IOSR Journal of Mathematics(IOSR-JM) is an open access 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.
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
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.
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
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.
TIU CET Review Math Session 4 Coordinate Geometryyoungeinstein
College Entrance Test Review
Math Session 4 Coordinate Geometry
Formulas for the Slope of a line, Midpoint, Distance between any two points,
Equations of a Line
A Quantified Approach for large Dataset Compression in Association MiningIOSR Journals
Abstract: With the rapid development of computer and information technology in the last several decades, an
enormous amount of data in science and engineering will continuously be generated in massive scale; data
compression is needed to reduce the cost and storage space. Compression and discovering association rules by
identifying relationships among sets of items in a transaction database is an important problem in Data Mining.
Finding frequent itemsets is computationally the most expensive step in association rule discovery and therefore
it has attracted significant research attention. However, existing compression algorithms are not appropriate in
data mining for large data sets. In this research a new approach is describe in which the original dataset is
sorted in lexicographical order and desired number of groups are formed to generate the quantification tables.
These quantification tables are used to generate the compressed dataset, which is more efficient algorithm for
mining complete frequent itemsets from compressed dataset. The experimental results show that the proposed
algorithm performs better when comparing it with the mining merge algorithm with different supports and
execution time.
Keywords: Apriori Algorithm, mining merge Algorithm, quantification table
Self-Medication of Anti-Biotics amongst University Students of Islamabad: Pre...IOSR Journals
The prevalence and pattern of self-medication with antibiotics among undergraduate and graduate community of students at different universities of Islamabad was evaluated using structured self-medication administered questionnaire. This cross-sectional, study was conducted in March 2013. A convenience sample was taken from 4 non-medical universities of the city of Islamabad, Pakistan. Data was analyzed using SPSS v14 and associations were tested using the Chi square test. A total of 210 questionnaires were randomly distributed with a respondent rate of 100%. The prevalence of self-medication was found to be 77.03% (Female: Male Ratio=1:1.14). The major reasons given for self-medicating with antibiotics were; 33.63% assumed knowledge on antibiotics (P=0.478), 26.64% prior experience on use (P=0.378), while 9.17% admitted lack of time to go for consultation (P=0.130). Majority of respondents however, self-medicate with antibiotics occasionally. The most reported antibiotic class (48.58%) was the β-lactams (as amoxicillin) while co-trimoxazole was rarely used (2.23%). The most reported condition for self-medication was respiratory tract infections (59%). About almost half of the respondents (46.79%) purchased the drugs from drug stores. These findings highlight the needs for planning interventions to promote the judicious use of antibiotics within the student population
TIU CET Review Math Session 4 Coordinate Geometryyoungeinstein
College Entrance Test Review
Math Session 4 Coordinate Geometry
Formulas for the Slope of a line, Midpoint, Distance between any two points,
Equations of a Line
A Quantified Approach for large Dataset Compression in Association MiningIOSR Journals
Abstract: With the rapid development of computer and information technology in the last several decades, an
enormous amount of data in science and engineering will continuously be generated in massive scale; data
compression is needed to reduce the cost and storage space. Compression and discovering association rules by
identifying relationships among sets of items in a transaction database is an important problem in Data Mining.
Finding frequent itemsets is computationally the most expensive step in association rule discovery and therefore
it has attracted significant research attention. However, existing compression algorithms are not appropriate in
data mining for large data sets. In this research a new approach is describe in which the original dataset is
sorted in lexicographical order and desired number of groups are formed to generate the quantification tables.
These quantification tables are used to generate the compressed dataset, which is more efficient algorithm for
mining complete frequent itemsets from compressed dataset. The experimental results show that the proposed
algorithm performs better when comparing it with the mining merge algorithm with different supports and
execution time.
