This document discusses conformal mapping, which maps curves and regions in such a way that preserves angles and their directions. It provides examples of conformal mappings:
1) The mapping w = ez maps a vertical line in the z-plane to a circle in the w-plane, with the phase angle increasing along the circle.
2) The mapping ω = eiθ0(z-z0)/(z-z0) maps an area in the upper half z-plane to the interior of a unit circle in the ω-plane. Points on the x-axis in z are mapped to the boundary of the circle.
Collinearity Equations
Kinds of product that can be derived by the collinearity equation
- Space Resection By Collinearity
- Space Intersection By Collinearity
- Interior Orientation
- Relative Orientation
- Absolute Orientation
- Self-Calibration
Collinearity Equations
Kinds of product that can be derived by the collinearity equation
- Space Resection By Collinearity
- Space Intersection By Collinearity
- Interior Orientation
- Relative Orientation
- Absolute Orientation
- Self-Calibration
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Chapter 12
Section 12.1: Three-Dimensional Coordinate Systems
We locate a point on a number line as one coordinate, in the plane as an ordered pair, and in
space as an ordered triple. So we call number line as one dimensional, plane as two
dimensional, and space as three dimensional co – ordinate system.
In three dimensional, there is origin (0, 0, 0) and there are three axes – x -, y - , and z – axis. X –
and y – axes are horizontal and z – axis is vertical. These three axes divide the space into eight
equal parts, called the octants. In addition, these three axes divide the space into three
coordinate planes.
– The xy-plane contains the x- and y-axes. The equation is z = 0.
– The yz-plane contains the y- and z-axes. The equation is x = 0.
– The xz-plane contains the x- and z-axes. The equation is y = 0.
If P is any point in space, let:
– a be the (directed) distance from the yz-plane to P.
– b be the distance from the xz-plane to P.
– c be the distance from the xy-plane to P.
Then the point P by the ordered triple of real numbers (a, b, c), where a, b, and c are the
coordinates of P.
– a is the x-coordinate.
– b is the y-coordinate.
– c is the z-coordinate.
– Thus, to locate a point (a, b, c) in space, start from the origin (0, 0, 0) and move a
units along the x-axis. Then, move b units parallel to the y-axis. Finally, move c
units parallel to the z-axis.
The three dimensional Cartesian co – ordinate system follows the right hand rule.
Examples:
Plot the points (2,3,4), (2, -3, 4), (-2, -3, 4), (2, -3, -4), and (-2, -3, -4).
The Cartesian product x x = {(x, y, z) | x, y, z in } is the set of all ordered triples of
real numbers and is denoted by 3 .
Note:
1. In 2 – dimension, an equation in x and y represents a curve in the plane 2 . In 3 –
dimension, an equation in x, y, and z represents a surface in space 3 .
2. When we see an equation, we must understand from the context that it is a curve in the
plane or a surface in space. For example, y = 5 is a line in 2 �but it is a plane in 3 �
������
3. in space, if k, l, & m are constants, then
– x = k represents a plane parallel to the yz-plane ( a vertical plane).
– y = k is a plane parallel to the xz-plane ( a vertical plane).
– z = k is a plane parallel to the xy-plane ( a horizontal plane).
– x = k & y = l is a line.
– x = k & z = m is a line.
– y = l & z = m is a line.
– x = k, y = l and z = m is a point.
Examples: Describe and sketch y = x in 3
Example:
Solve:
Which of the points P(6, 2, 3), Q(-5, -1, 4), and R(0, 3, 8) is closest to the xz – plane? Which point
lies in the yz – plane?
Distance between two points in space:
We simply extend the formula from 2 to . 3 . The distance |p1 p2 | between the points
P1(x1,y1, z1) and P2(x2, y2, z2) is: 2 2 21 2 2 1 ...
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space. The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
One Modification which Increases Performance of N-Dimensional Rotation Matrix...AI Publications
This article presents one modification of algorithm for generation of n-dimensional rotation matrix M, which rotates given n-dimensional vector X to the direction of given n-dimensional vector Y. Algorithm, named N-dimensional Rotation Matrix Generation Algorithm (NRMG) includes rotations of given vectors X and Y to the direction of coordinate axis x1 using two-dimensional rotations in coordinate planes. Proposed modification decreases the number of needed two-dimensional rotations to 2(Lw-1) were Lw is the number of corresponding components of the two given vectors, that are not equal.
1 ijcmp mar-2018-3-one modification which increasesAI Publications
This article presents one modification of algorithm for generation of n-dimensional rotation matrix M, which rotates given n-dimensional vector X to the direction of given n-dimensional vector Y. Algorithm, named N-dimensional Rotation Matrix Generation Algorithm (NRMG) includes rotations of given vectors X and Y to the direction of coordinate axis x1 using two-dimensional rotations in coordinate planes. Proposed modification decreases the number of needed two-dimensional rotations to 2(Lw-1) were Lw is the number of corresponding components of the two given vectors, that are not equal.
