components to find the components of the resultant. Fuck trigonometry. As we can see in the picture on the left, the ancient Egyptians not only tried to work with right triangles in building the pyramids but also in. as a way of applying geometry to astronomy. Lessons, Essential Questions, Focus, Get Started, Activities, Explore, Bringing Ideas Together and Lesson Summary. In this hub you will be shown how to choose the correct one. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy. The trig functions, equations, and definitions you will learn will most likely come up again so don’t miss the SINE or go off on a Tangent in your studies. Although the educational system presents numerous opportunities for students to enjoy developing new skills,. I have spent hours researching this and I have found that trigonometry has affected astronomy as well as many other professions but once I read over the information it does not ever tell me how. Prepare the activity sheet by cutting up the sheets into squares, and ideally laminating them. You will see a line that says “Radian” and “Degree”. But always remember the basics of surveying and do not rush the activity because you might repeat the field work if you didn’t get the correct data because of rushing. Suppose an object in a plane (the floor,. He observes the top of the pole at an angle of 60 °. Fortunately, there is another model, the unit circle (shown below) used to visualize and study the trig functions. In order to do this you will need to know two of. Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! Right-Angled Triangle. These triangles are named by the measures of their angles, and are known as 45o-45o-90o triangles and 30o-60o-90o triangles. Angle Measure Angles can be measured in 2 ways, in degrees or in radians. You might wonder, "Why do we care? Does this ever show up in real life?" The answer is most definitely yes. The Unit Circle is probably the most important tool you’ll use in both Pre-Calculus, and then later (occasionally) in Calculus. See more ideas about Trigonometry, Teaching math and Precalculus. Trigonometry is first introduced at about 150BC by a Hellenistic mathematician, Hipparchus. athematics Year 9 Satisfactory Work sample 1 2014 dition Page 3 of 40 Measurement: Trigonometry Year 9 Mathematics achievement standard The parts of the achievement standard targeted in the assessment task are highlighted. A unit overview outlining the five lesson packs and assessment pack for Trigonometry. algebra trigonometry When you algebra trigonometry do these, leave some lines that you will come and fill in with details later. Ricci left for China in 1581 and arrived in Macao in 1582. Though the ancient Greeks, such as Hipparchus. The three fundamental trig functions are sine, cosine, and tangent. Use this lesson as a refresher of what trig ratios are and how they work. We can find an unknown angle in a right-angled triangle, as long as we know the lengths of two of its sides. Mathematics From Outdoors and Outdoor Trigonometry continue the theme exploring more complex mathematical topics such as Pythagoras' theorem, trigonometry and circles. Find $$\cos\alpha$$, $$\tan\alpha$$, $$\cot\alpha$$, if $$\sin\alpha = {\frac{5}{13}}$$ and $${\frac{\pi}{2}} \alpha \pi$$. Procedure: Step 1. The Trigonometry And Pre-Calculus Tutor is a 5 hour DVD course geared to fully prepare a student to enter university level Calculus. You may not want to discuss power with your students in great detail if they are just beginning to study voltage and current in AC circuits, because power is a sufficiently confusing subject on its own. Measure the angle of the point of observation from the ground say h= 1m. The triangle of most interest is the right-angled triangle. Solve the right triangle below. I used to work in telecommunications, and used trigonometry to calculate the height of trees. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy. Cosine theta equals x, sine theta equals y and tangent theta equals y over x, where x and y are the coordinates of the point on the terminal side of the angle. This, in turn, shows that the Babylonians were the first to study trigonometry and not the ancient Greeks, and that they proved the Pythagorean theorem a thousand years before the famed Greek mathematician Pythagoras was even born. Fortunately, I’ve found your company. Special Right Triangles and Right Triangle Trigonometry Reporting Category Triangles Topic Investigating special right triangles and right triangle trigonometry Primary SOL G. Trigonometry For this unit there are 3 level of worksheets. Apply Meet our faculty We have student-run publications, writing contests and more to spark your creativity. Which problem would you like to work on?. Chilkoot pass: Stark, the winter of 2004 1925, in cooperation with other teachers and site administrators must respond to a bar mid morning. The relation between the sides and angles of a right triangle is the basis for trigonometry. 