Tuesday, August 31, 2010

NXT Tribot

The tribot was build exactly (99.99%, tell you what's not exact later) follows the Lego documentation, I used it to experiment how things are detected using the ultrasonic sensor. While thinking for the project, I read Krystian Majewski article about a NXT radar, it inspire me to make the tribot scan its surroundings, look for an object and then go and grab it. Just want to share it here for all the fun I have in making it.

After I build the tribot, I have an idea of scanning the surroundings with the ultrasonic sensor and look for something, say, within 10 inch of distance. I first tried to do this by incrementing the B/C motors a degree a time and scan the distance, however, it seems that the data read back fluctuates a lot and it is difficult to determine when it "sees" an object. To get better understanding of the problem, I build a simple program in NXT-G to control the B/C motors to rotate slowly and scan the readings, save them to a file in the NXT and uploaded to my PC, plot them using Excel and the graph on the right is what I found.




Then I tried another approach, the B/C motors were controlled to rotate anitclockwise slowly, at the same time, scans the surroundings for objects with distance less than 10 inch. This is not accurate and it only serves to get a rough starting point, then, I use an iteration method to converge the result. The purpose is to find the point where the ultrasonic cone first intersects the object edge, then we will rotate the robot a known amount so that the robot faces the center of the object and grab it.


When the first point was found, the robot is stopped immediately. Depends on the speed of motors and other unknown reasons, the first point could be close to the edge of the object or already get inside the object a certain amount.










The robot is then rotated back (clockwise) a certain angle, check the ultrasonic again to see if it intersect the object. If intersection occurs, it means that the robot overrruns too much before detecting the object the first time. As a result, our next rotation should be also clockwise with the same angle as we have to find a point where the ultrasonic cone does not intersect the object.



If no intersection occurs, it means that the object edge is somewhere between this point and the point we first found it in our previous step. We shall then rotate anti-clockwise for half the amount and check again for intersection.



We can summarize the method as below:


1) Rotate anticlockwise to find the first point

2) Rotate clockwise for a certain degree and scan, if intersect, further rotate the same degree clockwise and scan again, until we found the point with no intersection.

3) If step 2 does not intersect, rotate it anti-clockwise for half the angle in step 2 and check ultrasonic distance. If no intersection, repeat again until we found intersection point. If intersection occurs, rotate it clockwise for half the angle and check for ultrasonic intersection.

4) Steps 2 and 3 are repeated according to the result of ultrasonic intersection with the object. It will eventually converge to a point where the robot motors no longer rotate.

The video below shows how it happens.



I take a small bottle as object for grabbing, the blue ball which comes with the nxt kit is too short, and ultrasonic sensor did not detect the ball well. I put a few coins inside the bottle to make it heavier, so that it will not fall down when the tribot collide with it. The bottle I use is a bit too tall for the purpose, and hit against the ultrasonic sensor when the tribot approach it, but I don't have a right size one at home.

A minor modification to the tribot is to remove the orange color part in the grab area (see picture below the right one), so that it grabs the bottle better.



After having all the fun with the iteration method, I summarize my discoveries here:


1) Ultrasonic sensing does not provide an accurate or sharp data on the border of the object, the accuracy is even worst when detecting distance while the robot is moving.

2) A flat smooth floor is can avoid getting extra noise (although they are always there) and confuse the sensor reading.

3) To run the robot at low speed, you have to use the motor block with motor power control selected. Otherwise, the motor may not move at all. Motor power control is only available on the motor block, but not the move block.

4) The program written detects only one of the edge of the object, and hardcode an angle to further rotate the motors so that it aligns with the center of the object. It is better to memorize the angle of the first edge found, and detect the second edge, then find the mid point of the object. It is difficult to do it now as using the NXT-G a more complicated program runs more slowly on my computer, also, without a compass sensor, just by memorizing the rotations of the motor cannot control it accurately (e.g. you record a +90 deg rotation, then you run a -90 deg rotation it does not go back to the exact orientation).

