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Conic section

A conic section is a curve formed by the intersection of a plane with a right circular cone, resulting in nondegenerate cases such as the circle, ellipse, parabola, or hyperbola.^[1] These curves arise when the plane intersects one or both nappes of the double cone, with the specific type determined by the angle of intersection relative to the cone's vertex angle./11:Parametric_Equations_and_Polar_Coordinates/11.05:Conic_Sections) The four primary types of conic sections are distinguished by their geometric properties and the eccentricity e, a measure of how much the curve deviates from a circle: the circle has e = [0](/page/0), defined as the set of points equidistant from a center; the ellipse has $0 < e < 1, consisting of points where the sum of distances to two foci is constant; the parabola has e = 1, formed by points equidistant from a focus and a directrix line; and the hyperbola has e > 1, where the difference of distances to two foci is constant./09:Curves_in_the_Plane/9.01:Conic_Sections) All conic sections can be represented by the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where the coefficients determine the type through the discriminant B^2 - 4AC.^[1] Conic sections have been studied since ancient times, with the Greek mathematician Apollonius of Perga providing the first systematic treatment in his eight-volume work Conics around 200 BCE, where he coined the modern names for the ellipse, parabola, and hyperbola and explored their properties geometrically.^[2] Initially pursued for pure mathematical interest, their significance expanded in the 17th century with applications to planetary orbits under Kepler's laws and Newton's inverse-square law of gravitation, unifying celestial mechanics.^[1] Today, conic sections find practical uses in physics, engineering, and optics, such as modeling satellite trajectories, designing parabolic reflectors, and analyzing elliptical paths in astronomy./11:Parametric_Equations_and_Polar_Coordinates/11.05:Conic_Sections)

Euclidean Geometry

Definition

A conic section is a curve formed by the intersection of a plane with the surface of a right circular double cone, consisting of two nappes joined at a common vertex.^[3] The type of curve depends on the orientation of the plane relative to the cone's axis and generators: a plane perpendicular to the axis yields a circle, a special ellipse; a plane tilted at an angle less than the cone's semi-vertical angle produces an ellipse; a plane parallel to a generator results in a parabola; and a plane intersecting both nappes at an angle steeper than the generator forms a hyperbola.^[3] These intersections provide a geometric origin for the primary non-degenerate conic sections, excluding cases where the plane passes through the vertex to produce degenerate forms like a point or lines.^[3] The discovery of conic sections as cone slices is attributed to the ancient Greek mathematician Menaechmus around 350 BC, who identified them while investigating methods to duplicate the cube.^[4] Conic sections are alternatively defined as the locus of points in a plane where the ratio of the distance from a fixed point, called the focus, to the distance from a fixed line, called the directrix, remains constant; this constant ratio is the eccentricity e.^[3] For sections derived from a double cone, this focus-directrix property is rigorously established using Dandelin spheres—inscribed spheres tangent to the intersecting plane and to the cone along circles—where the points of tangency on the plane correspond to the foci, and the distance ratios align with the cone's geometry. Conic sections are classified by their eccentricity e: a circle has e = 0; an ellipse has $0 < e < 1; a parabola has e = 1; and a hyperbola has e > 1.^[3] This parameter quantifies the deviation from circularity and unifies the geometric and locus definitions across all types.^[3]