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Chapter 12
Section 12.1: Three-Dimensional Coordinate Systems
We locate a point on a number line as one coordinate, in the plane as an ordered pair, and in
space as an ordered triple. So we call number line as one dimensional, plane as two
dimensional, and space as three dimensional co – ordinate system.
In three dimensional, there is origin (0, 0, 0) and there are three axes – x -, y - , and z – axis. X –
and y – axes are horizontal and z – axis is vertical. These three axes divide the space into eight
equal parts, called the octants. In addition, these three axes divide the space into three
coordinate planes.
– The xy-plane contains the x- and y-axes. The equation is z = 0.
– The yz-plane contains the y- and z-axes. The equation is x = 0.
– The xz-plane contains the x- and z-axes. The equation is y = 0.
If P is any point in space, let:
– a be the (directed) distance from the yz-plane to P.
– b be the distance from the xz-plane to P.
– c be the distance from the xy-plane to P.
Then the point P by the ordered triple of real numbers (a, b, c), where a, b, and c are the
coordinates of P.
– a is the x-coordinate.
– b is the y-coordinate.
– c is the z-coordinate.
– Thus, to locate a point (a, b, c) in space, start from the origin (0, 0, 0) and move a
units along the x-axis. Then, move b units parallel to the y-axis. Finally, move c
units parallel to the z-axis.
The three dimensional Cartesian co – ordinate system follows the right hand rule.
Examples:
Plot the points (2,3,4), (2, -3, 4), (-2, -3, 4), (2, -3, -4), and (-2, -3, -4).
The Cartesian product x x = {(x, y, z) | x, y, z in } is the set of all ordered triples of
real numbers and is denoted by 3 .
Note:
1. In 2 – dimension, an equation in x and y represents a curve in the plane 2 . In 3 –
dimension, an equation in x, y, and z represents a surface in space 3 .
2. When we see an equation, we must understand from the context that it is a curve in the
plane or a surface in space. For example, y = 5 is a line in 2 �but it is a plane in 3 �
������
3. in space, if k, l, & m are constants, then
– x = k represents a plane parallel to the yz-plane ( a vertical plane).
– y = k is a plane parallel to the xz-plane ( a vertical plane).
– z = k is a plane parallel to the xy-plane ( a horizontal plane).
– x = k & y = l is a line.
– x = k & z = m is a line.
– y = l & z = m is a line.
– x = k, y = l and z = m is a point.
Examples: Describe and sketch y = x in 3
Example:
Solve:
Which of the points P(6, 2, 3), Q(-5, -1, 4), and R(0, 3, 8) is closest to the xz – plane? Which point
lies in the yz – plane?
Distance between two points in space:
We simply extend the formula from 2 to . 3 . The distance |p1 p2 | between the points
P1(x1,y1, z1) and P2(x2, y2, z2) is: 2 2 21 2 2 1 ...
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space. The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
One Modification which Increases Performance of N-Dimensional Rotation Matrix...AI Publications
This article presents one modification of algorithm for generation of n-dimensional rotation matrix M, which rotates given n-dimensional vector X to the direction of given n-dimensional vector Y. Algorithm, named N-dimensional Rotation Matrix Generation Algorithm (NRMG) includes rotations of given vectors X and Y to the direction of coordinate axis x1 using two-dimensional rotations in coordinate planes. Proposed modification decreases the number of needed two-dimensional rotations to 2(Lw-1) were Lw is the number of corresponding components of the two given vectors, that are not equal.
1 ijcmp mar-2018-3-one modification which increasesAI Publications
This article presents one modification of algorithm for generation of n-dimensional rotation matrix M, which rotates given n-dimensional vector X to the direction of given n-dimensional vector Y. Algorithm, named N-dimensional Rotation Matrix Generation Algorithm (NRMG) includes rotations of given vectors X and Y to the direction of coordinate axis x1 using two-dimensional rotations in coordinate planes. Proposed modification decreases the number of needed two-dimensional rotations to 2(Lw-1) were Lw is the number of corresponding components of the two given vectors, that are not equal.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Overview on Edible Vaccine: Pros & Cons with Mechanism
lec41.ppt
1. NPTEL – Physics – Mathematical Physics - 1
Lecture 41
Conformal Mapping
In this section we shall introduce the concept of conformal mapping. Mainly we shall
focus on the connection between such mappings and harmonic functions. We shall
conclude our discussion with a short note on the application of conformal mappings on
physical problems.
Mapping of functions: Mappings that preserve angles and their directions, but not
necessarily the scale are called conformal mapping. Formally we can explain in
the following way. Consider the set of equations,
𝑢 = 𝑢(𝑥, 𝑦) (1)
𝑣 = 𝑣(𝑥, 𝑦)
Where 𝑢 and 𝑣 are functions of 𝑥 and 𝑦 . The above equations represent mapping
of points between (𝑢, 𝑣) and (𝑥, 𝑦) planes. An inverse transformation can trivially
be defined as,
𝑥 = 𝑥(𝑢, 𝑣) (2)
𝑦 = 𝑦(𝑢, 𝑣)
It can happen that each point in the plane corresponds to each point in the 𝑥𝑦 plane or a
set of points in one plane correspond to a set in the other plane. In this case each point or
the set of points is said to be images of each other.
