This document summarizes research on finding the n-diameter configuration of points on different shapes. It discusses finding the 3-diameter configuration on a line, ellipse, and square. For the ellipse, the points are conjectured to alternate sides. Simulation found the center of mass trajectory is a smaller ellipse and the points are almost symmetric. For a triangle in a square, the center of mass traces a small rectangle and the 3-diameter product is maximized at the rectangle's vertices. Future work includes expanding to general n-diameters and improving calculation speed.
The author has derived the formula to analytically compute all the important parameters of a disphenoid (isosceles tetrahedron with four congruent acute-triangular faces) such as volume, surface area, vertical height, radii of inscribed & circumscribed spheres, solid angle subtended at each vertex, coordinates of vertices, in-centre, circum-centre & centroid of a disphenoid for the optimal configuration in 3D space. The author has also proved the important conclusions related to a disphenoid by mathematical derivations using 3D coordinate geometry.
Mathematical Analysis of Spherical Rectangle (Application of HCR's Theory of ...Harish Chandra Rajpoot
All the articles have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the formula are very practical & simple to apply in case of any spherical rectangle to calculate all its important parameters such as solid angle, surface area covered, interior angles etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formula can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the center of the sphere such as normal height, angle between the consecutive lateral edges, area of plane rectangular base etc.
Volume & surface area of right circular cone cut by a plane parallel to its s...Harish Chandra Rajpoot
All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.
The author has derived the formula to analytically compute all the important parameters of a disphenoid (isosceles tetrahedron with four congruent acute-triangular faces) such as volume, surface area, vertical height, radii of inscribed & circumscribed spheres, solid angle subtended at each vertex, coordinates of vertices, in-centre, circum-centre & centroid of a disphenoid for the optimal configuration in 3D space. The author has also proved the important conclusions related to a disphenoid by mathematical derivations using 3D coordinate geometry.
Mathematical Analysis of Spherical Rectangle (Application of HCR's Theory of ...Harish Chandra Rajpoot
All the articles have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the formula are very practical & simple to apply in case of any spherical rectangle to calculate all its important parameters such as solid angle, surface area covered, interior angles etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formula can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the center of the sphere such as normal height, angle between the consecutive lateral edges, area of plane rectangular base etc.
Volume & surface area of right circular cone cut by a plane parallel to its s...Harish Chandra Rajpoot
All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.
HCR's Infinite or Convergence Series (Calculations of Volume, Surface Area of...Harish Chandra Rajpoot
Three series had been derived by the author, using double-integration in polar co-ordinates, binomial expansion and β & γ-functions for determining the volume, surface-area & perimeter of elliptical-section of oblique frustum of a right circular cone. All these three series are in form of discrete summation of infinite terms which converge into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case studies & practical computations.
Derivations of inscribed & circumscribed radii for three externally touching ...Harish Chandra Rajpoot
All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles. Here is also the derivation of a general formula for computing the length of common chord of two intersecting circles.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
An orthogonal system is one in which the coordinates arc mutually perpendicular
Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1
A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem
A hard problem in one coordi nate system may turn out to be easy in another system.
In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical.
Solid angle subtended by a rectangular plane at any point in the space Harish Chandra Rajpoot
This is the most general case for any location of given point in the space which is derived by using basic formula taken from the book "Advanced Geometry by H.C. Rajpoot". Its derivation & detailed explanation has been given in author's book of research articles of 3-D Geometry.
HCR's Infinite or Convergence Series (Calculations of Volume, Surface Area of...Harish Chandra Rajpoot
Three series had been derived by the author, using double-integration in polar co-ordinates, binomial expansion and β & γ-functions for determining the volume, surface-area & perimeter of elliptical-section of oblique frustum of a right circular cone. All these three series are in form of discrete summation of infinite terms which converge into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case studies & practical computations.
Derivations of inscribed & circumscribed radii for three externally touching ...Harish Chandra Rajpoot
All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles. Here is also the derivation of a general formula for computing the length of common chord of two intersecting circles.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
An orthogonal system is one in which the coordinates arc mutually perpendicular
Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1
A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem
A hard problem in one coordi nate system may turn out to be easy in another system.
In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical.
Solid angle subtended by a rectangular plane at any point in the space Harish Chandra Rajpoot
This is the most general case for any location of given point in the space which is derived by using basic formula taken from the book "Advanced Geometry by H.C. Rajpoot". Its derivation & detailed explanation has been given in author's book of research articles of 3-D Geometry.
She today gives all the scores of her hardcore creative potentiality to her God, Family, few old friends and three Gurus - Mr.Rakesh Shrestha, Mr.Sandeep Marwah (Managing Director, AAFT) & Mr.Arun Anand (Photography department, Head - AAFT).