Eccentricity, Focus, and Directrix

A conic section can be defined as the locus of points P in a plane such that the ratio of the distance from P to a fixed point F (the focus) to the distance from P to a fixed line D (the directrix) is a constant value e, known as the eccentricity.^[5] This focus-directrix property unifies the ellipse, parabola, and hyperbola, distinguishing them by the value of e: $0 \leq e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola.^[6] The eccentricity quantifies the "elongation" or deviation from a circle, with e = 0 corresponding to a circle as a special ellipse.^[7] The focus-directrix characterization arises geometrically from the intersection of a plane with a right circular cone, as demonstrated by the Dandelin spheres construction introduced by Germinal Pierre Dandelin in 1822.^[8] For an ellipse, two spheres are inscribed tangent to the cone along circles and to the intersecting plane at two distinct points, which become the foci; the directrices are the lines where the plane intersects the planes of these tangent circles.^[9] Along each generating line of the cone, the points of tangency with the spheres satisfy the constant sum of distances to the foci equal to the distance between the tangent circle planes. The eccentricity is determined by the cone's semi-vertical angle and the orientation of the intersecting plane. For a hyperbola, a similar construction uses two spheres, one on each nappe, yielding foci and directrices with the constant difference of distances, and the eccentricity likewise depends on the cone and plane geometry.^[8] In the parabolic case, a single sphere suffices, tangent at the focus, with the directrix as the intersection line, resulting in e = 1.^[5] In standard forms aligned with the major or transverse axis, the eccentricity for an ellipse is given by e = c/a, where a is the semi-major axis and c = \sqrt{a^2 - b^2} is the distance from the center to each focus (with b the semi-minor axis).^[7] For a hyperbola, e = c/a > 1, where a is the semi-transverse axis and c = \sqrt{a^2 + b^2} (with b the semi-conjugate axis).^[7] A parabola has e = 1 by definition, with the focus at a distance p (the focal length or parameter) from the vertex. The corresponding directrices, assuming horizontal orientation, are x = \pm a/e for the ellipse and hyperbola (two lines symmetric about the center), and x = -p for the parabola.^[10]

Standard Forms in Cartesian Coordinates

The standard forms of conic sections in Cartesian coordinates assume the curves are aligned with the coordinate axes, with centers or vertices translated from the origin as needed. These forms provide simplified algebraic representations for ellipses, parabolas, and hyperbolas, facilitating analysis of their geometric properties.^[7] For an ellipse centered at (h, k) with its major axis parallel to the x-axis, the equation is

\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,

where a > b > 0, a is the length of the semi-major axis, and b is the length of the semi-minor axis.^[11] If the major axis is parallel to the y-axis, the equation becomes

\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1,

with a > b > 0. The distance from the center to each focus is given by c = \sqrt{a^2 - b^2}, and the eccentricity is e = c/a < 1.^[12] A circle is a special case of the ellipse where a = b = r, yielding the equation

(x - h)^2 + (y - k)^2 = r^2,

with (h, k) as the center and r > 0 as the radius; here, the eccentricity is zero.^[13] The standard form for a parabola with vertex at (h, k) and axis parallel to the x-axis (opening right if p > 0) is

(y - k)^2 = 4p(x - h),

where p \neq 0 is the focal parameter, representing the distance from the vertex to the focus. For the axis parallel to the y-axis (opening up if p > 0), it is

(x - h)^2 = 4p(y - k).

The focus is at (h + p, k) and the directrix is the line x = h - p for the horizontal case.^[7] This parabolic form derives from the geometric definition: the set of points equidistant from a fixed focus and directrix. Consider the standard parabola y^2 = 4px with vertex at the origin, focus at (p, 0), and directrix x = -p. For a point (x, y) on the curve, the distance to the focus equals the distance to the directrix:

\sqrt{(x - p)^2 + y^2} = |x + p|.

Squaring both sides yields

(x - p)^2 + y^2 = (x + p)^2,

which simplifies to

x^2 - 2px + p^2 + y^2 = x^2 + 2px + p^2 \implies y^2 = 4px.

Translations and reflections produce the general axis-aligned forms.^[12] For a hyperbola centered at (h, k) with transverse axis parallel to the x-axis, the equation is

\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1,

where a > 0, b > 0, a is half the length of the transverse axis, and b relates to the asymptotes. If the transverse axis is parallel to the y-axis, it is

\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.

The foci are at a distance c = \sqrt{a^2 + b^2} from the center, with eccentricity e = c/a > 1.^[7]

General Cartesian Form

The general Cartesian form of a conic section is given by the second-degree equation