As per the transformation defined by Eqs. (1) and (2) a region or a curve in the uv plane
can be mapped into region or a curve respectively in the 𝑥𝑦 plane. The Jacobian for the
transformation in Eq. (1) is given by
𝜕𝑢 𝜕𝑢
𝜕(𝑢, 𝑣)
𝜕(𝑥, 𝑦)
= |
𝜕𝑥 𝜕𝑦
𝜕𝑣 𝜕𝑣
𝜕𝑥 𝜕𝑦
|
=
𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣
𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥
− (3)
Joint initiative of IITs and IISc – Funded by MHRD Page 57 of 66
2. NPTEL – Physics – Mathematical Physics - 1
By transformation defined by Eq. (1) let a point (𝑥0, 𝑦0) in the 𝑥𝑦 plane is
mapped onto a point (𝑢0, 𝑣0) in the uv plane and further two curves 𝐶1 and 𝐶2 in 𝑥𝑦
plane map onto two other curves 𝐶̃1and 𝐶̃2 in the 𝑢𝑣 plane. Let us assume that the curves
meet at (𝑥0, 𝑦0) and (𝑢0, 𝑣0) and further the angles surrounded by the respective set
of curves are same in both magnitude and direction as shown in the figure.
In this case, the mapping is said to be conformal at (𝑥0, 𝑦0). In some other
mapping, where only the magnitude of the angle is preserved but not the direction
is called as isogonal mapping. The results can trivially be extended to the case of
complex variables, where it can be stated as in the following.
Let a curve 𝐶 be presented by a parametric equation,
𝑧 = 𝑧(𝑡) (𝑎 ≤ 𝑡 ≤ 𝑏)
And 𝑓(𝑧) be a function defined at all points z on 𝐶.
Then
(4)
𝑤 = 𝑓[𝑧(𝑡)] (5)
is the equation of the image 𝑐̃ of 𝐶 under the transformation 𝑤 = 𝑓(𝑧) (in the
same sense of Eq. (1)). This transformation is said to be conformal at a point 𝑧0 if
𝑓(𝑧) is analytic there and 𝑓′(𝑧) ≠ 0. Also this transformation is conformal
at each point in the neighborhood of 𝑧0.
Example 1 : Mapping of a curve
The mapping 𝜔 = 𝑒𝑧 is conformal throughout the entire z plane since 𝑓′(𝑧) =
𝑒𝑧 ≠ 0 for all 𝑧. Now consider two lines 𝑥 = 𝐶1 and 𝑦 = 𝐶2 in the z plane as
shown in figure. The images of these curves 𝐶1 and 𝐶2 can be obtained as
follows.
Joint initiative of IITs and IISc – Funded by MHRD Page 58 of 66
3. NPTEL – Physics – Mathematical Physics - 1
𝜔 = 𝑒𝑧 can be written in polar form as
𝜌𝑒𝑖𝜑 = 𝑒𝑥+𝑖𝑦 with 𝜔 = 𝜌𝑒𝑖𝜑
and 𝑧 = 𝑥 + 𝑖𝑦
thus 𝜌 = 𝑒𝑥 and 𝜑 = 𝑦
The image of a point 𝑧 = (𝑐, 𝑦) on a vertical
line 𝑥 = 𝐶1 has the following polar
coordinates
𝜌 = 𝑒𝑐1
𝜑 = 𝑦 in the y- plane
So the amplitude is constant and the phase
angle increases from 0 to 2𝜋. Thus the image
of the line 𝑥 = 𝐶1 in the uv plane is a circle of
radius 𝑒𝑐1 moving counterclockwise along the
circle as shown in fig. (2)
Each point on the circle is the image
of an
infinite number of points, spaced 2𝜋 units
apart alone the line.
Example 2 : Mapping of an area
Let us take the shaded region in the upper half plane as shown in the figure (Fig (iii)).
Joint initiative of IITs and IISc – Funded by MHRD Page 59 of 66
4. NPTEL – Physics – Mathematical Physics - 1
Let 𝑧0 be an arbitrary point P in the upper
half of the z- plane denoted by R
The transformation 𝜔 = 𝑒𝑖θ0 (𝑧−𝑧0
)
𝑧−̅𝑧̅0̅
Maps onto the interior of a region 𝑅′ of unit circle |𝜔| = 1. Each point on the x axis, A,
B, C, D, E, F etc is mapped onto the boundary of the circle as shown in the figure
The angle 𝜃0 is determined by making one particular point on the 𝑥 − axis correspond to
a given point on the circle. Again as in example 1, 𝜔 moves anticlockwise along the unit
circle.
Joint initiative of IITs and IISc – Funded by MHRD Page 60 of 66