This document presents a fast algorithm for drawing ellipses on a grid of discrete pixels using only integer arithmetic. It builds on Bresenham's line algorithm and circle algorithm. The ellipse algorithm calculates points in two sets in the first quadrant, incrementing one coordinate and testing whether to decrement the other based on an error function. It defines variables like error, slope terms, and stopping criteria that are updated iteratively to efficiently determine the next pixel coordinates. The algorithm achieves 4-point symmetry by calling a subroutine to plot corresponding points in the other quadrants for each calculated point.
An approach to compare graphs using vertex degrees - graph isomorphismSandeep Kunkunuru
A hypothetical approach with an example. No proof is provided. Approach tries to use a seemingly mandatory rule for two graphs to be isomorphic, it isnt proved or argued that the rule is sufficient by itself for isomorphism, that's essentially hoped for.
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
Applications of boolean algebra minterm and maxterm expansionskanyuma jitjumnong
This chapter discusses applications of Boolean algebra including converting English sentences to Boolean equations, using truth tables to design combinational logic, and minterm and maxterm expansions. It provides examples of translating sentences like "The alarm will ring if the alarm switch is on and the door is open, or it is after 6pm and the window is open" into Boolean equations. It also demonstrates designing a combinational logic circuit using a truth table and deriving the minterm and maxterm expansions from the truth table. Minterms and maxterms are defined as products and sums of literals, where each variable appears once. General minterm and maxterm expansions are provided for functions with n variables.
The document discusses trigonometric identities that are important for working with trigonometric functions in calculus. It covers basic identities like reciprocal, quotient, Pythagorean, cofunction, and odd-even identities. Examples are provided of using identities to simplify expressions and solve trigonometric equations.
1. The document discusses geometric proofs in geometry. It covers topics like writing justifications for each step, using definitions and theorems as reasons, and completing two-column proofs.
2. Examples are provided to demonstrate writing justifications, filling in blanks in two-column proofs, and writing full proofs from given plans.
3. The key aspects of writing geometric proofs are to justify each logical step and use appropriate reasons like definitions, postulates, properties, and theorems. Students must also draw diagrams and state the given information and what is to be proved.
The document discusses the midpoint and distance formulas. It defines the midpoint as the point that divides a segment into two congruent segments. The midpoint formula is given as (x1 + x2)/2, (y1 + y2)/2. The distance formula is given as the square root of (x2 - x1)2 + (y2 - y1)2, which is the length of the hypotenuse of a right triangle formed by the differences in the x- and y-coordinates. Homework problems from page 19 numbers 4 through 36 are assigned, excluding number 24.
This document discusses multiple angle identities in trigonometry. It covers double angle identities, power-reducing identities, half-angle identities, and examples of using identities to solve trigonometric equations and find heights of triangles. Specific identities covered include sin2x, cos2x, tan2x, and expressions for reducing powers of trig functions like sin4x to single powers.
(1) An ellipse is defined by the equation (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the major and minor axes.
(2) There are two methods to scan convert an ellipse - the polynomial method and trigonometric method.
(3) The midpoint ellipse algorithm uses a decision parameter p to recursively scan convert the ellipse pixel by pixel in a manner similar to the midpoint circle algorithm.
An approach to compare graphs using vertex degrees - graph isomorphismSandeep Kunkunuru
A hypothetical approach with an example. No proof is provided. Approach tries to use a seemingly mandatory rule for two graphs to be isomorphic, it isnt proved or argued that the rule is sufficient by itself for isomorphism, that's essentially hoped for.
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
Applications of boolean algebra minterm and maxterm expansionskanyuma jitjumnong
This chapter discusses applications of Boolean algebra including converting English sentences to Boolean equations, using truth tables to design combinational logic, and minterm and maxterm expansions. It provides examples of translating sentences like "The alarm will ring if the alarm switch is on and the door is open, or it is after 6pm and the window is open" into Boolean equations. It also demonstrates designing a combinational logic circuit using a truth table and deriving the minterm and maxterm expansions from the truth table. Minterms and maxterms are defined as products and sums of literals, where each variable appears once. General minterm and maxterm expansions are provided for functions with n variables.