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,

where A, B, C, D, E, and F are real constants, and not all of A, B, and C are zero.^[14]^[15] This equation encompasses all non-degenerate conic sections—ellipses, parabolas, and hyperbolas—as well as degenerate cases like points, lines, or pairs of lines, depending on the coefficients.^[16]^[17] The type of conic section represented by this equation, assuming it is non-degenerate, is determined by the discriminant B^2 - 4AC: if B^2 - 4AC < 0, it is an ellipse (or circle if B = 0 and A = C > 0); if B^2 - 4AC = 0, it is a parabola; and if B^2 - 4AC > 0, it is a hyperbola.^[15]^[16]^[17] This classification holds because the discriminant distinguishes the nature of the quadratic terms, reflecting the curvature and asymptotic behavior of the curve.^[18] When the conic is rotated relative to the coordinate axes (i.e., B \neq 0), the xy term can be eliminated through a rotation of axes by an angle \theta satisfying \cot 2\theta = \frac{A - C}{B}.^[19]^[20]^[15] The rotation formulas are x = x' \cos \theta - y' \sin \theta and y = x' \sin \theta + y' \cos \theta, which substitute into the original equation to yield a new quadratic without the x'y' term, simplifying classification and analysis.^[19]^[21] If \frac{A - C}{B} = 0, then \theta = 45^\circ.^[20] To shift the conic to its center (for ellipses and hyperbolas), a translation of axes is performed by solving the system of partial derivatives set to zero: \frac{\partial}{\partial x}(Ax^2 + Bxy + Cy^2 + Dx + Ey + F) = 2Ax + By + D = 0 and \frac{\partial}{\partial y}(Ax^2 + Bxy + Cy^2 + Dx + Ey + F) = Bx + 2Cy + E = 0, yielding the center coordinates (h, k).^[22]^[23] This method works because the center is the point where the gradients balance, minimizing the quadratic form.^[22] The translation x = x' + h, y = y' + k then centers the equation at the origin in the new coordinates.^[23] Under affine transformations, certain quantities remain invariant and aid in classifying conics: the trace A + C, which relates to the overall scaling, and the determinant AC - \frac{B^2}{4}, which is proportional to the negative of the discriminant and preserves the conic type.^[24]^[25] These invariants ensure that the geometric essence—elliptic, parabolic, or hyperbolic—is unchanged despite shearing or stretching.^[24] After rotation and translation, the general form reduces to a standard aligned equation for further properties.^[15]

Polar Coordinates

In polar coordinates, conic sections are conveniently expressed with the focus at the pole (origin), which highlights their geometric properties relative to the focus and directrix. This representation is particularly useful for analyzing shapes where the focus plays a central role, such as in certain geometric constructions.^[26] The polar equation arises directly from the focus-directrix definition of a conic section, where a point P on the conic satisfies the condition that its distance to the focus F equals e times its distance to the directrix, with e being the eccentricity. Place the focus at the origin and assume the directrix is the vertical line x = -d (to the left of the focus), where d > 0 is the distance from the focus to the directrix. For a point P with polar coordinates (r, \theta), the distance to the focus is r, and the distance to the directrix is d + r \cos \theta. Thus, the definition yields

r = e (d + r \cos \theta).

Solving for r,

r - e r \cos \theta = e d, \quad r (1 - e \cos \theta) = e d, \quad r = \frac{e d}{1 - e \cos \theta}.

To align with the standard form where the conic opens away from the directrix, the equation is often written as

r = \frac{e d}{1 + e \cos \theta}

by adjusting the orientation (directrix to the right or sign convention). Here, e d is the semi-latus rectum l, the distance from the focus to the conic along the line perpendicular to the axis through the focus, so the general form simplifies to

r = \frac{l}{1 + e \cos \theta}.

^[27]^[28] For conics with a vertical directrix (perpendicular to the polar axis), the equation uses \sin \theta instead of \cos \theta, yielding r = \frac{l}{1 + e \sin \theta} (or with minus sign for orientation). Specific cases follow by substituting e: for a parabola (e = 1), r = \frac{2p}{1 + \cos \theta}, where p is the distance from the vertex to the focus (and l = 2p); for an ellipse ($0 < e < 1), the form describes the closed curve; for a hyperbola (e > 1), it captures the two branches.^[26]^[29] The latus rectum is the chord through the focus perpendicular to the major axis (or axis of symmetry), with length $2l = \frac{2 b^2}{a} for both ellipses and hyperbolas, where a is the semi-major axis and b the semi-minor axis (or analogous for hyperbola). This length equals the semi-latus rectum doubled and relates directly to the polar parameter l. For parabolas, the latus rectum length is $4p.^[16]^[30]

Geometric Properties

Conic sections exhibit several distinctive geometric properties that highlight their shared characteristics as curves derived from plane-cone intersections. One fundamental property is the equation of the tangent line at a point on the curve. For an ellipse given by the standard equation \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, the tangent at the point (x_0, y_0) on the ellipse is

\frac{x x_0}{a^2} + \frac{y y_0}{b^2} = 1.

