Fact-checked by Grok 2 weeks ago

Conjunction

A conjunction is a term used in various fields. In , it is a that connects words, phrases, or clauses to form more complex sentences and express relationships such as addition, contrast, or cause. In astronomy, a conjunction occurs when two or more celestial bodies appear close together in the sky from an observer's on . In logic, conjunction is a (often symbolized as ∧ or "and") that is true only if both operands are true. The term also has applications in other areas, such as and , as detailed in subsequent sections.

Etymology and Overview

Origin of the Term

The word "conjunction" derives from the Latin coniunctiō, meaning "a joining together" or "," which stems from the verb coniungō (to join or unite), combining con- (together) with iungō (to join). This root emphasized the act of connecting elements, whether physical, conceptual, or relational, and was widely employed in to denote various forms of linkage. The term evolved through conjunction, borrowed directly from Latin, entering around the late 14th century, where it first appeared in contexts related to and . In medieval manuscripts and early English texts, it retained its core sense of , often applied to the alignment of bodies or the syntactic of words in sentences. Classical Roman authors influenced its rhetorical application, embedding the concept within the traditions of oratory and logic. This classical usage bridged linguistic and intellectual connections, prefiguring the term's adoption in English scholarly discourse. Chaucer's works from around 1385, such as The Complaint of Mars, depict the astronomical meeting of planets like Mars and Venus, illustrating the term's early astrological significance in late medieval English poetry, as one of its earliest notable appearances in English literature. Across disciplines, this foundational meaning of connection persists in grammar, astronomy, and logic.

Broad Applications

Conjunction denotes the state of being joined, combined, or occurring together, encompassing both physical and abstract unions. According to , it is defined as "the act or an instance of conjoining: the state of being conjoined," applicable to various forms of connection or coincidence. The Oxford Learner's Dictionary similarly describes it as "a of events, etc., that happen at the same time; the act of combining events, etc., that happen at the same time," emphasizing its broad applicability beyond specific contexts. Across disciplines, conjunction embodies the theme of or linkage, facilitating the of elements that might otherwise remain separate. In everyday , phrases like "in conjunction with" illustrate this by denoting or collaborative occurrence, as in events planned together for mutual effect. This interdisciplinary utility positions conjunction as a foundational for understanding relational dynamics, from collaborative efforts in practical settings to conceptual mergers in intellectual pursuits, without prescribing the method of . The term's usage has evolved historically from literal interpretations of tangible joining to more abstract significances in scientific and philosophical realms. Deriving from Latin coniunctio, meaning "a joining together," it initially connoted unions such as marriages or physical combinations in the late 14th century. In medieval , for instance, coniunctio symbolized the essential union of opposites, like disparate substances merged to achieve transformation, marking a shift toward symbolic processes. By the , this progression enabled conjunction's adoption in diverse abstract frameworks, reflecting advancements in thought that prioritize relational and coincidental structures over mere physical adjacency.

Grammar

Coordinating Conjunctions

Coordinating conjunctions are words that connect elements of equal grammatical rank, such as words, phrases, or independent clauses, within a sentence. The seven primary coordinating conjunctions in English are for, and, nor, but, or, yet, and so. These conjunctions ensure syntactic balance by linking items that share the same function, distinguishing them from subordinating conjunctions that introduce dependent clauses. The FANBOYS—derived from the first letters of for, and, nor, but, or, yet, and so—serves as a mnemonic device to aid in memorization, particularly in educational settings like writing centers and courses. This tool is widely used in to help students identify and apply coordinating conjunctions when constructing compound sentences or lists. Usage rules for coordinating conjunctions emphasize maintaining parallel structure, where connected elements must follow the same grammatical form to avoid imbalance. For instance, when joining two independent clauses, a precedes the conjunction to separate them clearly. No comma is needed when linking words or phrases of equal weight within a series. Examples illustrate these principles: "I like tea and coffee" connects two nouns of equal rank without a comma. In contrast, "She ran, but he walked" joins two independent clauses, requiring a comma before the conjunction. These patterns trace back to Old English equivalents, such as ond for "and," ac for "but," and oþþe for "or," which evolved amid shifts from inflected, flexible word order to Modern English's fixed syntax and expanded conjunctive roles.