1 The trigonometric circle Take an x-axis and an y-axis (orthonormal) and let O be the origin. Trigonometry is one of the most useful topics in mathematics, and these thorough, detailed worksheets will give students a solid foundation in it. Physicists. The Plus teacher packages are designed to give teachers (and students) easy access to Plus content on a particular subject area. At some point (I suspect from a previous private tutor. Mansfield added that the mathematics on the tablet are more advanced than even modern trigonometry. You need to enable JavaScript in your browser to work in this site. Trigonometry & Calculus - powered by WebMath. So if you're trying to find the trig functions of angles that aren't part of right triangles, we're going to see that we're going to have to construct right triangles, but let's just focus on the right triangles for now. I've only just started learning how to use this program and I haven't been able to find anything so far that makes sense to me. Get this from a library! Mathematics at work : practical applications of arithmetic, algebra, geometry, trigonometry, and logarithms to the step-by-step solutions of mechanical problems, with formulas commonly used in engineering practice and a concise review of basic mathematical principles. Influential works from the 4 t h 4^{th} 4 t h & the 5 t h 5^{th} 5 t h century AD, known as Siddhantas first defined the sine as the relationship between half an angle & half a chord, while also defining the cosine. The Unit Circle is basically a visual representation of certain “ special angles ”, for which the exact values of the trig functions are known. NWACC Dept of Mathematics Dept Final Exam Review for Trig - Part 1 Trigonometry, 10th Edition; Lial, Hornsby, Schneider Spring 2013 3 23. The hypotenuse is the longest side. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. Labelling the sides. Trigonometry According to Ptolemy. The following sheets list the key concepts which are taught in the specified math course. They are equal to the radian measures of the angles and , respectively. Trigonometry; Trigonometry flashcards, diagrams and study guides. Trigonometry, the branch of mathematics that describes the relationship between the angles and lengths of triangles, helped early explorers plot the stars and navigate the seas. The Political Corner. Easily review trigonometry as you work at a computer! Note: in many places, I will mention algebraic identities. We can find an unknown angle in a right-angled triangle, as long as we know the lengths of two of its sides. In our final trigonometry example, we will use Excel to examine the trig identity Notice in the screen shot below that this identity holds true when q is given in radians and degrees. The subject originated in the Islamic Caliphates of the Middle East, North Africa and Spain during the 8th to 14th centuries. ARCHITECT WHO USES TRIGONOMETRY? 1Wednesday, March 26, 14 2. Students in the Principles course also solve problems involving acute triangles. Our expert trigonometry equation solver is a professional who has years of experience in this field. In the third set, students will work with right triangles in general. Use the fundamental, quotient, Pythagorean, co-function, and even/odd identities to verify trigonometric identities. In today's post I focus on ideas and resources for teaching trigonometry. Problem : A student stands on the ground at point, which is 10 m away from the foot of a pole. “It is a fascinating mathematical work that demonstrates undoubted genius. a Triangle. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own. It grew, it developed,it evolved over time. Secant is the reciprocal of cosine. Unit 7 – Trigonometric Functions Trigonometry is essentially the study of how lengths vary compared to the rotations or angles that create the length. 528Hz Tranquility Music For Self Healing & Mindfulness Love Yourself - Light Music For The Soul - Duration: 3:00:06. A keen aptitude for math improves critical thinking and promotes problem-solving abilities. Sinusoidal Waves as Sound We've studied the graph of in some detail in class. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. Let's work here on our understanding of "invention". A way of remembering how to compute the sine, cosine, and tangent of an angle. Mechanical Engineers. I am currently enrolled in a Trigonometry class. Trigonometry is where you apply your knowledge of triangles from geometry and use the resulting formulas to help you solve problems. How to calculate an apparent dip from a real dip (and viceversa) using orthographic projection and trigonometry by Jorge Any student of geology in any university in the world learn during its degree the relationship between the real dip and the infinite apparent dips that a plane contains. Trigonometry is a mathematical method used to define relations between elements of a triangle. Guild Of Light - Tranquility Music. COMING SOON - Inverse functions. My 4th grader scored in the 98th percentile and my 7th grader in the 86th percentile on the national assessment tests. For instance: cos2(t) = (cos(t))2 sin12(t) = (sin(t)) 1 2 = p sin(t) So, for instance, when nding the derivative for something like the. Trigonometry specifically. The side opposite the right angle is called the hypotenuse (side $c$ in the figure). For a familiarization with SOHCAHTOA or help with more advanced equations, contact an on-demand Trigonometry tutor from Chegg Tutors. Fortunately, I’ve found your company. Trigonometric functions calculator. It makes it homework help trigonometry impossible for other people to find out that you used our essay writer service. You might be surprised to find that the heart of Trigonometry lies in Geometry.
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### Tip 60: Package your code with Autotools