This equation arises from the condition that the line intersects the ellipse at exactly one point, ensuring tangency.^[31] A remarkable optical property shared by certain conic sections is their reflection behavior, which follows from the focus-directrix definition. In an ellipse, a ray originating from one focus reflects off the curve such that it passes through the other focus, as the tangent makes equal angles with the lines to the two foci. Similarly, for a parabola, incoming parallel rays reflect toward the single focus, with the tangent forming equal angles between the parallel ray and the line to the focus. These properties underpin applications in optics but stem purely from the geometric configuration.^[32] The areas enclosed or bounded by conic sections also reveal key geometric insights. The area of an ellipse with semi-major axis a and semi-minor axis b is \pi a b, which can be derived by integrating the curve or recognizing it as a stretched circle. For a parabolic segment—the region bounded by a parabola and a chord connecting two points on it—the area is \frac{2}{3} times the base (chord length) multiplied by the height (perpendicular distance from the chord to the vertex), as established through the method of exhaustion. This result shows the segment area exceeds that of the inscribed triangle by one-third.^[33]^[34] Hyperbolas possess linear asymptotes that guide their shape at infinity. For the standard hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opening horizontally, the asymptotes are the lines y = \pm \frac{b}{a} x, obtained by setting the constant term to zero in the equation. These lines pass through the center and approach the branches without intersecting them.^[35] The ellipse further demonstrates a construction property known as the string property: the sum of distances from any point on the ellipse to the two foci remains constant and equal to $2a, the major axis length. This can be visualized using a string of length $2a pinned at the foci, with a pencil tracing the curve while keeping the string taut.^[32]

Historical Development

Ancient Greek Contributions

The study of conic sections originated in ancient Greece during the fourth century BCE, primarily as a geometric tool to solve classical problems. Menaechmus, a contemporary of Aristotle and pupil of Eudoxus, is credited with the first systematic investigation of conics around 350 BCE. He discovered these curves by intersecting planes with cones of different angles, identifying the parabola, ellipse, and hyperbola as distinct sections while attempting to solve the Delian problem of doubling the cube.^[36]^[4]^[37] Euclid, active around 300 BCE, provided one of the earliest written references to conic sections in his seminal work Elements, though his treatment was limited and preparatory. In Book II, Proposition 5 and Book VI, he alluded to their properties in the context of geometric constructions, such as applications of areas that implicitly generate conic loci, but he did not develop a comprehensive theory. Euclid's now-lost treatise Conics likely contained more detailed elements, serving as a foundational reference for later mathematicians like Archimedes, yet it focused on basic principles without exploring advanced theorems.^[38]^[39]^[40] The most influential contribution came from Apollonius of Perga, who around 200 BCE authored the definitive Greek treatise Conics in eight books, synthesizing and expanding prior work into a rigorous geometric framework. In the first four surviving books, Apollonius rigorously defined the ellipse, parabola, and hyperbola through plane sections of right circular cones at various angles relative to the axis, coining their modern names and deriving their fundamental properties using synthetic geometry. He proved key theorems on tangents and on asymptotes for hyperbolas, establishing their limiting behaviors as lines approached the curve. The later books, partially preserved through Arabic translations, applied these to practical problems such as constructing sundials and trisecting angles using conic intersections. Apollonius' work emphasized deriving properties from axioms without coordinates and remained the standard reference for over a millennium.^[41]^[42]^[43]^[44] Later, in the early 4th century CE, Pappus of Alexandria advanced the study in his Collection, providing the first explicit focus-directrix definition for conic sections: the locus of points where the ratio of the distance to a fixed point (focus) and a fixed line (directrix) is constant, with this ratio being the eccentricity (0 for circle, 0<e<1 for ellipse, e=1 for parabola, e>1 for hyperbola). This property unified the conics and facilitated later applications in optics and astronomy.^[45]