Subordinating and Correlative Conjunctions

Subordinating conjunctions are words or phrases that introduce s, connecting them to independent clauses to express relationships such as time, cause, condition, or concession. These conjunctions create complex s by subordinating one idea to another, making the dependent clause unable to stand alone as a complete . Unlike coordinating conjunctions, which link clauses of equal importance, subordinating conjunctions establish a hierarchical structure. Subordinating conjunctions are categorized based on the logical relationships they signal. Temporal conjunctions indicate time, such as after, before, when, while, until, and since; for example, "The boy prayed as the ground shook." Causal conjunctions denote reason or cause, including because, since, as, and in order that; an example is "Johnny is skinny because he doesn’t work out." Concessive conjunctions express contrast or concession, like although, though, even though, and while; for instance, "I could not outrun the werewolf, although I tried." Conditional conjunctions specify conditions, such as if, unless, provided that, and even if. Place-related ones include where and wherever. Common errors with subordinating conjunctions include creating sentence fragments by placing the dependent clause alone, such as "Although I tried," or run-on sentences without proper punctuation when combining clauses. Correlative conjunctions consist of paired words or phrases that work together to connect elements of equal grammatical weight, often emphasizing balance or alternatives in a sentence. They require parallel structure in the connected elements to maintain clarity and correctness. Common correlative pairs include both...and, which links two similar ideas, as in "Both trees and flowers grow colorfully during summer"; either...or, for alternatives, like "I will either go for a walk or play "; and neither...nor, to negate both options, for example, "It was neither rainy nor foggy outside." Other pairs are not only...but also, emphasizing addition, such as "Her report was not only detailed but also well supported"; whether...or, presenting choices, as in "Have they decided whether they will come with us or stay home?"; and as...as, for comparisons, like " isn’t as fun as ping pong." Additional pairs include no sooner...than and just as...so. Misuse of correlative conjunctions often involves failing to pair them correctly or violating parallelism, such as saying "Neither...or" instead of "Neither...nor," or mixing structures like "Either you will do your homework, or failing the class." Another frequent error is inserting unnecessary commas between the paired elements, as in "Either I get out of the sun, or it’s going to start smelling like ," which should omit the . These errors can disrupt sentence flow and logical emphasis.

Astronomy

Definition and Observation

In astronomy, a conjunction is defined as the alignment of two celestial bodies, such as , stars, or the , such that they share the same (in the ) or ecliptic (in the system), resulting in their apparent proximity as viewed from . This event creates the illusion of closeness in the sky, even though the bodies may be separated by vast distances in space, with the term deriving from the Latin conjunctio, meaning "a joining together." From a geocentric perspective, which centers on 's position, conjunctions are apparent alignments based on the observer's , whereas heliocentric views consider actual orbital positions relative to , revealing that true physical alignments are rare due to differing orbital inclinations and speeds. For instance, an inferior conjunction with an inner planet like occurs when both and the planet lie on the same side of in their orbits, but the geocentric view emphasizes the angular separation as seen from our planet. Observing conjunctions involves measuring the angular separation between the , typically considered close if less than 1° (about twice the apparent diameter of the ), using telescopes or to resolve finer details and enhance visibility of fainter objects. Modern software tools like Stellarium or NASA's Solar System Dynamics ephemeris calculators predict conjunction timings by simulating , allowing astronomers to plan observations based on precise orbital data. Historically, ancient astronomers employed instruments such as astrolabes to measure altitudes and azimuths for detecting conjunctions, which were crucial for calendar-making and . Visibility of conjunctions depends on several factors, including the intrinsic brightness () of the involved bodies, their elongation from to avoid glare during twilight, and local atmospheric conditions like clarity and . For example, a conjunction involving and is often spectacular due to their high brightness, but if occurring near , it may require pre-dawn or post-sunset viewing under clear skies to be observable to the . Professional observatories use (CCD) cameras for quantitative imaging, capturing data on separation and color contrasts to study relative motions.