The document discusses trigonometric identities that are important for working with trigonometric functions in calculus. It covers basic identities like reciprocal, quotient, Pythagorean, cofunction, and odd-even identities. Examples are provided of using identities to simplify expressions and solve trigonometric equations.
1. The document discusses geometric proofs in geometry. It covers topics like writing justifications for each step, using definitions and theorems as reasons, and completing two-column proofs.
2. Examples are provided to demonstrate writing justifications, filling in blanks in two-column proofs, and writing full proofs from given plans.
3. The key aspects of writing geometric proofs are to justify each logical step and use appropriate reasons like definitions, postulates, properties, and theorems. Students must also draw diagrams and state the given information and what is to be proved.
The document discusses the midpoint and distance formulas. It defines the midpoint as the point that divides a segment into two congruent segments. The midpoint formula is given as (x1 + x2)/2, (y1 + y2)/2. The distance formula is given as the square root of (x2 - x1)2 + (y2 - y1)2, which is the length of the hypotenuse of a right triangle formed by the differences in the x- and y-coordinates. Homework problems from page 19 numbers 4 through 36 are assigned, excluding number 24.
This document discusses multiple angle identities in trigonometry. It covers double angle identities, power-reducing identities, half-angle identities, and examples of using identities to solve trigonometric equations and find heights of triangles. Specific identities covered include sin2x, cos2x, tan2x, and expressions for reducing powers of trig functions like sin4x to single powers.
(1) An ellipse is defined by the equation (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the major and minor axes.
(2) There are two methods to scan convert an ellipse - the polynomial method and trigonometric method.
(3) The midpoint ellipse algorithm uses a decision parameter p to recursively scan convert the ellipse pixel by pixel in a manner similar to the midpoint circle algorithm.
This document provides an overview of using Karnaugh maps to minimize Boolean expressions. It begins with an example of minimizing a 3-variable expression using both algebraic manipulation and a Karnaugh map. Key concepts discussed include grouping adjacent 1s on the map, where adjacency includes horizontal, vertical, and diagonal neighbors. Larger groups that encompass both values of a variable eliminate that term from the minimized expression. The document then demonstrates 4-variable and don't-care maps. It concludes with practice problems for the reader to write minimized expressions and truth tables for additional maps.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
Projection of a Vector upon a Plane from an Arbitrary Angle, via Geometric (C...James Smith
We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.
The document defines key geometry concepts like point, line, plane, angle, and different types of angles. It discusses pairs of angles formed when lines intersect or a transversal crosses parallel lines. Specifically:
1) When two lines intersect, pairs of vertically opposite angles are equal.
2) When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles on the same side of the transversal are supplementary.
3) The sum of the angles of any triangle is 180 degrees.
MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES - POLYGONAL CHAINS and POLYGONS - DEFINITION - MEASURES of the SIDES - TYPES of TRIANGLES - EQUILATERAL ACUTE - ISOSCELES ACUTE - ISOSCELES RIGHT - ISOSCELES OBTUSE - SCALENE ACUTE - SCALENE RIGHT - SCALENE OBTUSE - SUM of the INTERIOR ANGLES - EXTERIOR ANGLES THEOREM - SUM of the EXTERIOR ANGLES - PROOFS STEP by STEP
This document provides an overview of Mathematica, including information about its creator Stephen Wolfram, notational conventions, built-in mathematical constants, and keyboard shortcuts for entering symbols. It also includes examples evaluating mathematical expressions in Mathematica, such as verifying Safford's famous calculation of 133,491,850,208,566,925,016,658,299,941,583,225 as the square of 365,365,365,365,365,365 and calculating 100 digits of pi and e.
The document discusses several fundamental concepts of algebra including:
1. Different types of numbers such as integers, rational numbers, and irrational numbers.
2. Properties of operations like addition, subtraction, multiplication, and division.
3. Exponent rules for simplifying expressions with exponents like multiplying terms with the same base.
1. The document provides the weightage or marks distribution for different units in Class XI. The highest weightage is given to Algebra with 37 marks, followed by Sets and Functions with 29 marks.
2. The document then provides definitions and derivations of some fundamental trigonometric identities involving sine, cosine, tangent, cotangent, secant and cosecant functions. Key identities such as sin^2(θ) + cos^2(θ) = 1 are derived from basic definitions.