Chandni has always been a photography globe trotter and carries different cultures in her vibrant personality.She speaks, "what i learned all the span of my work while capturing the most natural and whimsical images is that
i could feel the spiritual presence of the universal powers in everything i struggle to produce & the beautiful blessings being showered on my creativity."
As the use of the phrase "fair trade" has increased on food labels, fully committed fair traders are asking "What's Next?".
Our panel discusses this and asks pressing questions, such as:
- What do we expect of businesses that go beyond baseline standards to seek deep change? And, how do we communicate these expectations to each other and consumers?
- We will be asking fair traders and entrepreneurs who go beyond baseline fair trade what is the single best way to increase impact for farmers?
- As we work to increase impact, we also must support this work by creating a value-add for consumers. Does impact automatically translate into a value add? Why/why not? If not, how can we tip that balance?
- Among the proliferation of certifications, labels, and “feel good” terminology, how do brands communicate about impact in a meaningful way? What is their message?
- How do brands--like those part of OSC2--go beyond fair trade and how can they work together to strengthen impact?
OSC2 Compostable Packaging Coalition: an OverviewPluot Consulting
This group's purpose is to remove petroleum-based packaging from landfills and oceans. Click through to learn about our goals, how we're going about it, our members who are making it happen, and our early successes.
Museum Paper Rubric50 pointsRubric below is a chart form of .docxgilpinleeanna
Museum Paper Rubric
50 points
Rubric below is a chart form of the instructions on the Museum Paper assignment sheet.
Category
Description
Points
First Step
10 points
Selfie with image (5)
5
Evocative, detailed description and comparisons (5)
3
Second Step
6 points
3 scholarly sources (6)
6
Third Step
26 points
Clear statement with individual analysis (10)
7
Selective and objective detail that supports statement (10)
8
Writing: flow, readability (6)
6
Format
8 points
Length 3-4 pages (3)
3
Illustration/image included at end (2)
2
Chicago style end note citations (3)
3
Total
50 points
Comments
Good research about Shiva and iconography. Your particular thesis - your personal analysis - should be the core of the essay rather than a historical overview, and the research content would revolve around your thesis statement. For example, your descriptions include descriptions of iconography showing his power, so that could be a direction for your thesis statement. Shiva is a Hindu deity, so it is unlikely that this would be "Shiva Buddha" (since Buddha is part of Buddhism). To avoid generalizations such as the third eye and red mark (page 2 middle) , it would be stronger to explain what that third eye meant.
43
MATH 220 - CT Scan Project
(in class)
Directions This project is due Thursday April 26 at the beginning of class. There are two
parts to this project - the first is an introduction to computed tomography scan, or CT scan, and
works through a sample project while explaining how CT scan images are produced. If there is
time, this will be worked through in class. The second part is an individual project, in which you
are given the output of a two-dimensional CT scan and you are to determine what the picture is.
For this project, we do not ask that you summarize the statement of each problem, nor do
we want you to turn in this paper. Please turn in one sheet of paper with the answers to each
question clearly written. Answer each question using complete sentences. Answers that
simply indicate a single number, a single equation, etc, will be given no credit. Although you may
work in groups or even as a class, your responses should be in your own words. Any indication
of plagiarism, such as duplicate sentences, will be treated as a violation of academic integrity,
resulting in a zero on this project and the dishonesty reported to the Office of Academic Integrity.
This project will be worth 3% of your course grade. Technology allowed. The data is given in a
file that assumes you are using MatLab.
CT Scan Project
The radiation x-ray was discovered by a German physicist, Wilhelm Roentgen, who did not
know what it was, so he simply called it “X-radiation”. A single x-ray passing through a body is
absorbed at rates depending upon the material it goes through. Thus if an x-ray is passed through
bone, a certain amount of intensity units is absorbed, while if it passes through soft material, a
different amount of intensity u ...
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1. Background and Project Description Line Ellipse Rectangle Future Work
Interacting Particles
Midterm Report
University of Illinois at Urbana-Champaign
Department of Mathematics
Illinois Geometry Lab
October 29, 2015
2. Background and Project Description Line Ellipse Rectangle Future Work
Background
n-diameter
A generalization of the diameter. For n points, find them such
that the geometric mean of the distances between them is
maximized.
Equivalently, maximize the product of the distances between
points.
Our Project
Find the n-diameter and the configuration of points that gives
it for a given shape.
This may be very expensive to calculate; it may be better to
make an approximation by finding a configuration with equal
arc lengths or equal Euclidean distances.