Islamic Scholars

During the Islamic Golden Age, scholars in the Abbasid Caliphate and beyond extended the study of conic sections from synthetic geometry to algebraic and applied domains, particularly in solving equations and optical design. Building on the foundational work of Apollonius and Pappus, they integrated algebra with geometric constructions, enabling practical advancements in mathematics and science. Muhammad ibn Musa al-Khwarizmi (c. 780–850 CE), often regarded as the father of algebra, introduced systematic algebraic methods in his treatise Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, where he provided geometric solutions to quadratic equations using constructions such as completing the square with areas represented by rectangles and squares. These methods resolved problems such as finding square roots and completing the square, laying the groundwork for later algebraic treatments of conics in Islamic mathematics.^[46]^[47] Omar Khayyam (1048–1131 CE) made a pivotal advancement by applying conic sections to higher-degree equations in his Treatise on Demonstration of Problems of Algebra. He classified 25 types of cubic equations and solved 14 of them geometrically by determining the intersection points of conics, such as a rectangular hyperbola with a circle or a parabola with a hyperbola, yielding positive real roots without numerical approximation. This geometric-algebraic synthesis represented a significant innovation, bridging Diophantine analysis with conic properties and influencing subsequent European algebra.^[48]^[49] In astronomy, Ibn al-Shatir (1304–1375 CE), the last major muwaqqit of the Great Mosque of Damascus, refined Ptolemaic models in his Nihayat al-Sul fi Tasyir al-Aflak. He employed conic-based adjustments in planetary theories, including eccentrics and the Tusi couple to eliminate the equant, creating accurate geocentric models for the Moon and planets that served as a direct precursor to Kepler's elliptical orbits. These innovations improved predictive tables and demonstrated conics' utility in modeling celestial motions.^[50] Islamic contributions to optics further highlighted conics' practical value. Ibn Sahl (c. 940–1000 CE) explored anaclastic instruments in his On Burning Mirrors and Lenses, using hyperbolas as conic sections to design plano-hyperbolic lenses that focus parallel rays to a single point, thereby deriving the law of refraction through geometric analysis of ray paths in transparent media.^[51] Similarly, Ibn al-Haytham, known as Alhazen (c. 965–1040 CE), advanced catoptrics in his monumental Kitab al-Manazir (Book of Optics) and a dedicated treatise on parabolic burning mirrors. He investigated reflection properties of parabolic mirrors to concentrate solar rays for ignition, resolving spherical aberrations and employing conic intersections to solve reflection problems, such as finding reflection points on curved surfaces for optimal focusing.^[52]

European Revival

The revival of conic sections in Europe during the Renaissance was significantly advanced by the publication of ancient Greek texts, building on Latin translations of Arabic manuscripts that had preserved this knowledge through the medieval period. A pivotal event occurred in 1566 when Italian mathematician and physician Federico Commandino edited and published the first complete Latin edition of Apollonius of Perga's Conics (Books I–IV), rendering the work accessible to European scholars and reigniting interest in the geometric properties of ellipses, parabolas, and hyperbolas.^[53] In the late 16th century, French mathematician François Viète integrated conic sections into astronomical calculations, applying techniques like angular sections to model celestial phenomena in works such as his 1593 Zeteticorum libri V, which explored conic properties algebraically. This astronomical application culminated with Johannes Kepler's groundbreaking use of ellipses in 1609, where he demonstrated in Astronomia Nova that planetary orbits, including Mars', follow elliptical paths with the Sun at one focus, revolutionizing heliocentric models and establishing conics as essential for describing gravitational motion.^[54] The 17th century marked a shift toward analytic approaches, with René Descartes' 1637 La Géométrie introducing coordinate geometry that equated conic sections to quadratic equations in two variables, enabling algebraic manipulation of their curves and intersections. Independently, Pierre de Fermat developed methods around 1636–1638 for finding tangents to conic sections, such as parabolas and ellipses, by approximating secant lines and minimizing algebraic expressions, laying groundwork for differential calculus applied to these curves.^[55] Advancements continued into the 18th and 19th centuries, as Leonhard Euler introduced parametric equations for conic sections in his 1748 Introductio in analysin infinitorum, representing ellipses and hyperbolas using trigonometric functions like x = a \cos t, y = b \sin t to facilitate integration and series expansions. Carl Friedrich Gauss applied the method of least squares in 1809 to fit elliptical orbits to astronomical observations in Theoria Motus Corporum Coelestium, minimizing errors in planetary data to predict positions accurately. Finally, Jean-Victor Poncelet's 1822 Traité des propriétés projectives des figures advanced the study of conics through projective transformations, emphasizing invariant properties under perspective, which unified their geometric interpretations.^[56]^[57]