Types of Astronomical Conjunctions

Astronomical conjunctions are classified primarily based on the relative positions of the planets involved with respect to Earth and the Sun. For inferior planets—Mercury and Venus, which orbit closer to the Sun than Earth—these events include inferior conjunctions, where the planet passes between Earth and the Sun, and superior conjunctions, where the planet is aligned on the opposite side of the Sun from Earth. During an inferior conjunction, the inner planet's disk may transit the Sun if the alignment is precise, though this is rare and requires specific orbital inclinations. Superior conjunctions for inferior planets occur when they are maximally distant from Earth, hidden behind the solar disk. For superior planets—those orbiting beyond Earth, such as Mars, , Saturn, , and —only superior conjunctions are possible, as these bodies can never pass between Earth and the Sun. In a superior conjunction, the outer planet aligns with the Sun from Earth's perspective, appearing to pass behind it, which temporarily makes the planet unobservable due to solar glare. These events mark the points in the outer planet's synodic cycle when it shifts from evening to morning sky visibility. Related to conjunctions but distinct are appulses, which describe the closest apparent approach of two celestial bodies in the sky without requiring exact alignment in or longitude; appulses often occur near conjunctions but emphasize minimum separation rather than longitudinal equality. Special cases of conjunctions highlight notable configurations involving multiple bodies. A great conjunction refers specifically to the alignment of and Saturn, occurring approximately every 20 years due to their differing orbital periods—Jupiter's 11.86 years and Saturn's 29.46 years—allowing Jupiter to lap Saturn periodically in Earth's view. The most recent great conjunction took place on December 21, 2020, when the planets appeared just 0.1 degrees apart, the closest since 1623, with the next occurring on November 4, 2040. Triple conjunctions involve three bodies aligning closely over a short period, often resulting from the relative motions during a superior planet's passage relative to inner ones; types include those between two inferior (rarely fully visible), an inferior and a superior planet (about every 10 years), or two superior (also roughly decadal). A more recent example is the rare triple conjunction of Saturn and Neptune in 2025 (encounters on May 20, August 6, and October 24), though Neptune's faintness (magnitude ~7.8) limits naked-eye visibility; spectacular alignments, like the 'planetary parade' of June 24, 2022, with Mercury, Venus, Mars, Jupiter, and Saturn visible in the morning sky, also highlight multiple close approaches. Predictions of conjunction intervals rely on basic , particularly Kepler's third law, which relates a 's orbital period P to its semi-major axis a via P^2 \propto a^3, enabling computation of sidereal periods from distances. These periods inform the synodic period S between (period E = 1 year) and another (period P), calculated as $1/S = |1/E - 1/P| for the time between successive conjunctions or oppositions; for superior planets, S = 1 / (1/E - 1/P). This formula, derived from relative angular velocities, allows astronomers to forecast events like Mercury's conjunctions every 116 days or Jupiter's every 399 days.

Logic

The Logical Operator

In logic, conjunction is a fundamental binary connective that combines two propositions into a single compound proposition, which is true if and only if both component propositions are true. This truth-functional operation, often denoted as AND, evaluates to false in all other cases, capturing the intuitive notion of simultaneous truth in both inputs. The standard symbolic representation of conjunction is the wedge symbol \wedge, though variations include the dot \cdot and the ampersand & , particularly in programming and some algebraic contexts. These notations emphasize conjunction's role as a multiplicative-like operation in logical systems, distinct from natural language ambiguities. The concept of conjunction as a propositional connective originates in Stoic logic, the earliest system of propositional logic developed in ancient Greece. It was formalized as a binary operation in modern terms by George Boole in his 1854 treatise An Investigation of the Laws of Thought, which laid the foundations of Boolean algebra and propositional logic. For example, the proposition "It is raining \wedge the ground is wet" holds true only when both conditions are satisfied simultaneously. This logical use shares a conceptual similarity with grammatical conjunctions that link clauses, though it prioritizes truth values over syntax.

Properties and Applications

The truth table for the logical conjunction P \land Q defines its semantics, where the output is true only if both inputs are true:
PQP \land Q
TrueTrueTrue
TrueFalseFalse
FalseTrueFalse
FalseFalseFalse
Logical conjunction satisfies several fundamental algebraic properties in Boolean algebra. It is commutative, meaning P \land Q \equiv Q \land P. It is also associative, so (P \land Q) \land R \equiv P \land (Q \land R). Conjunction is idempotent, with P \land P \equiv P. Additionally, it distributes over disjunction: P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R). A related property from De Morgan's laws states that the negation of a conjunction is equivalent to the disjunction of the negations: \neg (P \land Q) \equiv \neg P \lor \neg Q. In , conjunction underpins the bitwise AND operation, which applies the bit by bit to binary representations of integers for tasks like masking and permission checks. It also appears in conditional statements, where the logical AND ensures multiple conditions must all hold true before executing a code block. In set theory, conjunction corresponds to the operation A \cap B, which collects elements belonging to both sets A and B, mirroring the requirement that both propositions hold. In proof theory, the conjunction introduction rule permits deriving P \land Q directly from separate premises establishing P and Q, facilitating the combination of independent proofs.

