Slides from when I was teaching CS4052 Computer Graphics at Trinity College Dublin in Ireland.
These slides aren't used any more so they may as well be available to the public!
There are some mistakes in the slides, I'll try to comment below these.
This is the third lecture - on using linear algebra for transformations.
Lesson 6: Polar, Cylindrical, and Spherical coordinatesMatthew Leingang
"The fact that space is three-dimensional is due to nature. The way we measure it is due to us." Cartesian coordinates are one familiar way to do that, but other coordinate systems exist which are more useful in other situations.
Un sistema de coordenadas es un conjunto de valores que permiten definir la posición de cualquier punto de un espacio vectorial.
Estos sistemas de coordenadas son de suma importancia ya que para resolver problemas de electrotástica, magnetostática y
campos variables en el tiempo, tenemos que tener un conocimiento previo de cómo utilizarlos y cómo hacer cambios de bases
vectoriales entre ellos para que la resolución de los problemas sea menos compleja.
TRANSFORMACIÓN DE COORDENADAS
Para adentrarnos en el tema de transformación de coordenadas, considero que es importante conocer primeramente, la definición y/u concepto
de lo que es un sistema de coordenadas, así que iniciando desde este punto, tenemos que:
Un sistema de coordenadas es un sistema que utiliza uno o más números (coordenadas) para determinar unívocamente la posición de un
punto u objeto geométrico.
El orden en que se escriben las coordenadas es significativo y a veces se las identifica por su posición en una tupla ordenada; también se las
puede representar con letras, como por ejemplo (la coordenada-x). El estudio de los sistemas de coordenadas es objeto de la geometría
analítica, permite formular los problemas geométricos de forma "numérica“.
Teniendo esto en claro, podemos definir a aquello que se conoce como Transformación de coordenadas… Entonces, tenemos que:
La transformación de coordenadas es una operación por la cual una relación, expresión o figura se cambia en otra siguiendo una ley dada.
Analíticamente, la ley se expresa por una o mas ecuaciones llamadas ecuaciones de transformación.
También se define como el cambio de posición de los ejes de referencia en un sistema de coordenadas, ya sea por traslación, rotación, o ambas. El propósito de dicho cambio por lo general es simplificar la ecuación de una curva para manejo posterior.
TRANSFORMACIÓN DE COORDENADAS RECTANGULARES A POLARES
Primero definiremos a cada sistema de coordenadas…
Coordenadas Rectangulares:son aquellas que nos permiten determinar la ubicación de un punto mediante dos distancias y refiriéndolas a una dirección base y a un punto base.
Slides from when I was teaching CS4052 Computer Graphics at Trinity College Dublin in Ireland.
These slides aren't used any more so they may as well be available to the public!
There are some mistakes in the slides, I'll try to comment below these.
This is the third lecture - on using linear algebra for transformations.
Lesson 6: Polar, Cylindrical, and Spherical coordinatesMatthew Leingang
"The fact that space is three-dimensional is due to nature. The way we measure it is due to us." Cartesian coordinates are one familiar way to do that, but other coordinate systems exist which are more useful in other situations.
Un sistema de coordenadas es un conjunto de valores que permiten definir la posición de cualquier punto de un espacio vectorial.
Estos sistemas de coordenadas son de suma importancia ya que para resolver problemas de electrotástica, magnetostática y
campos variables en el tiempo, tenemos que tener un conocimiento previo de cómo utilizarlos y cómo hacer cambios de bases
vectoriales entre ellos para que la resolución de los problemas sea menos compleja.
TRANSFORMACIÓN DE COORDENADAS
Para adentrarnos en el tema de transformación de coordenadas, considero que es importante conocer primeramente, la definición y/u concepto
de lo que es un sistema de coordenadas, así que iniciando desde este punto, tenemos que:
Un sistema de coordenadas es un sistema que utiliza uno o más números (coordenadas) para determinar unívocamente la posición de un
punto u objeto geométrico.
El orden en que se escriben las coordenadas es significativo y a veces se las identifica por su posición en una tupla ordenada; también se las
puede representar con letras, como por ejemplo (la coordenada-x). El estudio de los sistemas de coordenadas es objeto de la geometría
analítica, permite formular los problemas geométricos de forma "numérica“.
Teniendo esto en claro, podemos definir a aquello que se conoce como Transformación de coordenadas… Entonces, tenemos que:
La transformación de coordenadas es una operación por la cual una relación, expresión o figura se cambia en otra siguiendo una ley dada.
Analíticamente, la ley se expresa por una o mas ecuaciones llamadas ecuaciones de transformación.
También se define como el cambio de posición de los ejes de referencia en un sistema de coordenadas, ya sea por traslación, rotación, o ambas. El propósito de dicho cambio por lo general es simplificar la ecuación de una curva para manejo posterior.
TRANSFORMACIÓN DE COORDENADAS RECTANGULARES A POLARES
Primero definiremos a cada sistema de coordenadas…
Coordenadas Rectangulares:son aquellas que nos permiten determinar la ubicación de un punto mediante dos distancias y refiriéndolas a una dirección base y a un punto base.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
2. Trigonometry (from Greek trigōnon
"triangle" + metron "measure")[1] is a
branch of mathematics that deals with
triangles,
Trigonometry deals with relationships
between the sides and the angles of
triangles and with the trigonometric
functions, which describe those
relationships.