3. Background and Project Description Line Ellipse Rectangle Future Work
Line
Intuitively, one might think that the maximizing configuration on
the line would be to divide the line into equal segments. However,
this is not so. For example,
f = (1 0) (x 0) (1 x) [(1 x) 0] [1 (1 x)] [x (1 x)]
= x2
(x 1)2
(2x 1)
df
dx
= 2
f
x
+ 2
f
x 1
+ 2
f
2x 1
= 0
0 = 2x2
2x x + 1 + 2x2
x + x2
x
0 = 5x2
5x + 1
x =
1
2
✓
1 ±
1
p
5
◆
This problem for the line has already been solved in general.
4. Background and Project Description Line Ellipse Rectangle Future Work
General Problem
Ellipse
For an ellipse with unital major axis and minor axis of length a (so
a < 1), note that it becomes more line-like as a ! 0. For example,
for n = 3, if we assume that it must be symmetric about the
x-axis, the solution can be found to be points at
p1 = (a, 0)
p2 = x,
r
1
⇣x
a
⌘2
!
p3 = x,
r
1
⇣x
a
⌘2
!
x = a
"
a2 + 3
p
25a4 18a2 + 9
6 (a2 1)
#
As a ! 0, x ! 0 and so
5. Background and Project Description Line Ellipse Rectangle Future Work
General Problem
Ellipse
As a ! 0, x ! 0 and so
p1 = (a, 0)
p2 = (0, 1)
p3 = (0, 1)
Note that these are solution is not necessarily symmetric for
the n = 3 case.
However, our numerical solution returned a symmetric result,
so this should be a good assumption.
Conjecture: The points for the n-diameter on the ellipse will
alternate sides.
6. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.9
7. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.8
8. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.7
9. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.6
10. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.5
11. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.4
12. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.3
13. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.2
14. Background and Project Description Line Ellipse Rectangle Future Work
Images
Ellipse - a = 0.1
15. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Arc length approximation
Research on the movement of three particles on a given ellipse
Hold the arc length between every two particles the same
Attempt to find the distribution of particles with maximum
product of diameters
Find the trajectory of center of mass when three
particles moving
16. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Conjecture
(1) The trajectory of center of mass may be some ’beautiful’
figure
(2) The center of mass may never fall on the original
(3) The distribution of three particles with maximum product
of diameters may be symmetric
17. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Methodology
Choose the fixed ellipse with expression
x2
52
+
y2
32
= 1
Apply numerical method and recursion in calculation and
programming
Conduct simulation with help of Java and Mathematica
18. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
Figure: Particles distribution with one at rightmost
19. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
20. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
21. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
22. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
23. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
24. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
25. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
Figure: Trajectory of center of mass.
26. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Figures
Figure: Distribution of particles with maximum product of diameters.
27. Background and Project Description Line Ellipse Rectangle Future Work
Arc length approximation
Results and Findings
Trajectory of center of mass is an ellipse with the same shape
as the ellipse that we are researching on, except a much
smaller size
The center of mass never falls on the original
The distribution of three particles with maximum product of
diameters is almost symmetric, with one particle falling on the
lowest position on ellipse. (not exactly symmetric may due to
precision in calculation)
28. Background and Project Description Line Ellipse Rectangle Future Work
Triangle CoM
Modeling the Trajectory of the Center of Mass
Given a unit square and 3 points on its edges, A, B, and C, that
divide the perimeter of the square into three equal-length pieces,
we want to identify the center of mass of the triangle 4ABC and
model the trajectory of its center of mass in terms of the location
of the three points.
Due to the symmetric nature of the square, it su ces to consider
the movement of one of the three points on one edge
As A travels on the edges of the square in a full cycle, the CoM of
4ABC moves around the center of the square on a small rectangle
for 3 cycles. The vertices of the small rectangle is given by
(4
9 , 4
9 ), (4
9 , 5
9 ), (5
9 , 5
9 ), and (5
9 , 4
9 ).
33. Background and Project Description Line Ellipse Rectangle Future Work
Triangle CoM
3-Diameter Problem
Given the triangle 4ABC given above, we are interested in
maximizing the product of the three diameters, i.e. AB · BC · AC.
It can be shown that the product is maximized when the CoM of
4ABC lies on the 4 vertices of the small rectangle.
34. Background and Project Description Line Ellipse Rectangle Future Work
Mapping
Finding n-diameters on rectangles
We can represent the location
on the rectangle by a mapping
f : R ! R2. For example,
f (0) = (0, 0)
f (7) = (6, 1)
f (12) = (4, 4)
f (19) = (0, 1)
f (22) = (2, 0)
35. Background and Project Description Line Ellipse Rectangle Future Work
Future Work
Expand the research from 3-diameters to general n
Improve speed of calculation
Change the restriction same arc length to same length
of diameter