Practical Applications

In Physics and Astronomy

Conic sections play a fundamental role in describing planetary and satellite orbits under gravitational forces. Kepler's first law states that planets orbit the Sun in elliptical paths with the Sun located at one focus of the ellipse. This was established through observations between 1609 and 1619. The second law, known as the law of equal areas, asserts that a line segment joining a planet to the Sun sweeps out equal areas in equal intervals of time, implying conservation of angular momentum. Kepler's third law relates the orbital period T to the semi-major axis a of the ellipse via the proportionality T^2 \propto a^3.^[58]/Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/13%3A_Gravitation/13.06%3A_Kepler%27s_Laws_of_Planetary_Motion) Isaac Newton's inverse square law of universal gravitation provides a theoretical foundation for these empirical laws, demonstrating that orbits under such a central force are conic sections—ellipses for bound orbits, parabolas for marginal escapes, and hyperbolas for unbound trajectories. The specific type of conic depends on the total mechanical energy E: negative for ellipses, zero for parabolas, and positive for hyperbolas. The eccentricity e is determined by the energy and angular momentum, given by e = \sqrt{1 + \frac{2 E h^2}{\mu^2}}, where h is the specific angular momentum and \mu is the standard gravitational parameter. This derivation unifies Kepler's laws within classical mechanics, showing how the focus-directrix property aligns with the gravitational potential.^[59]/25%3A_Celestial_Mechanics/25.04%3A_Energy_Diagram_Effective_Potential_Energy_and_Orbits) In optics, the reflective properties of conic sections enable precise focusing and wave propagation. Parabolic mirrors reflect parallel rays—such as incoming light from distant stars—to a single focal point, a principle exploited in reflecting telescopes to minimize spherical aberration and gather light efficiently. This occurs because any ray parallel to the parabola's axis reflects through the focus, following the reflection law where the incident angle equals the reflected angle. Elliptical geometries exhibit a complementary property: rays originating from one focus reflect off the ellipse and converge to the other focus, as seen in whispering galleries where sound waves or light propagate along curved walls with minimal loss, concentrating acoustic or optical energy between foci./02%3A_Geometrical_Optics/2.05%3A_Perfect_Imaging_by_Conic_Sections)/10%3A_Analytic_Geometry/10.01%3A_The_Ellipse) Modern applications extend conic section concepts into relativistic regimes and oscillatory systems. Near black holes, general relativity modifies Newtonian conic orbits, introducing precession; for instance, particles on hyperbolic trajectories can escape after close approaches, but photon paths form unstable "conic-like" geodesics that spiral or deflect strongly due to spacetime curvature. In wave mechanics, Lissajous figures—traced by two perpendicular simple harmonic oscillations of equal frequency but phase difference—form ellipses, illustrating how conic paths emerge in coupled oscillatory phenomena like those in atomic spectra or mechanical vibrations.^[60]^[61]