The NXT-G program is provided here, welcome any comments for improvement.



Angle of intersection

Free algebra worksheets

Usually algebra textbooks provide lots of problems to practice the algebraic concepts and techniques, but some of you may still benefit from resources for free (or mostly so) printable algebra worksheets. Please see the list below, which I've originally compiled for my HomeschoolMath.net site.

Algebra worksheets

Worksheet Builder
Great and free worksheet maker software with nearly 7,000 built-in algebra and geometry questions.
www.jmap.org/JMAP_WORKSHEET_BUILDER_INSTALLATION_FILES.htm

Free Algebra Worksheets from KUTA Software
Free worksheets (PDF) for equations, exponents, inequalities, polynomials, radical & rational expressions and more.
www.kutasoftware.com/free.html

AlgebraHelp.com worksheets
Interactive worksheets that are checked online for most algebra 1 topics.
www.algebrahelp.com/worksheets/

Math.Com algebra worksheets generator
Generate worksheets for: linear equations, systems of equations, and quadratic equations.
www.math.com/students/worksheet/algebra_sp.htm

LessonCorner worksheets
These free worksheets include a few topics such as calculations with polynomials, factoring, and graphing lines.
www.lessoncorner.com/worksheets/

Algebra Fun Sheets
Worksheets that integrate algebra skills with fun activities including sudoku, word finds, riddles, color patterns, crosswords, games, matching cards, etc. A subscription is required.
www.algebrafunsheets.com

About.com Algebra Worksheets
An assorted collection of free algebra worksheets and answers. These pages are not very well organized, but they have lots of worksheets.
math.about.com/od/algebraworksheets/Algebra_Worksheets.htm

Algebra Worksheets from MathWorksheetCenter
Lots of worksheets for over 100 algebra topics. A few are free; most are accessible only by one-year a subscription.
www.mathworksheetscenter.com/mathskills/algebra/

A few fun algebra worksheets
These are for graphing linear equations and linear inequalities.

Online Math Work
Free multiple-choice worksheets for pre-algebra and algebra 1 topics. You can do them online, or copy to a word processor to print.
www.mathonlinework.com




Lastly... my own algebra worksheet collections, which aren't free but there are many free samples:

Algebra 1-A worksheets cover Algebra 1-B worksheets cover Math Mammoth Algebra 1 Worksheets Collection
A two-part collection (A and B) of 137 quality algebra worksheets covering all the topics in a typical algebra 1 curriculum. These worksheets are hand-crafted and contain lots of word problems and other variable problems. Free samples available. $11.50.
www.mathmammoth.com/worksheets/algebra_1.php


Summer Math Program


algebra 1 homework help free

AP CALCULUS



Dd loved her Calculus learning experiences through a variety of sources. She has read plenty of Calculus books (some are listed below under Reading list) and worked on plenty of Calculus problems. She also liked the latest AOPS Calculus book very much, and there is a wonderfulcourse that goes with the book. She also enjoyed reading Life of Fred Calculus book. I am listing a few resources for AP Calculus preparation.

AP Calculus consists of two exams-AP Calculus AB and AP Calculus BC. There are various online and distance courses available to learn Calculus, some of which I have listed below.CollegeBoard has in depth information on the syllabus, learning materials and AP exam information. Read the AP Calculus AB and AP Calculus BC course descriptions to get an idea as to what is involved in doing well on the AP exams. Check out the past AP Calculus AB and AP Calculus BC Free Response(FR) questions.

  • Websites, tutorials, lecture notes, cheat sheets and more
Calculus.Org is a wonderful in depth website that has links and resources on AP and General Calculus.

Ask Mr. Calculus is a Calculus help page with past AP exam Free Response questions, interactive Calculus learning pages, Visual Calculus etc.

Visual Calculus nice site with great tutorials.

Karl's Calculus Tutor online tutorials and lessons.

Drexel University's Calculus resources.

Calculus Lecture Notes from the University of Toronto at Scarborough College. Includes many past exams.

Graphics for the Calculus classroom. Great visual explanations.