Other Uses

In Astrology

In astrology, a conjunction occurs when two or more planets are aligned within an of approximately 8-10 degrees, causing their energies to blend and intensify, often amplifying the characteristics of the involved celestial bodies. This alignment is considered one of the most potent aspects, as it fuses the influences of the planets without separation, potentially leading to a unified expression of their qualities in a . The concept of conjunctions traces its origins to ancient and around the second millennium BCE, where scholars initially focused on planetary alignments, particularly conjunctions with , as omens for earthly events. This practice evolved in , with Ptolemy's Tetrabiblos (2nd century ) classifying conjunctions as "bodily" relations when planets share the same , integrating them into broader systems of aspects and signs to interpret influences on human affairs. Interpretations of conjunctions vary based on the planets involved: benefic planets like and in conjunction often signify harmony, expansion, and good fortune, such as opportunities for growth or relationships, while malefic planets like Mars and Saturn can indicate tension, obstacles, or conflict due to their clashing energies. In charts, conjunctions reflect inherent personality traits or life themes formed at birth, blending the planets' influences permanently, whereas conjunctions—when current planetary positions align with natal points—trigger temporary events or activations, often bringing focus to specific areas of life. A notable example is the Jupiter-Saturn conjunction on December 21, 2020, known as the "" or "," which occurred in Aquarius and was interpreted astrologically as a pivotal shift toward , social reform, and the dawn of the Age of Aquarius, influencing global themes of collective change and technological advancement.