3. Suggestions are provided for proving trigonometric identities which include starting from one side and making it look like the other side, using pythagorean identities, and expressing everything in terms of sine and
This document provides an overview of real numbers and their properties. It defines natural numbers, integers, rational numbers, and irrational numbers. It describes operations on real numbers like addition, subtraction, multiplication, and division. Key properties discussed include the commutative, associative, and distributive properties. It also covers fractions, negatives, and finding a common denominator to add fractions.
This document is a solution to a physics problem set composed and formatted by E.A. Baltz and M. Strovink. It contains solutions to 6 problems using vector algebra and trigonometry. The document uses concepts like the law of cosines, dot products, cross products, and vector identities to break vectors into components and calculate angles between vectors. It also applies these concepts to problems involving vectors representing locations on a sphere and wind resistance problems for airplanes.
This document presents a multi-part math problem involving a polygon with sides of decreasing length forming an arithmetic progression. It provides information about the polygon and asks the reader to analyze several propositions related to distances traveled by a formic along the polygon's path. It then asks the reader to identify which of the propositions are true.
The document defines sets and set operations like union, intersection, and difference. It discusses the properties of real numbers including rational and irrational numbers. It also covers topics like absolute value, inequalities, distance between points, and the midpoint formula. Examples are provided to illustrate definitions of circles, parabolas, and graphing conic sections cut from a cone.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
This document provides an introduction to foundational algebra concepts including real numbers, integers, opposites, rational and irrational numbers, variables, absolute value, and graphing numbers on a number line. It includes vocabulary definitions and examples of evaluating expressions, comparing numbers, graphing sets of numbers, and solving absolute value expressions. The homework assigned is to complete odd problems 1-43 on page 54.
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
This document provides information about angles and how to identify, measure, and classify different types of angles. It discusses key angle terms like acute, obtuse, right, and straight angles. It also explains how to use a protractor to measure angles and introduces different angle relationships formed when two lines are intersected by a transversal, such as corresponding angles, alternate interior angles, and vertical angles. Students are provided examples to estimate and measure angles, identify angle components, and investigate angle properties.
1) Solving linear inequalities involves using the same concepts as solving linear equations to isolate the variable, but inequalities have infinite solutions rather than a single solution.
2) Inequality signs (<, >, ≤, ≥) indicate whether values are less than, greater than, less than or equal to, or greater than or equal to another value.
3) Graphing linear inequalities on a number line or coordinate plane involves graphing the corresponding equality and then shading the appropriate side based on the inequality sign and whether it includes the endpoint or not.
The document discusses parse trees, which are data structures that represent the grammatical structure of sentences or expressions. Parse trees are important because they allow recursive functions to operate on expressions. The document provides examples of drawing parse trees for mathematical and logical expressions to demonstrate how operators and operands are represented as nodes. It explains that parse trees are used by computer compilers to represent program code structure.
This document provides an overview of using Karnaugh maps to minimize Boolean expressions. It begins with an example of minimizing a 3-variable expression using both algebraic manipulation and a Karnaugh map. Key concepts discussed include grouping adjacent 1s on the map, where adjacency includes horizontal, vertical, and diagonal neighbors. Larger groups that encompass both values of a variable eliminate that term from the minimized expression. The document then demonstrates 4-variable and don't-care maps. It concludes with practice problems for the reader to write minimized expressions and truth tables for additional maps.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
Projection of a Vector upon a Plane from an Arbitrary Angle, via Geometric (C...James Smith
We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.
The document defines key geometry concepts like point, line, plane, angle, and different types of angles. It discusses pairs of angles formed when lines intersect or a transversal crosses parallel lines. Specifically:
1) When two lines intersect, pairs of vertically opposite angles are equal.
2) When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles on the same side of the transversal are supplementary.
3) The sum of the angles of any triangle is 180 degrees.
MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES - POLYGONAL CHAINS and POLYGONS - DEFINITION - MEASURES of the SIDES - TYPES of TRIANGLES - EQUILATERAL ACUTE - ISOSCELES ACUTE - ISOSCELES RIGHT - ISOSCELES OBTUSE - SCALENE ACUTE - SCALENE RIGHT - SCALENE OBTUSE - SUM of the INTERIOR ANGLES - EXTERIOR ANGLES THEOREM - SUM of the EXTERIOR ANGLES - PROOFS STEP by STEP
This document provides an overview of Mathematica, including information about its creator Stephen Wolfram, notational conventions, built-in mathematical constants, and keyboard shortcuts for entering symbols. It also includes examples evaluating mathematical expressions in Mathematica, such as verifying Safford's famous calculation of 133,491,850,208,566,925,016,658,299,941,583,225 as the square of 365,365,365,365,365,365 and calculating 100 digits of pi and e.
The document discusses several fundamental concepts of algebra including:
1. Different types of numbers such as integers, rational numbers, and irrational numbers.
2. Properties of operations like addition, subtraction, multiplication, and division.
3. Exponent rules for simplifying expressions with exponents like multiplying terms with the same base.
1. The document provides the weightage or marks distribution for different units in Class XI. The highest weightage is given to Algebra with 37 marks, followed by Sets and Functions with 29 marks.
2. The document then provides definitions and derivations of some fundamental trigonometric identities involving sine, cosine, tangent, cotangent, secant and cosecant functions. Key identities such as sin^2(θ) + cos^2(θ) = 1 are derived from basic definitions.
3. Suggestions are provided for proving trigonometric identities which include starting from one side and making it look like the other side, using pythagorean identities, and expressing everything in terms of sine and
This document provides an overview of real numbers and their properties. It defines natural numbers, integers, rational numbers, and irrational numbers. It describes operations on real numbers like addition, subtraction, multiplication, and division. Key properties discussed include the commutative, associative, and distributive properties. It also covers fractions, negatives, and finding a common denominator to add fractions.
This document is a solution to a physics problem set composed and formatted by E.A. Baltz and M. Strovink. It contains solutions to 6 problems using vector algebra and trigonometry. The document uses concepts like the law of cosines, dot products, cross products, and vector identities to break vectors into components and calculate angles between vectors. It also applies these concepts to problems involving vectors representing locations on a sphere and wind resistance problems for airplanes.
This document presents a multi-part math problem involving a polygon with sides of decreasing length forming an arithmetic progression. It provides information about the polygon and asks the reader to analyze several propositions related to distances traveled by a formic along the polygon's path. It then asks the reader to identify which of the propositions are true.
The document defines sets and set operations like union, intersection, and difference. It discusses the properties of real numbers including rational and irrational numbers. It also covers topics like absolute value, inequalities, distance between points, and the midpoint formula. Examples are provided to illustrate definitions of circles, parabolas, and graphing conic sections cut from a cone.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
This document provides an introduction to foundational algebra concepts including real numbers, integers, opposites, rational and irrational numbers, variables, absolute value, and graphing numbers on a number line. It includes vocabulary definitions and examples of evaluating expressions, comparing numbers, graphing sets of numbers, and solving absolute value expressions. The homework assigned is to complete odd problems 1-43 on page 54.
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
This document provides information about angles and how to identify, measure, and classify different types of angles. It discusses key angle terms like acute, obtuse, right, and straight angles. It also explains how to use a protractor to measure angles and introduces different angle relationships formed when two lines are intersected by a transversal, such as corresponding angles, alternate interior angles, and vertical angles. Students are provided examples to estimate and measure angles, identify angle components, and investigate angle properties.
1) Solving linear inequalities involves using the same concepts as solving linear equations to isolate the variable, but inequalities have infinite solutions rather than a single solution.
2) Inequality signs (<, >, ≤, ≥) indicate whether values are less than, greater than, less than or equal to, or greater than or equal to another value.
3) Graphing linear inequalities on a number line or coordinate plane involves graphing the corresponding equality and then shading the appropriate side based on the inequality sign and whether it includes the endpoint or not.
The document discusses parse trees, which are data structures that represent the grammatical structure of sentences or expressions. Parse trees are important because they allow recursive functions to operate on expressions. The document provides examples of drawing parse trees for mathematical and logical expressions to demonstrate how operators and operands are represented as nodes. It explains that parse trees are used by computer compilers to represent program code structure.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Training: ISO/IEC 27001 Information Security Management System - EN | PECB
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Setup Warehouse & Location in Odoo 17 Inventory
Belipse
1. A Fast Bresenham Type Algorithm
For Drawing Ellipses
by
John Kennedy
Mathematics Department
Santa Monica College
1900 Pico Blvd.
Santa Monica, CA 90405
rkennedy@ix.netcom.com
2. Except for this comment explaining that it is blank for
some deliberate reason, this page is intentionally blank!