Keywords: Apriori Algorithm, mining merge Algorithm, quantification table
Self-Medication of Anti-Biotics amongst University Students of Islamabad: Pre...IOSR Journals
The prevalence and pattern of self-medication with antibiotics among undergraduate and graduate community of students at different universities of Islamabad was evaluated using structured self-medication administered questionnaire. This cross-sectional, study was conducted in March 2013. A convenience sample was taken from 4 non-medical universities of the city of Islamabad, Pakistan. Data was analyzed using SPSS v14 and associations were tested using the Chi square test. A total of 210 questionnaires were randomly distributed with a respondent rate of 100%. The prevalence of self-medication was found to be 77.03% (Female: Male Ratio=1:1.14). The major reasons given for self-medicating with antibiotics were; 33.63% assumed knowledge on antibiotics (P=0.478), 26.64% prior experience on use (P=0.378), while 9.17% admitted lack of time to go for consultation (P=0.130). Majority of respondents however, self-medicate with antibiotics occasionally. The most reported antibiotic class (48.58%) was the β-lactams (as amoxicillin) while co-trimoxazole was rarely used (2.23%). The most reported condition for self-medication was respiratory tract infections (59%). About almost half of the respondents (46.79%) purchased the drugs from drug stores. These findings highlight the needs for planning interventions to promote the judicious use of antibiotics within the student population
Profile of Trace Elements in Selected Medicinal Plants of North East IndiaIOSR Journals
Trace elements like Mn, Fe, Cu, Zn and major elements K and Ca were quantified in ten selected medicinal plants of North East India by using Proton Induced X- ray Emission (PIXE) technique. No toxic heavy metals such as As, Hg, Pb and Cd were detected. The concentration (ppm) of the elements in the studied plants was found to be as follows: manganese(10 to1800 ), iron(27 to 836), copper(6 to140), zinc(10 to 160), potassium(14120 to 76950) and calcium(1660 to 32030). The levels of trace metals present in the plants was found to be beyond the safety standards of WHO in edible plants but around the permissible range for consumed medicinal herbs as defined for different countries.
Effect of astaxanthin on ethylene glycol induced nephrolithiasisIOSR Journals
Nephrolithiasis is one of the most common and painful of urological disorders with a high prevalence rate. The role of calcium oxalate crystals, which are the predominant component of kidney stones in generating oxidative stress, have been clearly demonstrated in previous studies. Astaxanthin, found in marine organisms is a dietary xanthophyll carotenoid with enhanced antioxidative properties and pharmacological effects. In the present study, we have investigated the effect of this natural antioxidant, at a daily dose of 25mg/kg in experimental calcium oxalate nephrolithiasis in male Wistar rats. Liver function markers, hepatic antioxidants, albumin creatinine ratios, renal calcium content and changes in body and kidney weight have been studied to evaluate the effect of this carotenoid in vivo. The effect of citrate, a component of most pharmaceutical drugs for management of nephrolithiasis has also been evaluated for the purpose of comparison with astaxanthin treatment. Astaxanthin is seen to exert a protective effect on the liver and kidney tissues in ethylene glycol treated rats by improving the liver function, restoring the activity of the hepatic antioxidant enzymes, decreasing the albumin creatinine ratios and calcium levels and maintaining the organ to body weight ratio. Our results also indicate that astaxanthin administration is more beneficial than citrate treatment
Analyzing Employee’s Heart rate using Nonlinear Cellular Automata modelIOSR Journals
Non-linear Cellular Automata model is a simulation tool which can be used to diagnosis the intensity of the disease. This paper aims to study the Heart rate behavior between normal respiratory patients and healthy controls/unhealthy controls. We also discuss about Heart Rate Variability (HRV) of employee’s through non-linear Cellular Automata model. Cellular Automata model gives us striking results for further studies
Notions via β*-open sets in topological spacesIOSR Journals
In this paper, first we define β*-open sets and β*-interior in topological spaces.J.Antony Rex Rodrigo[3] has studied the topological properties of 𝜂 * -derived, 𝜂 * -border, 𝜂 * -frontier and 𝜂 * exterior of a set using the concept of 𝜂 * -open following M.Caldas,S.Jafari and T.Noiri[5]. By the same technique the concept of β*-derived, β*-border, β*-frontier and β*exterior of a set using the concept of β*-open sets are introduced.Some interesting results that shows the relationships between these concepts are brought about
Optimal Estimating Sequence for a Hilbert Space Valued ParameterIOSR Journals
Some optimality criteria used in estimation of parameters in finite dimensional space has been extended to a separable Hilbert space. Different optimality criteria and their equivalence are established for estimating sequence rather than estimator. An illustrious example is provided with the estimation of the mean of a Gaussian process
A study of the chemical composition and the biological active components of N...IOSR Journals
Nigella Sativa (N.S.) is an annual herbaceous plant from Ranunculaceae family producing small black seeds with aromatic odor and taste. Fenugreek (Trigonella foenum-graecum L.) , belongs to the subfamily papilionacae of the family Leguminosae (bean family, Fabaceae). The plant is an aromatic herbaceous annual, widely cultivated in Mediterranean countries and Asia.