References

  1. [1]
  2. [2]
    What Are Conjunctions? Definition and Examples - Grammarly
    Jan 15, 2025 · A conjunction is a word that links other words, phrases, or clauses together. There are three main types of conjunctions: coordinating, correlative, and ...Coordinating Conjunctions · Subordinating conjunctions · Correlative conjunctions
  3. [3]
    Using Conjunctions | Definition, Rules & Examples - Scribbr
    Rating 4.0 (3,676) Oct 23, 2022 · A conjunction is a word that is used to connect words, phrases, and clauses. There are many conjunctions in the English language, but some common ones include ...
  4. [4]
    Conjunction - Etymology, Origin & Meaning
    Originating from late 14th-century Old French and Latin, "conjunction" means a joining or meeting, especially of individuals or celestial bodies in the sky.
  5. [5]
    conjunction: meaning, translation - WordSense Dictionary
    Origin & history. From Old French conjonction, from Latin coniūnctiō ("joining"), from coniungere ("to join"). Pronunciation. IPA ...
  6. [6]
  7. [7]
    Conjunction Definition & Meaning | YourDictionary
    Origin of Conjunction​​ From Old French conjonction, from Latin coniūnctiō (“joining”), from coniungere (“to join”).Missing: etymology | Show results with:etymology
  8. [8]
    The eight parts | What Graeco-Roman Grammar Was About
    Qualitas had been coined by Cicero ... The term too was transparent: in Greek a sundesmos or 'binding together'; in Latin a conjunctio or 'conjoining'.
  9. [9]
    Forms and Celestial Motion in Chaucer's "Complaint of Mars" - jstor
    without the presence of an observer applying this standard to celestial motion, a conjunc- tion has no distinct standing among every other configuration of the ...
  10. [10]
    Forms and Celestial Motion in Chaucer's Complaint of Mars | PMLA
    Oct 23, 2020 · Chaucer's short poem The Complaint of Mars (c. 1385) presents a conjunction between the planets Mars and Venus as if it were a love affair.Missing: alignments | Show results with:alignments
  11. [11]
    A Library Guide to Jung's Collected Works: Coniunctio
    Coniunctio: "An alchemical symbol of a union of unlike substances; a marrying of the OPPOSITES in an intercourse which has as its fruition the birth of a ...
  12. [12]
    Coordinating conjunctions - Graduate Writing Center
    English has seven coordinating conjunctions—for, and, nor, but, or, yet, so—which you can remember using the mnemonic FANBOYS: For indicates causation: “We left ...
  13. [13]
    Coordinating Conjunctions - Excelsior OWL
    Coordinating conjunctions coordinate or join two equal parts. They are particularly important because, when used with a comma, they can actually connect.Missing: examples | Show results with:examples
  14. [14]
    [PDF] FANBOYS.pdf - Allan Hancock College
    Using the word FANBOYS makes it easy to remember the coordinating conjunctions. When using a fanboy to connect two sentences, a comma goes before the fanboy.
  15. [15]
    [PDF] Coordinating Conjunctions
    A conjunction is the glue that holds words, phrases and clauses (both dependent and independent) together. A coordinating conjunction is a conjunction that ...
  16. [16]
    Parallel Structure - Purdue OWL
    The usual way to join parallel structures is with the use of coordinating conjunctions such as "and" or "or." Words and Phrases. With the -ing form (gerund) ...Missing: comma | Show results with:comma
  17. [17]
    Comma Rules | Style and Grammar | Academic Writing
    Comma Rules · Use commas to separate independent clauses when they are joined by any of these seven coordinating conjunctions: and, but, for, or, nor, so, yet.
  18. [18]
    Commas - Towson University
    Remember to place a comma before the coordinating conjunction joining the last two elements of the series. Otherwise, the intended meaning of the sentence may ...
  19. [19]
    None
    ### Summary of Subordinating Conjunctions from https://www.sjsu.edu/writingcenter/docs/handouts/Subordinating%20Conjunctions.pdf
  20. [20]
    Subordinating Conjunctions - Excelsior OWL
    Subordinating conjunctions connect unequal parts, making a phrase subordinate. Common examples include 'after', 'although', 'because', 'before', 'even though', ...Missing: categories | Show results with:categories
  21. [21]
    Correlative Conjunctions - Grossmont College
    Some conjunctions combine with other words to form what are called correlative conjunctions. They always occur as a pair of phrases, joining various sentence ...Missing: definition | Show results with:definition
  22. [22]
    [PDF] Correlative Conjunctions
    List of common correlative conjunctions: ○ both . . . and ■ Example: “​Both​ trees ​and​ flowers grow colorfully during summer.” ○ not only . . . but also ■ ...
  23. [23]
    Grammar and Punctuation | University of West Florida
    Note: Correlative conjunctions must be used as pairs (not only with but also, not just but) and not mixed with other conjunctions such as neither with or.
  24. [24]
    Unnecessary Comma | 20 Most Common Writing Errors
    Unnecessary commas often result from misapplying rules, such as with coordinating conjunctions, correlative conjunctions (except if/then), and non-essential ...
  25. [25]
    What is a planetary conjunction?
    Jan 18, 2024 · When the innermost planets (Mercury and Venus) are on the far side of the Sun it is called superior conjunction. When they are between the Earth ...
  26. [26]
    Appulse - In-The-Sky.org
    An appulse is when two or more astronomical objects – usually the planets or the Moon – appear close to one another in the sky.
  27. [27]
    The 'Great' Conjunction of Jupiter and Saturn - NASA
    Dec 15, 2020 · An especially vibrant planetary conjunction easily visible in the evening sky over the next two weeks as the bright planets Jupiter and Saturn come together.
  28. [28]
    Planetary Conjunction 2025 | Planets Close - Star Walk
    There are three main types of triple conjunctions: Between inferior planets*: These occur almost every time Venus passes behind the Sun (superior conjunction), ...What is a planetary conjunction? · What is a triple conjunction?