3. Fast Ellipse Drawing
There is a well-known algorithm for plotting straight lines on a display device or a plotter where
the grid over which the line is drawn consists of discrete points or pixels. In working with a lattice of
points it is useful to avoid floating point arithmetic. One of the first published algorithms was by Jack
Bresenham who worked for I.B.M. ("*'&Ñ. The main idea in the algorithm is to analyze and manipulate
the linear equation so that only integer arithmetic is used in all the calculations. Integer arithmetic has
the advantages of speed and precision; working with floating point values requires more time and
memory and such values would need to be rounded to integers anyway. In this paper we consider the
more difficult problem of approximating the plot of an ellipse on a grid of discrete pixels, using only
integer arithmetic.
Before reading this paper it is suggested you read the paper by the same author on drawing circles.
The key ideas in the previous paper give all the details for circles, and there is much similarity between
drawing ellipses and circles. This paper will discuss the basic differences between circles and ellipses
but it assumes complete familiarity with the circle algorithm.
# #
Assume B# C# œ " represents the real variable equation of an ellipse which is to be plotted
+ ,
using a grid of discrete pixels where each pixel has integer coordinates. There is no loss of generality
here since once the points are determined they may be translated to any elliptical center that is not the
origin (!ß !).
We will compare errors associated with the B and C coordinates of the points that we are plotting.
Although we plot points of the form T (B3 ß C3 ), these points usually do not exactly satisfy the ellipses'
defining equation. In order to avoid dividing we re-write the above equation in the form
,# † B# +# † C# œ ,# † +#
For a given point T (B3 ß C3 ), the quantity
,# † B# +# † C3 ,# † +#
3
#
is a measure telling where T lies in relation to the true ellipse. If this quantity is negative it means T
lies inside the true ellipse, and if this quantity is positive it means T is outside the true ellipse. When
this quantity is ! (which may be rare but does happen) T is exactly on the ellipse. The expression
l ,# † B# +# † C3 ,# † +# l
3
#
is a more practical measure of the error. The absolute value will be needed when comparing two such
errors. We define a function which we call the I663:=/I<<9< which is an error measure for each
plotted point.
I663:=/I<<9<(B3 ß C3 ) œ l ,# † B# +# † C3 ,# † +# l
3
#
Fast Ellipse Drawing 1
4. The ellipse plotting algorithm differs from the circle algorithm in that the ellipses' symmetry allows
only % simultaneous points to be plotted at a time. We still need only calculate points in the first
quadrant, but we need to complete a full *!° , not just %&° . In fact, the criterion for where to make the
break is determined by the slope of the tangent line to the ellipse.
In the first quadrant the ellipse tangent line slope is always negative. Starting on the B-axis and
wrapping in a counterclockwise direction the slope is large and negative which means the C -coordinates
increase faster than the B-coordinates. But once the tangent line slope passes through the value " then
the B-coordinates start changing faster than the C -coordinates.
Thus we will calculate two sets of points in the first quadrant. The first set starts on the positive B-
axis and wraps in a counterclockwise direction until the tangent line slope reaches ". The second set
will start on the positive C-axis and wrap in a clockwise direction until the tangent line slope again
reaches the value ". See the figure at the end of this paper.
For the first set of points we will always increment C and we will test when to decrement B. This
process is very similar to that for circles. For the second set of points we will always increment B and
decide when to decrement C .
Our test decision as to when to decrease B for the first set of points is based on the comparison of the
two values,
I663:=/I<<9<(B3 "ß C3 ") and I663:=/I<<9<(B3 ß C3 ")
Then we choose either B3 " or B3 as our new B-coordinate depending on which of the two
I663:=/I<<9< values is the smallest.
In any event, we need to know how the function I663:=/I<<9<(B3 ß C3 ) changes for each possible
change in its arguments. We do not need to calculate I663:=/I<<9< anew for each next point, if we just
keep track of how the value changes as the arguments change.