Aim:- to extract and study the biological active components of fixed oils of N.Sativa and Fenugreek seeds.
Materials and methods:Fixed oil of the N.S. and the F.S seeds were extracted and characterized using infrared spectroscopic techniques (Tensor27- PRUKER). Biological activity test was applied on the bacteria (Bacillus pumilus, E.coli, and Pseudo M.).
Results: Both studied fixed oils showed identical antimicrobial activity.
Conclusion:- this study showed an identical similarity between the active biological components of both studied materials (N.Sativa and Fenugreek seeds) in spite of their different botanical origin, leading to a matched biological activity. This finding may be useful in replacing one herbal seeds instead of the other according to their availability when applying these seeds for their known therapeutic uses.
Characterizing Erythrophleum Suaveolens Charcoal as a Viable Alternative Fuel...IOSR Journals
An experimental study was conducted to characterize erythrophleum suaveolens (Gwaska) charcoal. The test was conducted for proximate analysis (involving the determination of moisture content, ash, volatile matter and fixed carbon) and ultimate analysis (involving the determination of carbon, hydrogen, oxygen, nitrogen sulphur and calorific value) of erythrophleum suaveolens charcoal. The determined values of moisture, ash, volatile matter and fixed carbon were 0.94%, 6.13%, 6.77% and 86.16% respectively. Also the determined values of carbon, hydrogen, oxygen, nitrogen, sulphur and calorific value were 77.5%, 9%, 5.48%, 1.89%, 0.003% and 7158.6995 Kcal/Kg respectively. Therefore, the gwaska charcoal satisfies the blast furnace requirements for moisture, ash and sulphur in Nigeria. However, its volatile matter exceeds the specified limit except for Indian standard practice. The erythrophleum suaveolens charcoal’s thermal properties showed that it could compete favourably with coke and therefore can be an excellent reducing fuel for the production of iron.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
Frenet Curves and Successor Curves: Generic Parametrizations of the Helix and...Toni Menninger
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices.
This paper shows constructively how to solve the natural equations explicitly for an infinite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant helices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices.
The present paper gives for the first time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix.
The paper also, for the benefit of novices and teachers, provides a novel and generalized presentation of the theory of Frenet curves, which is not restricted to curves with positive curvature. Bishop frames are examined along with Frenet frames and Darboux frames as a useful tool in the theory of space curves. The closed curve problem receives attention as well.
Is ellipse really a section of cone. The question intrigued me for 20 odd years after leaving high school. Finally got the proof on a cremation ground. Only thereafter I came to know of Dandelin spheres. But this proof uses only bare basics within the scope of high school course of Analytical geometry.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
From Siloed Products to Connected Ecosystem: Building a Sustainable and Scala...
G0624353
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53
www.iosrjournals.org
www.iosrjournals.org 43 | Page
Combination of Cubic and Quartic Plane Curve
C.Dayanithi
Research Scholar, Cmj University, Megalaya
Abstract
The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid.
A cross-section of the set of unistochastic matrices of order three forms a deltoid.
The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid.
The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six.
The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the
Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the
curve in 1856.
The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with
vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic
to the triangle's sides.
I. Introduction
Various combinations of coefficients in the above equation give rise to various important families of curves as
listed below.
1. Bicorn curve
2. Klein quartic
3. Bullet-nose curve
4. Lemniscate of Bernoulli
5. Cartesian oval
6. Lemniscate of Gerono
7. Cassini oval
8. Lüroth quartic
9. Deltoid curve
10. Spiric section
11. Hippopede
12. Toric section
13. Kampyle of Eudoxus
14. Trott curve
II. Bicorn curve
In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational
quartic curve defined by the equation
It has two cusps and is symmetric about the y-axis.
The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real
plane, and a double point in the complex projective plane at x=0, z=0 . If we move x=0 and z=0 to the origin
2. Combination Of Cubic And Quartic Plane Curve
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substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve,
we obtain
This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i
and z=1.