  29. [29]
    Orbits and Kepler's Laws - NASA Science
    May 21, 2024 · Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit.
  30. [30]
    Planetary Opposition and Conjunction - Forsyth Astronomical Society
    Oct 16, 2022 · Superior Conjunction is when the planet is on the opposite side of the Sun from the Earth (0o elongation). · Inferior Conjunction is when the ...
  31. [31]
  32. [32]
    Sentence Connectives in Formal Logic
    May 4, 2010 · Sentence connectives in formal logic simulate natural language sentence-linking words like 'and' and 'or', and are fundamental operations in ...Sequents and Valuations · Rules and Connectives · Selected Existence Questions
  33. [33]
    George Boole - Stanford Encyclopedia of Philosophy
    Apr 21, 2010 · George Boole (1815–1864) was an English mathematician and a founder of the algebraic tradition in logic. He worked as a schoolmaster in ...The Context and Background... · The Mathematical Analysis of... · Boole's Methods
  34. [34]
    The Mathematics of Boolean Algebra
    Jul 5, 2002 · The Boolean operations then express the ordinary truth tables for disjunction (with +), conjunction (with ·) and negation (with −). An ...
  35. [35]
    Bitwise Operators in Python
    Jul 28, 2025 · You can use bitwise operators to implement algorithms such as compression, encryption, and error detection as well as to control physical devices.
  36. [36]
    Conditionals with if/else & Booleans | AP CSP (article)
    The condition is a Boolean expression: an expression that evaluates to either true or false . Boolean values are another type of data type in programming ...Conditionals With If, Else... · A Simple Conditional · Comparison Operators
  37. [37]
    [PDF] Logic and Sets
    The intersection of two sets A and B, written A ∩ B, is the set of ele- ments contained in both sets. A ∩ B = {x : x ∈ A and x ∈ B}. Note the relation between ...
  38. [38]
    Proof-Theoretic Semantics - Stanford Encyclopedia of Philosophy
    Dec 5, 2012 · For example, when principles of harmony are discussed, and one considers the standard introduction rule for conjunction A B A ∧ B.
  39. [39]
    Aspects and Orbs - Understanding the Planetary Interplay
    In a conjunction, the two or more planets are very close together - up to 10 degrees apart. Planets in conjunction fuse their energies together into a complete, ...
  40. [40]
    Astrological Aspects, Their Orb and Meaning - Astrotheme
    As indicated by its name, it is formed when two planets are on the same degree, or when they are very close to each other. The conjunction takes a positive or ...
  41. [41]
  42. [42]
    Conjunction Aspect in Astrology: Understanding the Most Powerful ...
    They observed that conjunctions of benefic planets like Jupiter and Venus created opportunities and blessings, while conjunctions involving malefic planets like ...Conjunction Aspect: The... · Solar Conjunctions And Their... · Lunar Conjunctions And...
  43. [43]
    The Meaning of the Aspects in Astrology
    The Conjunction: Planets and points that form a conjunction are energies that are united. They are blended; therefore, they act together.
  44. [44]
  45. [45]
    Intersection in Math | Definition, Symbol & Examples - Study.com
    The intersection of two mathematical objects is where they overlap. For geometric objects, the intersection is a point or set of points where the objects cross.
  46. [46]
    Intersection Points - (Honors Geometry) - Fiveable
    Intersection points are the specific coordinates where two or more geometric figures, such as lines, curves, or shapes, meet or cross each other.
  47. [47]
    How to Calculate and Interpret Intersection Points - House of Math
    Intersection point is a point where two graphs meets each other in the same coordinate system. Learn how to find the points where functions intersect.Missing: curves analytic simultaneous
  48. [48]
    3.2: Intersections - Mathematics LibreTexts
    May 9, 2021 · In this section, we will address three different methods for finding the points of intersection for two graphs.
  49. [49]
    Transversal Intersection -- from Wolfram MathWorld
    Intersection in homology only makes sense because an intersection can be made to be transversal. is positive or negative.
  50. [50]
    Tangential intersection - (Honors Geometry) - Fiveable
    Definition. A tangential intersection occurs when a line or curve touches a circle at exactly one point, known as the point of tangency, without crossing it ...
  51. [51]
    Differential geometry of transversal intersection curves of ...
    In transversal intersection, the normals of the surfaces at the intersection point are linearly independent, while as in nontransversal intersection, the ...
  52. [52]
    Disjunction and conjunction in Euclid's elements - Persée
    A straight line meeting parallel straight lines makes both the alternate angles equal to one another and the external one equal to the internal and opposed ...
  53. [53]
    Conjunction and Disjunction in Euclid's Elements - Academia.edu
    ABSTRACT : A survey is presented of the several ways disjunctive and conjunctive connectors are employed in the Elements. The presence of prepositive ētoi in ...Cite This Paper · Faqs · References (19)
  54. [54]
    [PDF] Clipping and Intersection - UT Computer Science
    Line clipping: Remove portion of line segment outside window. • Line segments can straddle the region boundary. • The Liang-Barsky algorithm efficiently clips ...
  55. [55]
    [PDF] Bézout's Theorem: A taste of algebraic geometry
    One of the “Great Theorems” in algebraic geometry is Bézout's Theorem, which explains the intersections of polynomial curves in the (projective) plane.
  56. [56]
    Bézout's Theorem -- from Wolfram MathWorld
    Bézout's theorem for curves states that, in general, two algebraic curves of degrees m and n intersect in m·n points and cannot meet in more than m·n points.