Calculus tutorials from The Connected Curriculum Project, Dept of Mathematics, Carroll College, Montana.

Calculus Help.com interactive Calculus help with some humor injected lessons.

Video Calculus from the University of T! ennessee Math Department at Knoxville. It is an excellent reso! urce tha t includes tutorials, drills and programs for pre-calculus and calculus.

Calculus Applets interactive applets for single variable calculus.

PowerPoint lectures of AP Calculus AB and BC.


Calculus on the web from Temple University.

MIT AP Calculus courses. Free opencourseware.


Calculus Cheat sheets from Paul's online Math notes.

Calculus course information links, resources on AP Calculus.


History of Calculus wonderful site on History of Calculus.


Math archives on Calculus.

Free Undergraduate Calculus courses from Opencoursware.

Free Online Calculus courses from William Smith.

Calculus Page Problems from eCalculus.org.

Ask Dr. Callahan Calculus 1 Teacher's Guide PDF.


  • Graphing Calculator Resources
Dh ! and I ne ver used a Graphing Calculator in India even though we had to learn much more advanced Calculus than the AP Calculus courses here. There were memorization and manipulations involved in solving advanced Calculus problems without having to resort to a calculator of any sorts. I am not sure if Graphing Calculators are used these days in India by all high school kids. We very much doubt that it would be practiced considering the prohibitive cost of these calculators to be able to afford by all Indian high school kids. We have exposed dd to both the version of learning with and without a graphing calculator, although the latter learning is not necessary to do so here in the US. She loves her Graphing Calculator although she enjoys working without them too.

Dd uses the above pictured Graphing Calculator for advance mathematics, mathematical programming, 3D graphing, along with numerous other tasks. There are numerous resources on the web regarding Graphing Calculators. I have listed a few below.

Check out the Education Technology page in Texas Instruments to learn all about Graph! ing Calc ulators



Graphing Calculator pages check out the various programs, tips and resources.

Calculus Calculator programs yet another site with numerous info.

Calculator pages on various calculators.

Graphing Calculators in Calculus see how it is used for Calculus.


  • Articles on Calculus learning.

Learning Calculus Prepared by: Susan Hermiller, Melanie Martin, Eric York.



How to succeed in AP Calculus from a high school site.

Read about The Calculus Trap by Richard Rusczyk, which talks about the best course of path for math learning for the very gifted student.

Why Do We Study Calculus? An interesting article on the importance of Calculus.

  • Teacher's sites




  • Calculus Course Choices
There are many online free Calculus courses as well as fee based ones. One can also take the Calculus course through the local community college or by hiring a tutor. Dd learned much of her Calculus through self learning by working through Spivak Calculus and Stewart Calculusbooks. She loves the AOPS Calculus book as well. Find the most appropriate course path that fits your child's needs and learning styles.

Art of Problem Solving ! (AOPS) Calculus Course My dd loves AOPS courses and books. This is great for a very mathy kid. It is an intense, challenging course that prepares a student for AP Calculus exam.

Math Archives Calculus courses online. A compiled list of varied Calculus courses available on the web.

Calculus courses from Hippocampus.






Aleks Math review for AP Calculus

Thinkwell Calculus Courses for both AB and BC.

EPGY AP Calculus courses.

Apex AP Calculus AB course.

Chalk Dust Calculus Courses.

Life of Fred Calculus book is a fun way to learn Calculus.

Change and Motion: Calculus Made Clear from the Teaching Company.

  • Calculus DVDs
Teaching Company Calculus Lectures in DVD and Audio formats.



Calculus is harder than Algebra

A few sources of confusion to be aware of.