Figure A transformed bicorn with a = 1
The parametric equations of a bicorn curve are:
and with
III. Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of
genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-
preserving automorphisms, and 336 automorphisms if orientation may be reversed. As such, the Klein quartic is
the Hurwitz surface of lowest possible genus; given Hurwitz's automorphisms theorem. Its (orientation-
preserving) automorphism group is isomorphic to PSL(2,7), the second-smallest non-abelian simple group. The
quartic was first described in (Klein 1878b).
Klein's quartic occurs in many branches of mathematics, in contexts including representation theory,
homology theory, octonion multiplication, Fermat's last theorem, and the Stark–Heegner theorem on imaginary
quadratic number fields of class number one; for a survey of properties.
IV. Bullet-nose curve
In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the
equation
The bullet curve has three double points in the real projective plane, at x=0 and y=0, x=0 and z=0, and y=0 and
z=0, and is therefore a unicursal (rational) curve of genus zero.
If
3. Combination Of Cubic And Quartic Plane Curve
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then
are the two branches of the bullet curve at the origin.
V. Lemniscate of Bernoulli
In geometry, the lemniscateof Bernoulli is a plane curve defined from two given points F1 and F2, known as
foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2
. The curve has a shape similar
to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is Latin for "pendant ribbon". It is a
special case of the Cassini oval and is a rational algebraic curve of degree 4.
The lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is
the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini
oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the
curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.
This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at
the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form
of Watt's linkage, with the lengths of the three bars of the linkage and the distance between its endpoints chosen
to form a crossed square.[
Its Cartesian equation is (up to translation and rotation):
In polar coordinates:
As parametric equation:
In two-center bipolar coordinates:
In rational polar coordinates:
VI. Cartesian oval
In geometry, a Cartesian oval, named after René Descartes, is a plane curve, the set of points that have
the same linear combination of distances from two fixed points.
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Let P and Q be fixed points in the plane, and let d(P,S) and d(Q,S) denote the Euclidean distances from these
points to a third variable point S. Let m and a be arbitrary real numbers. Then the Cartesian oval is the locus of
points S satisfying d(P,S) + m d(Q,S) = a. The two ovals formed by the four equations d(P,S) + m d(Q,S) = ± a
and d(P,S) − m d(Q,S) = ± a are closely related; together they form a quartic plane curve called the ovals of
Descartes.
The ovals of Descartes were first studied by René Descartes in 1637, in connection with their
applications in optics.
These curves were also studied by Newton beginning in 1664. One method of drawing certain specific
Cartesian ovals, already used by Descartes, is analogous to a standard construction of an ellipse by stretched
thread. If one stretches a thread from a pin at one focus to wrap around a pin at a second focus, and ties the free
end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval
with a 2:1 ratio between the distances from the two foci. However, Newton rejected such constructions as
insufficiently rigorous. He defined the oval as the solution to a differential equation, constructed its subnormals,
and again investigated its optical properties.
The French mathematician Michel Chasles discovered in the 19th century that, if a Cartesian oval is
defined by two points P and Q, then there is in general a third point R on the same line such that the same oval is
also defined by any pair of these three points.
The set of points (x,y) satisfying the quartic polynomial equation
[(1 - m2
)(x2
+ y2
) + 2m2
cx + a2
− m2
c2
]2
= 4a2
(x2
+ y2
),
where c is the distance between the two fixed foci P = (0, 0) and Q = (c, 0), forms two ovals, the sets
of points satisfying the two of the four equations
d(P,S) ± m d(Q,S) = a,
d(P,S) ± m d(Q,S) = −a
that have real solutions. The two ovals are generally disjoint, except in the case that P or Q belongs to them. At
least one of the two perpendiculars to PQ through points P and Q cuts this quartic curve in four real points; it
follows from this that they are necessarily nested, with at least one of the two points P and Q contained in the
interiors of both of them. For a different parametrization and resulting quartic,
VII. Lemniscate of Gerono
In algebraic geometry, the lemniscateof Gerono, or lemnicate of Huygens, or figure-eight curve, is a
plane algebraic curve of degree four and genus zero shaped like an symbol, or figure eight. It has equation
It was studied by Camille-Christophe Gerono.
Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is
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Another representation is
which reveals that this lemniscate is a special case of a lissajous figure.
The dual curve, pictured below, has therefore a somewhat different character. Its equation is
VIII. Cassini oval
A Cassini oval is a quartic plane curve defined as the set (or locus) of points in the plane such that the
product of the distances to two fixed points is constant. This is related to an ellipse, for which the sum of the
distances is constant, rather than the product. They are the special case of polynomial lemniscates when the
polynomial used has degree 2.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in 1680.
Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
Let q1 and q2 be two fixed points in the plane and let b be a constant. Then a Cassini oval with foci q1 and q2 is
defined to be the locus of points p so that the product of the distance from p to q1 and the distance from p to q2 is
b2
. That is, if we define the function dist(x,y) to be the distance from a point x to a point y, then all points p on a
Cassini oval satisfy the equation
If the foci are (a, 0) and (−a, 0), then the equation of the curve is
When expanded this becomes
The equivalent polar equation is
Form of the curve
The shape of the curve depends, up to similarity, on e=b/a. When e>1, the curve is a single, connected loop
enclosing both foci. When e<1, the curve consists of two disconnected loops, each of which contains a focus.
When e=1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double
point (specifically, a crunode) at the origin. The limiting case of a → 0 (hence e → ), in which case the foci
coincide with each other, is a circle.
The curve always has x-intercepts at ±c where c2
=a2
+b2
. When e<1 there are two additional real x-
intercepts and when e>1 there are two real y-intercepts, all other x and y-intercepts being imaginary.
The curve has double points at the circular points at infinity, in other words the curve is bicircular. These points
are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve
has a point of inflection there. From this information and Plücker's formulas it is possible to deduce the Plücker
numbers for the case e≠1: Degree = 4, Class = 8, Number of nodes = 2, Number of cusps = 0, Number of double
tangents = 8, Number of points of inflection = 12, Genus = 1.
The tangents at the circular points are given by x±iy=±a which have real points of intersection at (±a, 0). So the
foci are, in fact, foci in the sense defined by Plücker.[6]
The circular points are points of inflection so these are
triple foci. When e≠1 the curve has class eight, which implies that there should be at total of eight real foci. Six
of these have been accounted for in the two triple foci and the remaining two are at
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So the additional foci are on the x-axis when the curve has two loops and on the y-axis when the curve has a
single loop.
Curves orthogonal to the Cassini ovals: Formed when the foci of the Cassini ovals are the points (a,0) and (-
a,0), equilateral hyperbolas centered at (0,0) after a rotation around (0,0) are made to pass through the foci.
IX. Lüroth quartic
In mathematics, a Lüroth quartic is a nonsingular quartic plane curve containing the 10 vertices of a
complete pentalateral. They were introduced by Jacob Lüroth (1869). Morley (1919) showed that the Lüroth
quartics form an open subset of a degree 54 hypersurface, called the Lüroth hypersurface, in the space P14
of all
quartics. Böhning & von Bothmer (2011) proved that the moduli space of Lüroth quartics is rational.
X. Deltoid curve
In geometry, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps. In
other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along
the inside of a circle with three times its radius. It can also be defined as a similar roulette where the radius of
the outer circle is 3
⁄2 times that of the rolling circle. It is named after the Greek letter delta which it resembles.
More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave
to the exterior, making the interior points a non-convex set.
A deltoid can be represented (up to rotation and translation) by the following parametric equations
where a is the radius of the rolling circle.
In complex coordinates this becomes
.
The variable t can be eliminated from these equations to give the Cartesian equation
and is therefore a plane algebraic curve of degree four. In polar coordinates this becomes
The curve has three singularities, cusps corresponding to . The parameterization above implies
that the curve is rational which implies it has genus zero.
A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency
travels around the deltoid twice while each end travels around it once.
The dual curve of the deltoid is
which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy,
giving the curve
with a double point at the origin of the real plane.
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XI. Spiric section
In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by
equations of the form
.
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Equivalently, spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x
and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and
the family of Cassini ovals. The name is from ζπειρα meaning torus in ancient Greek.
A spiric section is sometimes defined as the curve of intersection of a torus and a plane parallel to its rotational
symmetry axis. However, this definition does not include all of the curves given by the previous definition
unless imaginary planes are allowed.