level: your code is good enough to share

Part of a series of tips on POSIX and C. Start from the tip intro page, or get 21st Century C, the book based on this series.

The Autotools are what make it possible for you to download a library or program, and run

./configure
make
sudo make install

(and nothing else) to set it up. Please recognize what a miracle of modern science this is: the developer has no idea what the name of your compiler is, what sort of computer you have, where you keep your programs and libraries (/usr/bin? /sw? /cygdrive/c/bin?), and whatever quirks your machine demonstrates, and yet configure sorted everything out so that make could run seamlessly. And so, Autotools is the center of how anything gets distributd in the modern day. If you want anybody who is not your personal friend to use your code (or if you want a Linux distro to include your program in their package manager), then you need to have Autotools generate the build for you.

You will quickly realize how complicated the Autotools can get, but the basics are darn simple. By the end of this, we will have written six lines of packaging text and run four commands, and will have a complete (but rudimentary) package ready for distribution.

Here's how I imagine it all happening. The actual history is different from the sequence here. These are distinct packages, and there is a reason to run any of them without the other. But all that is irrelevant, and the purpose of this little dialogue is to help you think of the several tools as a unified whole working toward a unified goal.

P1: I love make. It's so nice that I can write down all the little steps to building my project in one place.

P2: Yes, automation is great. Everything should be automated, all the time.

P1: Yeah, I started adding lots of steps to my makefile, so users can type make to just produce the program, make install to install, make check to run tests, and so on. It's a lot of work to write all those makefile targets, but so smooth when it's all assembled.

P2: OK, I shall write a system--it will be called Automake--that will automatically generate makefiles from a very short pre-Makefile.

P1: That's great. Producing shared libraries is especially annoying, because every system has a different procedure.

P2: It is annoying. Given the system information, I shall write a program for generating the scripts needed to produce shared libraries from source code, and then put those into Automade makefiles.

P1: Wow, so all I have to do is tell you my operating system, and whether my compiler is named cc or clang or gcc or whatever, and you'll drop in the right code for the system I'm on?

P2: That's too much work. I will write a system called Autoconf that will be aware of every system out there and that will produce a report of everything Automake and your program needs to know about the system. Then Automake can use the list of environment variables in my report to produce a makefile.

P1: I am flabbergasted--you've automated the process of autogenerating Makefiles. But it sounds like we've just changed the work I have to do from inspecting the various platforms to writing configuration files for Autoconf and makefile templates for Automake.

P2: You're right. I shall write a tool, Autoscan, that will scan the Makefile.am you wrote for Automake, and autogenerate Autoconf's configure.ac for you.

P1: Now all you have to do is autogenerate Makefile.am.

P2: Yeah, whatever. RTFM and do it yourself.1

Each step in the story adds a little more automation to the step that came before it: Automake uses a simple script to generate makefiles (which already go pretty far in automating compilation over manual command-typing); Autoconf tests the environment and uses that information to run Automake; Autoscan checks your code for what you need to write to make Autoconf run. Libtool works in the background to assist Automake.

If you are doing something reasonably common (and compiling straight-up C code is the #1 most common task for the Autotools), then the system will do the right thing without your needing to go off the beaten path. The trouble shows up when you need to modify the defaults, at which point you're going to need to know what macro-generating macros to modify where.

Here's a demo, in which we get Autotools to take care of Hello, World. As per the count below, the script writes nine lines of text, and it produces a full package ready for distribution to the world.

This is a shell script you can copy/paste onto your command line (as long as you make sure there are no spaces after the backslashes). Of course, it won't run until you ask your package manager to install Autotools, Autoconf, Automake, and Libtool.

• The first few lines create a directory and write hello.c to it.
• Then we need to hand-write Makefile.in, which is two lines long, and four files that are required by the GNU coding standards (so GNU Autotools won't proceed without them).
• Getting Autotools up and running after this takes us three steps:
• Run autoscan, which produces configure.scan.
• Edit the file to give the specs of your project (name, version, contact email), and add the line AM_INIT_AUTOMAKE to initialize Automake. Yes, this is annoying, especially given that Autoscan used Automake's Makefile.in to gather info, so it is well aware that we want to use Automake. You could do this by hand; I used sed to directly stream the corrected version to configure.ac.
• Run autoreconf to use configure.ac to generate the files to ship out.