Sometimes I see students who used to get good grades in their Algebra classes but who are now struggling with Calculus.
They tell me they are confused by Calculus; they do not know when to apply the Chain Rule; they get uncomfortable around dx; they have no clue how to start the problems, and so on and so forth.
What most amazes them is they know they had no problems with Algebra, they understood Algebra well; so the question in their mind is: “What is it about Calculus that makes it so difficult?”
One problem is their implicit, non-accurate expectations. Sometimes they think they are doing something wrong because they have not gotten to the solution yet, aft! er filling in one page with equations. However, they may very well be on the right track. They are not doing anything wrong. They just never expected for the solution to take so much work, and time. They were expecting to arrive to the solution after three or four steps, like they used to do in their Algebra work. Instead, with Calculus problems they may need to go through ten or twelve similar steps.
Sometimes they ask:
“Another integral? You mean, this is not the result yet?”
Apparently they cannot believe it. I tell them:
“Look, we have to keep going. We are not there yet but we are getting close. O.K.? Think of this as a little marathon. You were used to run the 400 meters, now we are going for three miles. We just have to keep going.”
Another feature that makes Calculus harder than Algebra is the huge number of options when it comes to selecting routes to the solution, many of which may turn to be dead-ends. Calculus inc! orporates all the operations of Algebra, including exponents, ! roots, a nd logarithms; plus all trigonometric functions; and it makes heavy use of function composition. All these ingredients can be thrown into a problem involving limits, or derivatives, or integrals. So Calculus really pushes you to get your Algebra up to speed. And because the expressions get more complex, there are more forks in the road at almost every step of the way. This is confusing for many students.
Now, in my opinion, the major source of discomfort when doing the transition from Algebra to Calculus, is that students who like Algebra like also order, and neatness. This proves to be a disadvantage when it comes to acquiring an intuition for Calculus.
They tell me:
“I do not know what to do, where to start, when to apply what rule, none of that! In Algebra everything was more clear, more precise.”
I ask them:
“And you liked that, didn’t you? Having a set procedure to follow in an orderly manner, right?”
They say:!
“Yes!”
Then I say something like:
“Well, I hate to break the news to you but when it comes to Calculus, you are out there in the wild, and everything is moving, all the time, even if it does not seem that way. You have to get used to it.”
Many times they ask:
“But, why? Why is it that way?”
My explanation goes more or less like this:
“Well, Calculus was invented to deal with problems of a very physical nature: motion, volume, pressure, speed, weight, and the like, but not only that. The specific aim of Calculus was to provide answers as to how those physical magnitudes behave when they are changing, either growing or going down, and being related to each other at the same time. Originally, everything in Calculus happens in time, nothing stays the same, everything is changing, moving, decreasing, growing, building up, fading away, speeding up, slowing down, just like in the real world. So when you ! think of a variable x or y or v in Calculus, it is not just a number that is there, with some fixed value, but a number that wants to go somewhere else, a number that has already started to change, even if only a little tiny bit”
Somehow this bizarre wording paints a picture that makes sense for them, in that it puts their confusion into the proper perspective, eliminating a big part of it.
One of my students, after hearing this type of explanation, said:
“Oh! So, dx is the sneakiness of change! That’s why I don’t like it! But now I know I have to deal with it. I cannot just pretend it’s not there.”
And I said:
“That is exactly right. That is exactly what it is.”
Then she said:
“And, we always have to apply the Chain Rul! e, we cannot get rid of it, because the variables are connected to each other.”
And I said:
“Yes, the Chain Rule is always there.”
It is a big help to be able to tell students at least some of the reasons why they are confused, because when they do not even know the source of their confusion, everything gets exponentially more confusing for them.

calculus tutoring

Geometry Homework




        
Teacher: Andreas
Class: Geometry
Hour(s): 3,5,8
Due Date:
Assignment:

The following assignments are the required assignments for Geometry this quarter so far...

Geometry Assignments Chapter 5 and 6


5.1-5.2 Re! view
Page 246: 14-19, 21-25, 26 a and c, 27, 28, 32
Page 252: 28-30, 41, 42, 48, 55, 57

6.1-6.4 Review
Page 307: 7-15
Page 310: 1-10
Page 339: 1-11

Chapter 6 Review
Page 342: 1-19, skip 4 and 7

Chapter 6 Review Worksheet
Given out in class.