Spiric sections were first described by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed
to be the first toric sections to be described.
Equations
Start with the usual equation for the torus:
.
Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of
intersection gives
.
In this formula, the torus is formed by rotating a circle of radius a with its center following another circle of
radius b (not necessarily larger than a, self-intersection is permitted). The parameter c is the distance from the
intersecting plane to the axis of revolution. There are no spiric sections with c>b + a, since there is no
intersection; the plane is too far away from the torus to intersect it.
Expanding the equation gives the form seen in the definition
where
.
In polar coordinates this becomes
or
XII. Hippopede
In geometry, a hippopede (from ἱπποπέδη meaning "horse fetter" in ancient Greek) is a plane curve
determined by an equation of the form
,
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where it is assumed that c>0 and c>d since the remaining cases either reduce to a single point or can be put into
the given form with a rotation. Hippopedes are bicircular rational algebraic curves of degree 4 and symmetric
with respect to both the x and y axes. When d>0 the curve has an oval form and is often known as an oval of
Booth, and when d<0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a
lemniscate of Booth, after James Booth (1810–1878) who studied them. Hippopedes were also investigated by
Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For d = −c, the hippopede
corresponds to the lemniscate of Bernoulli.
Definition
Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the
plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in
turn is a type of toric section.
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If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting
hippopede in polar coordinates
or in Cartesian coordinates
.
Note that when a>b the torus intersects itself, so it does not resemble the usual picture of a torus.
XIII. Toric section
A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a
plane with a cone.
In general, toric sections are fourth-order (quartic) plane curves of the form
Spiric sections
A special case of a toric section is the spiric section, in which the intersecting plane is parallel to the
rotational symmetry axis of the torus. They were discovered by the ancient Greek geometer Perseus in roughly
150 BC. Well-known examples include the hippopede and the Cassini oval and their relatives, such as the
lemniscate of Bernoulli.
Villarceau circles
Another special case is the Villarceau circles, in which the intersection is a circle despite the lack of
any of the obvious sorts of symmetry that would entail a circular cross-section.
General toric sections
More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or
oblique to the rotational symmetry axis.
XIV. Kampyle of Eudoxus
The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a
curve, with a Cartesian equation of
from which the solution x = y = 0 should be excluded, or, in polar coordinates,
This quartic curve was studied by the Greek astronomer and mathematician Eudoxus of Cnidus (c. 408 BC –
c.347 BC) in relation to the classical problem of doubling the cube.
The Kampyle is symmetric about both the - and -axes. It crosses the -axis at and . It has
inflection points at
(four inflections, one in each quadrant). The top half of the curve is asymptotic to as , and
in fact can be written as
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where
is the th Catalan number.
XV. Trott curve
In real algebraic geometry, a general quartic plane curve has 28 bitangent lines, lines that are tangent to
the curve in two places. These lines exist in the complex projective plane, but it is possible to define curves for
which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane.
An explicit quartic with twenty-eight real bitangents was first given by Plücker (1839) As Plücker showed, the
number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real
bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel
lines Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting
a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at
infinity in the projective plane.
The Trott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying the degree four
polynomial equation
These points form a nonsingular quartic curve that has genus three and that has twenty-eight real
bitangents.
Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum
number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different
pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate
the two ovals, and two that do not. Additionally, each oval bounds a nonconvex region of the plane and has one
bitangent spanning the nonconvex portion of its boundary.
XVI. Conclusion
The dual curve to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the
primal curve.
The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form
where a, b, c, d, e and f are all zero or one and where
ad + be + ef = 1 (mod 2).
There are 64 choices for a, b, c, d, e and f, but only 28 of these choices produce an odd sum. One may
also interpret a, b, and c as the homogeneous coordinates of a point of the Fano plane and d, e, and f as the
coordinates of a line in the same finite projective plane; the condition that the sum is odd is equivalent to
requiring that the point and the line do not touch each other, and there are 28 different pairs of a point and a line
that do not touch.
The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a
triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of
the Fano plane. The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane
are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the
Coxeter graph
The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface, and to the
28 odd theta characteristics.
The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a
canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold, specifically a form of McKay
correspondence, and can be related to many further objects, including E7 and E8, as discussed at trinities.