    #Make a directory; write a hello world program to it.
mkdir -p autodemo
cd autodemo
cat > hello.c <<\
--------------
#include <stdio.h>

int main(){ printf("Hi.\n"); }
--------------

#Autoscan needs a Makefile.am.
cat > Makefile.am <<\
--------------
bin_PROGRAMS=hello
hello_SOURCES=hello.c
--------------

#GNU coding standards require these; a human has to write them:
echo 'No news' > NEWS
echo 'Just run it.' > README
echo 'Kernighan & Ritchie' > AUTHORS
echo 'None yet' > ChangeLog

#Autoscan creates configure.scan, then we edit a few
#things to make configure.ac.
autoscan
sed -e 's/FULL-PACKAGE-NAME/hello/' \
-e 's/VERSION/1/'   \
-e '10iAM_INIT_AUTOMAKE' \
<configure.scan > configure.ac

#Given configure.ac, run autoreconf to produce everything.
autoreconf -i -vv

So how much do all these macros do? The hello.c program itself is a leisurely three lines, Makefile.am is two lines, and we wrote four one-line files, for nine lines of user-written text. Your results may differ a little, but when I run wc -l * in the post-script directory, I find 8,920 lines of text, including a 4,700-line configure script. It's so bloated because it's so portable: thie script doesn't depend on Autotools, and can be run on any system with basic POSIX-compliance.

Run ./configure at the command prompt, and now you have the 600-line Makefile. Thanks to zsh's autocomplete, I can tell you there are 216 targets in this makefile. The default target, when you just type make on the command line, produces the executable, and sudo make install would install this program if you are so interested; run sudo make uninstall to clear it out.

As the author of the package, you will also be interested in make dist, which generates a tar file with everything a user would need to unpack and run the usual ./configure; make; sudo make install (without the aid of the Autotools system that you have on your development box). Also, make check verifies the tar file; don't forget that you can list multiple targets to make, like make dist check to produce the output package and then check it.

So, having run autoscan and autoreconf, two more commands and we've got a distributable package in one tar file:

./configure
make dist check

In the next few episodes, some notes on reading and producing the Makefile.am and config.ac files for when you need more than the defaults.

#### Footnotes

... yourself.1
RTFM is an acronym meaning Read The Manual.


text:
# scipy.interpolate.RectSphereBivariateSpline¶

class scipy.interpolate.RectSphereBivariateSpline(u, v, r, s=0.0, pole_continuity=False, pole_values=None, pole_exact=False, pole_flat=False)[source]

Bivariate spline approximation over a rectangular mesh on a sphere.

Can be used for smoothing data.

New in version 0.11.0.

Parameters
uarray_like

1-D array of colatitude coordinates in strictly ascending order. Coordinates must be given in radians and lie within the interval [0, pi].

varray_like

1-D array of longitude coordinates in strictly ascending order. Coordinates must be given in radians. First element (v[0]) must lie within the interval [-pi, pi). Last element (v[-1]) must satisfy v[-1] <= v[0] + 2*pi.

rarray_like

2-D array of data with shape (u.size, v.size).

sfloat, optional

Positive smoothing factor defined for estimation condition (s=0 is for interpolation).

pole_continuitybool or (bool, bool), optional

Order of continuity at the poles u=0 (pole_continuity[0]) and u=pi (pole_continuity[1]). The order of continuity at the pole will be 1 or 0 when this is True or False, respectively. Defaults to False.

pole_valuesfloat or (float, float), optional

Data values at the poles u=0 and u=pi. Either the whole parameter or each individual element can be None. Defaults to None.

pole_exactbool or (bool, bool), optional

Data value exactness at the poles u=0 and u=pi. If True, the value is considered to be the right function value, and it will be fitted exactly. If False, the value will be considered to be a data value just like the other data values. Defaults to False.

pole_flatbool or (bool, bool), optional

For the poles at u=0 and u=pi, specify whether or not the approximation has vanishing derivatives. Defaults to False.

BivariateSpline

a base class for bivariate splines.