        

online geometry homework

Geometry Homework

Drawing arcs is a common task in computer graphics. Arcs are typically drawn using an approximation by set of cubic Bezier splines. These splines are passed through the regular rendering pipeline where they are usually subdivided into a set of line segments and then rasterized. A circle, for example, can be represented by four cubic splines: one for each 90 degree arc.

An arc of up to 90 degrees can be approximated by a single cubic Bezier spline reasonably well. To find an appropriate approximating spline we set the start and end points of the ! spline to match the arc and then can find the middle two points using the following formula, where A is the start angle and B is the end angle:

h = 4/3 * tan ((B - A)/4)

pt1 = {center.x + r*cos(A), center.y + r*sin(A)}
pt2 = {center.x + r*cos(A) - h*r*sin(A), center.y + r*sin(A) + h*r*cos(A)}
pt3 = {center.x + r*cos(A) + h*r*sin(B), center.y + r*sin(A) - h*r*cos(B)}
pt4 = {center.x + r*cos(B), center.y + r*sin(B)}

This approach works quite well and is currently used by cairo. However, the problem with this approach is that it requires converting to polar coordinates and then back to Cartesian ones. This conversion is slow because it requires the use of trigonometric functions.

Fortunately, this paper gives a solution that avoids the conversion to polar coordinates:

ax = pt1.x - center.! x
ay = pt1.y - center.y
bx = pt4.x - center.x
b! y = pt4. y - center.y

q1 = ax*ax + ay*ay
q2 = q1 + ax*bx + ay*by

k2 = 4/3 (sqrt(2*q1*q2) - q2) / (ax*by - ay*bx)

pt2.x = pt1.x - k2*ay
pt2.y = pt1.y + k2*ax
pt3.x = pt4.x + k2*bx
pt3.y = pt4.y - k2*by

Unfortunately, there is no explanation, derivation or proof for this formula. Even more problematic, the formula for k2 becomes less stable as pt1 approaches pt4, becoming NaN when pt1 equals pt4.

Therefore, the homework problem has two parts:

1. Give an explanation or derivation for the formula for k2 provided above.
2. Provide a similar formulation for pt2 and pt3 that doesn't degenerate as pt1 and pt4 become close or prove that one doesn't exist.

The best answer will be credited in the new cairo stroking code.

geometry homework help free

Wednesday, August 25, 2010

The Heritage at Trump Place - 240 Riverside Boulevard, New York ...

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Mr. Trump sold off much of his stake in the mid-1990s to a group of .... of the 72nd Street subway station and to allocate 10 percent of the apartments for ...
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Costco - $10.89 after $3 OFF TruNature® Pure Wild Alaskan Salmon ...
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GAO-02-240 School Dropouts: Education Could Play a Stronger Role ...
File Format: PDF/Adobe Acrobat - Quick View
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by EC Play - 2002 - Related articles
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Best market for emerging designers The Market NYC - Time Out New York
Jul 19, 2010 ... Ad Space (120 x 240) ... TONY deal: Mention TONY for 10 percent off all purchases through July 31. 7. Eco-friendly tees you can live in ...
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Peter Greenberg: Larry King, Amtrak and Rule 240
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UFC 3-240-06A Wastewater Collection and Pumping
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UFC 3-240-06A. 16 January 2004. UNIFIED FACILITIES CRITERIA (UFC) ...... Depths of flow within 10 to 15 percent of critical are likely to be unstable and ...
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The Growth Of Physician Medical Malpractice Payments: Evidence ...
May 31, 2005 ... We found that growth of the top 10 percent of payments is smaller ... The growth of payments has leveled off in recent years: Between 2000 ...
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What Can You Do? | WaterSense | US EPA
... of water, while taking a five-minute shower uses 10 to 25 gallons. ... Solution: Turning off the tap while brushing your teeth in the morning and at bedtime can save up to 8 gallons of water per day, which equals 240 gallons a month! ... Some experts estimate that more than 50 percent of landscape water use ...
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