For the first set of points,
I663:=/I<<9<(B3 "ß C3 ") œ l ,# † ( B3 " )# +# † ( C3 " )# ,# † +# l
œ l ,# † B# ,# † #B3 ,# +# † C3 +# † #C3 +# ,# † +# l
3
#
œ l ,# † B# +# † C3 ,# † +# ,# † ( " #B3 ) +# † ( #C3 " ) l
3
#
while
I663:=/I<<9<(B3 ß C3 ") œ l ,# † B3 +# † ( C3 " )# ,# † +# l
#
œ l ,# † B# +# † C3 +# † #C3 +# ,# † +# l
3
#
œ l ,# † B# +# † C3 ,# † +# +# † ( #C3 " ) l
3
#
Fast Ellipse Drawing 2
5. The analysis of the following ellipse inequality is almost identical to that for the circle inequality
except we multiply the appropriate change factor by either +# or ,# . The end result is that
I663:=/I<<9<(B3 "ß C3 ") I663:=/I<<9<(B3 ß C3 ")
if and only if
# † c ,# † B# +# † C3 ,# † +# +# † ( #C3 " ) d ,# † ( " #B3 ) !
3
#
When this last inequality holds then we should decrement B when we plot the next point
T (B3" ß C3" ).
Having performed the above analysis we can begin to see the usefulness of defining three new
quantities.
G2+81/ œ ,# † ( " #B3 )
] G2+81/ œ +# † ( #C3 " )
I663:=/I<<9< œ ,# † B# +# † C3 ,# † +#
3
#
These three quantities may also be calculated recursively (i.e., iteratively). Since when B3 and C3
change, they change by „ ", the quantities G2+81/ and ] G2+81/ always change by exactly
„ # † ,# and „ # † +# respectively. The initial G2+81/ value is +# † ( " #+ ). The initial ] G2+81/
value is +# . The initial I663:=/I<<9< value is !. The program variable I663:=/I<<9< neither needs
nor uses an absolute value.
We can analyze the tangent line slope by implicitly differentiating the equation
,# † B# +# † C# œ ,# † +#
# † , # † B # † +# † C † C w œ !
Solving for C w yields
# † ,# † B
Cw œ
# † +# † C
Although one might normally expect to reduce the above fraction by dividing by #, it turns out to be
more efficient to not reduce this fraction.
Now we can determine where to break the two sets of calculated points in the first quadrant. The
first set corresponds to where C w ". So we consider the inequality
# † ,# † B
"
# † +# † C
Fast Ellipse Drawing 3
6. and we decide to continue to plot points while
# † ,# † B # † +# † C !.
Rather than perform the multiplications in this inequality each time through the while-loop test, we
again take advantage of the fact that whenever B and C change, they change by exactly " unit. So we
define two new values called W>9::381 and W>9::381] . Although we compute these values
recursively (iteratively), these values are really represented by the following equations.
W>9::381 œ # † ,# † B3 and W>9::381] œ # † +# † C3
When plotting the first set of points the initial values are given by W>9::381 œ # † ,# † + and
W>9::381] œ !. For the first set of points B decreases by " at each stage so W>9::381 has a
corresponding decrease of # † ,# . For the first set of points W>9::381] increases by # † +# .
When plotting the second set of points the initial values are given by W>9::381 œ ! and
W>9::381] œ # † +# † ,. For the second set of points B increases by " at each stage so W>9::381 has a
corresponding increase of # † ,# . For the second set of points W>9::381] decreases by # † +# each time
C decreases by ".