UnivariateSpline

a smooth univariate spline to fit a given set of data points.

SmoothBivariateSpline

a smoothing bivariate spline through the given points

LSQBivariateSpline

a bivariate spline using weighted least-squares fitting

SmoothSphereBivariateSpline

a smoothing bivariate spline in spherical coordinates

LSQSphereBivariateSpline

a bivariate spline in spherical coordinates using weighted least-squares fitting

RectBivariateSpline

a bivariate spline over a rectangular mesh.

bisplrep

a function to find a bivariate B-spline representation of a surface

bisplev

a function to evaluate a bivariate B-spline and its derivatives

Notes

Currently, only the smoothing spline approximation (iopt[0] = 0 and iopt[0] = 1 in the FITPACK routine) is supported. The exact least-squares spline approximation is not implemented yet.

When actually performing the interpolation, the requested v values must lie within the same length 2pi interval that the original v values were chosen from.

Examples

Suppose we have global data on a coarse grid

>>> lats = np.linspace(10, 170, 9) * np.pi / 180.
>>> lons = np.linspace(0, 350, 18) * np.pi / 180.
>>> data = np.dot(np.atleast_2d(90. - np.linspace(-80., 80., 18)).T,
...               np.atleast_2d(180. - np.abs(np.linspace(0., 350., 9)))).T

We want to interpolate it to a global one-degree grid

>>> new_lats = np.linspace(1, 180, 180) * np.pi / 180
>>> new_lons = np.linspace(1, 360, 360) * np.pi / 180
>>> new_lats, new_lons = np.meshgrid(new_lats, new_lons)

We need to set up the interpolator object

>>> from scipy.interpolate import RectSphereBivariateSpline
>>> lut = RectSphereBivariateSpline(lats, lons, data)

Finally we interpolate the data. The RectSphereBivariateSpline object only takes 1-D arrays as input, therefore we need to do some reshaping.

>>> data_interp = lut.ev(new_lats.ravel(),
...                      new_lons.ravel()).reshape((360, 180)).T

Looking at the original and the interpolated data, one can see that the interpolant reproduces the original data very well:

>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax1.imshow(data, interpolation='nearest')
>>> ax2.imshow(data_interp, interpolation='nearest')
>>> plt.show()

Choosing the optimal value of s can be a delicate task. Recommended values for s depend on the accuracy of the data values. If the user has an idea of the statistical errors on the data, she can also find a proper estimate for s. By assuming that, if she specifies the right s, the interpolator will use a spline f(u,v) which exactly reproduces the function underlying the data, she can evaluate sum((r(i,j)-s(u(i),v(j)))**2) to find a good estimate for this s. For example, if she knows that the statistical errors on her r(i,j)-values are not greater than 0.1, she may expect that a good s should have a value not larger than u.size * v.size * (0.1)**2.

If nothing is known about the statistical error in r(i,j), s must be determined by trial and error. The best is then to start with a very large value of s (to determine the least-squares polynomial and the corresponding upper bound fp0 for s) and then to progressively decrease the value of s (say by a factor 10 in the beginning, i.e. s = fp0 / 10, fp0 / 100, ... and more carefully as the approximation shows more detail) to obtain closer fits.

The interpolation results for different values of s give some insight into this process:

>>> fig2 = plt.figure()
>>> s = [3e9, 2e9, 1e9, 1e8]
>>> for ii in range(len(s)):
...     lut = RectSphereBivariateSpline(lats, lons, data, s=s[ii])
...     data_interp = lut.ev(new_lats.ravel(),
...                          new_lons.ravel()).reshape((360, 180)).T
...     ax = fig2.add_subplot(2, 2, ii+1)
...     ax.imshow(data_interp, interpolation='nearest')
...     ax.set_title("s = %g" % s[ii])
>>> plt.show()

Methods

 __call__(self, theta, phi[, dtheta, dphi, grid]) Evaluate the spline or its derivatives at given positions. ev(self, theta, phi[, dtheta, dphi]) Evaluate the spline at points get_coeffs(self) Return spline coefficients. get_knots(self) Return a tuple (tx,ty) where tx,ty contain knots positions of the spline with respect to x-, y-variable, respectively. get_residual(self) Return weighted sum of squared residuals of the spline approximation: sum ((w[i]*(z[i]-s(x[i],y[i])))**2,axis=0)

#### Previous topic

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#### Next topic

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