3. The Canadarm2 robotic manipulator on the
International Space Station is operated by controlling
the angles of its joints. Calculating the final position of
the astronaut at the end of the arm requires repeated
use of the trigonometric functions of those angles.
8. -1
-1
1
1
y
x
o
III
III IV
Identifying quadrants and angles in the
unit circle.
The Unit Circle is a circle of radius 1 unit with its
centre (0,0), on the Cartesian plane.The x-axis
and the y-axis divide the circle into four equal
quadrants.
11. An angle is measured from an
initial side which is the positive
x-axis to its terminal side.
12. There are 2 radians in a full rotation -
- once around the circle
There are 360° in a full rotation
To convert from degrees to radians or
radians to degrees, use the proportion
2360
deg
0
radiansrees
13. y
x
o
Terminal side
Initial side
When the angle is measured in an
anticlockwise direction from the initial side
to the terminal side, it is a positive angle.
Anti-clockwise
direction
14. y
x
o
Terminal side
Initial side
When the angle is measured in an
clockwise direction from the initial side to
the terminal side, it is a negative angle.
clockwise direction
15. Represent each of the following angles
in a Cartesian plane.Hence, state which
quadrant the angle is in.
0
45)(a
0
200)( b
3
1
)( c
3
2
)( d
17. Represent each of the following
angles in a Cartesian plane.Hence,
state which quadrant the angle is in.
0
400)(a
0
540)( b
4
15
)(c
5
11
)( d
20. The Unit Circle
Imagine a circle on the
coordinate plane, with
its center at the origin,
and a radius of 1.
Let P, ( x, y) is a point
on the circle somewhere
in quadrant I.
-
1
-
1
1
1 P
22. The Unit Circle
x
y
1
is the
angle of
rotation
From the trigonometric
ratios, this triangle gives
x
x
1
)cos(
x
y
)tan(
)sin,(cos P
y
y
1
)sin(
25. Find the value of each
trigonometric equation:
0
0
0
0
0
0
145tan)(
145cos)(
145sin))((
70tan)(
70cos)(
70sin)()(
iii
ii
ib
iii
ii
ia
0
0
0
0
0
0
314tan)(
314cos)(
314sin))((
210tan)(
210cos)(
210sin))((
iii
ii
id
iii
ii
ic
28. Quadrant III
Angle given,
Reference angle
For an angle, , in
quadrant III,
-
In quadrant III,
sin() is negative
cos() is negative
tan() is positive
P(-x,-y)
29. Quadrant IV
Angle given,
Reference angle
For an angle, , in
quadrant IV,
2
In quadrant IV,
sin() is negative
cos() is positive
tan() is negative
P(x,-y)
30. Add Sugar To Coffee
Use the phrase “Add Sugar To Coffee” to
remember the signs of the trig functions in
different quadrants.
AddSugar
To Coffee
All functions
are positive
Sine is positive
Tan is positive Cos is positive
AfterSchool
Terus Cabut
31. The relationship between positive angles
and negative angles in trigonometric
functions
)tan()tan(
)cos()cos(
)sin()sin(
Quadrant IV,
cos( )
positive
32. Given that and
Find the values of the following:
7660.0130sin 0
6428.0130cos 0
0
0
0
0
130cos)(
130sec)(
130cot)(
130tan)(
ecd
c
b
a
43. Solving Trigonometric
Equations
STEPS
Determine the quadrant the angle lies
based on the sign of value of the trigo.
function.
Find the reference angle using a
scientific calculator
Determine all the possible angle values
based on the given range.
44. Find the value of for each of the
following trigonometric equations for
00
3600
9238.2cos)(
1135.2tan)(
5478.0cos)(
2344.0sin)(
ecd
c
b
a
46. Find the values of x for each of the
following trigonometric equations for 00
3600 x
7883.0)152sin()(
3420.03cos)(
5344.02sin)(
0
xc
xb
xa
70. Solve the following equations for 00
3600 x
0
00
60cos2sin)(
0cos2cos)(
)30cos()30sin()(
xc
xxb
xxa
71. Given that and
and the angles A and B are in the
first quadrant and third quadrant
respectively, find the value of
tan(A-B)
13
5
sin A
5
3
sin
B
72. If and
express the following in terms
of m and/or n.
0
36cosm 0
44sinn
0
0
8cos)(
80sin)(
b
a
73. Given that ,
find the value of without
using a calculator
00
900,
13
5
cos
2
sin
101. Period = 1800
No Maximum and minimum values
Function undefined at 00
270,90x
102. y = tan
0o 90
o
180
o
270
o
360
o
-90o-180o-270o-360o
450
o
-
450o
x
103. Sketch the graph of each following
trigonometric functions for
xyc
xyb
xya
tan)(
2sin3)(
2cos2)(
20 x
104. The number of solutions to a trigonometric
equation involving non-trigonometric
expressions:
105. Separate the trigo. Expression from
the non-trigo. Expression.
Sketch the graphs of the to
functions on the same axis
Determine the number of points of
intersection =number of solutions.
106. Sketch the graph of
. On the same
axes, sketch the line
.Hence, find the number of
solutions to the equation
202cos3 xforxy
x
y
2
20022cos3 xforxx