Now we can give the ellipse plotting algorithm. Below, G , G] , and V+.3?= and ] V+.3?=
refer to the ellipse's center point coordinates and its horizontal and vertical radial values. So
V+.3?= œ + and ] V+.3?= œ ,. The constants X A9FW;?+</ and X A9EW;?+</ are the pre-
computed values # † ,# and # † +# .
procedure PlotEllipse(CX, CY, XRadius, YRadius : longint);
begin
var X, Y : longint;
XChange, YChange : longint;
EllipseError : longint;
TwoASquare, TwoBSquare : longint;
StoppingX, StoppingY : longint;
TwoASquare := 2*XRadius*XRadius;
TwoBSquare := 2*YRadius*YRadius;
X : œ XRadius;
Y := 0;
XChange : œ YRadius*YRadius*(1 2*XRadius);
YChange : œ XRadius*XRadius;
EllipseError := 0;
StoppingX := TwoBSquare*XRadius;
StoppingY := 0;
{ algorithm continues on the next page }
Fast Ellipse Drawing 4
7. while ( StoppingX StoppingY ) do {1st set of points,
yw > 1}
begin
Plot4EllipsePoints(X,Y); {subroutine appears later}
inc(Y);
inc(StoppingY, TwoASquare);
inc(EllipseError, YChange);
inc(YChange,TwoASquare);
if ((2*EllipseError + XChange) > 0 ) then
begin
dec(X);
dec(StoppingX, TwoBSquare);
inc(EllipseError, XChange);
inc(XChange,TwoBSquare)
end
end;
{ 1st point set is done; start the 2nd set of points }
X : œ 0;
Y := YRadius;
XChange : œ YRadius*YRadius;
YChange : œ XRadius*XRadius*(1 2*YRadius);
EllipseError := 0;
StoppingX := 0;
StoppingY := TwoASquare*YRadius;
while ( StoppingX Ÿ StoppingY ) do {2nd set of points, yw < 1}
begin
Plot4EllipsePoints(X,Y); {subroutine appears later}
inc(X);
inc(StoppingX, TwoBSquare);
inc(EllipseError, XChange);
inc(XChange,TwoBSquare);
if ((2*EllipseError + YChange) > 0 ) then
begin
dec(Y);
dec(StoppingY, TwoASquare);
inc(EllipseError, YChange);
inc(YChange,TwoASquare)
end
end
end; {procedure PlotEllipse}
Fast Ellipse Drawing 5
8. The subroutine called T 69>%I663:=/T 938>= takes advantage of the symmetry in the ellipse. We
only calculate the points in the first quadrant, but for each such point we actually plot % other points at
the same time as indicated in the figure below. The T 69>%I663:=/T 938>= subroutine would normally be
defined inside the above T 69>I663:=/ procedure. Also note that G and G] refer to the ellipse's
center point.
procedure Plot4EllipsePoints(X,Y : longint);
begin
PutPixel(CX+X, CY+Y); {point in quadrant 1}
PutPixel(CX-X, CY+Y); {point in quadrant 2}
PutPixel(CX-X, CY-Y); {point in quadrant 3}
PutPixel(CX+X, CY-Y) {point in quadrant 4}
end; {procedure Plot4EllipsePoints}
As shown in the figure below, when this subroutine plots the first set of points, the four points would
be like those numbered ", %, &, and ) in the figure. When plotting the second set of points, the four
points would be like those numbered #, $, ', and ( in the figure.
Y+
2nd plotting direction The tangent line slope
is clockwise at this point = -1
(-x2 , y2 ) (x , y )
2 2 2
3 2nd set of 1st plotting
points (x , y ) direction
1 1
(-x , y ) 1 is
1 1
counter-
4 1st set of clockwise
X- points
X+
5 8
(-x , -y ) (x , -y )
1 1 1 1
6 7
(-x 2 , -y2 ) (x , -y )
2 2
Y-
Figure 1. This figure indicates the two sets of points in the first quadrant that get plotted.
The plotting algorithm uses two sets with 4-point symmetry.
Fast Ellipse Drawing 6
9. REFERENCES:
". Jack Bresenham, Algorithm for Computer Control of a Digital Plotter, IBM Systems
Journal, Volume %, Number ", "*'&, pp. #&-$!.
#. Jack Bresenham, A Linear Algorithm for Incremental Display of Circular Arcs,
Communications of the ACM, Volume #!, Number #, February "*((, pp. "!!-"!'.
$. Jerry R. Van Aken, An Efficient Ellipse-Drawing Algorithm, I.E.E.E. Computer Graphics
& Applications, September "*)%, pp.#%-$&.
%. Michael Abrash, The Good, the Bad, and the Run-sliced, Dr. Dobbs Journal, Number "*%,
November "**#, pp."("-"('.
Fast Ellipse Drawing 7