In Engineering and Optics

In bridge design, parabolic arches are widely employed due to their structural efficiency in distributing loads and minimizing stresses. Under uniform loading, a parabolic arch shape ensures that forces are primarily axial compression along the curve, reducing bending moments and enabling lighter, more economical constructions compared to circular arches. For instance, tied-arch bridges utilize parabolic profiles to achieve optimal stress distribution, with the arch rib carrying compressive forces while horizontal ties handle tension. This design principle has been applied in numerous modern bridges, such as the Sydney Harbour Bridge, where the parabolic form contributes to stability under vehicular loads.^[62]^[63] Elliptical geometries find practical use in architectural acoustics, particularly in rooms designed as whispering galleries. In an elliptical chamber, sound waves originating at one focus reflect off the curved walls and converge at the opposite focus, allowing whispers to be heard clearly across the space despite distance. This effect leverages the ellipse's reflective property, where tangents at any point direct rays toward the second focus, enhancing audibility without amplification. Historic examples include the whispering gallery in St. Paul's Cathedral in London, where the elliptical dome facilitates this acoustic focusing for visitors standing at the foci.^[64] In optics and engineering, hyperbolic surfaces are utilized in lens and mirror designs to achieve wide-angle fields of view with reduced aberrations. Hyperbolic metalenses, for example, enable imaging over large angular ranges by compensating for off-axis distortions, making them suitable for applications like panoramic cameras and endoscopes. A combined hyperbolic mirror configuration can capture a field of view exceeding 180 degrees while maintaining resolution, as demonstrated in prototypes for wide-FOV imaging systems. This conic form's diverging properties allow for compact, high-performance optics that outperform traditional spherical lenses in angular coverage.^[65] Cycloidal gear profiles, derived from hypocycloid and epicycloid curves, are integral to mechanical engineering for smooth power transmission in clocks, pumps, and precision machinery. The hypocycloid generates the concave flanks of the gear tooth, ensuring constant velocity ratio and minimal backlash during meshing, which is superior to involute gears in low-speed, high-torque scenarios. These profiles arise from a point on a smaller circle rolling inside a larger fixed circle, producing the hypocycloid's cusp-free segments ideal for gear teeth. Early adoption in 17th-century clockworks by engineers like Christiaan Huygens highlighted their role in accurate timekeeping.^[66]^[67] The parabolic trajectory of projectiles under constant gravity represents a foundational engineering insight, first rigorously described by Galileo in 1638. In his Two New Sciences, Galileo demonstrated that a body projected horizontally combines uniform horizontal motion with vertically accelerated fall, yielding a parabolic path—essential for ballistics, artillery design, and trajectory prediction in mechanical systems. This model assumes negligible air resistance and uniform gravitational acceleration, providing the basis for calculating range and impact in engineering simulations.^[68]^[69] Contemporary engineering applications of parabolas include satellite dishes, where the reflector focuses incoming signals to a receiver at the focal point for efficient communication. The parabolic shape ensures that rays parallel to the axis converge precisely, maximizing signal strength in systems like GPS and television broadcasting. Designs typically feature diameters from 0.6 to 3 meters, with the focal length determining the feed placement for optimal gain.^[70]^[71] In automotive engineering, elliptical reflectors are employed in projector headlights to direct light from the source to a focal point before collimation through a lens, producing a sharp, controlled beam pattern. This conic configuration allows precise focusing of LED or halogen sources, improving illumination uniformity and reducing glare compared to purely parabolic designs. Modern vehicles, such as those with adaptive lighting systems, leverage elliptical cross-sections to achieve cut-off lines for oncoming traffic compliance.^[72]

Projective Geometry

Homogeneous Coordinates

Homogeneous coordinates provide a foundational framework for projective geometry, representing points in the projective plane \mathbb{RP}^2 as equivalence classes of triples [x : y : z], where x, y, z \in \mathbb{R} are not all zero, and two triples are equivalent if one is a scalar multiple of the other. This system embeds the Euclidean plane as a subset by identifying affine points (x, y) with [x : y : 1].^[73]^[74] Lines in this coordinate system are defined by linear equations of the form a x + b y + c z = 0, where [a : b : c] represents the line up to scalar multiple, dual to the point representation. To recover affine geometry, dehomogenization is performed by setting z = 1, yielding Cartesian coordinates (x/z, y/z) for points where z \neq 0, thus mapping the projective plane onto the affine plane while excluding the line at infinity.^[73]^[75] In homogeneous coordinates, a conic section is represented by a general quadratic equation

\begin{align*} a x^2 + b y^2 + c z^2 + d x y + e x z + f y z &= 0, \end{align*}

where the coefficients a, b, c, d, e, f define the conic up to scalar multiple. This form homogenizes the affine quadratic equation a x^2 + b y^2 + d x y + g x + h y + i = 0 by introducing the z terms, with g = e, h = f, and i = c in the dehomogenized case.^[73]^[75] The primary advantages of homogeneous coordinates for conic sections lie in their ability to incorporate points at infinity seamlessly, allowing parabolas and hyperbolas—which extend to infinity in affine space—to be treated uniformly with ellipses in the projective setting. Additionally, projective transformations, which preserve conic incidence properties, are realized as linear transformations on the homogeneous coordinates via $3 \times 3 invertible matrices acting on the triples [x : y : z]^T.^[74]^[73]

Points at Infinity

In projective geometry, the real projective plane \mathbb{RP}^2 extends the Euclidean plane by adjoining a line at infinity, which compactifies the space and identifies the directions of parallel lines as points where they intersect.^[73] This addition unifies the treatment of finite points and ideal points at infinity, allowing conic sections to be analyzed uniformly as curves in the projective plane.^[73] The behavior of a conic section at infinity is determined by its intersections with this line at infinity, providing a projective classification that distinguishes the classical types. An ellipse, being bounded and closed in the Euclidean plane, intersects the line at infinity in no real points.^[76] In contrast, a parabola touches the line at infinity at exactly one real point, which corresponds to the direction of the parabola's axis of symmetry.^[76] A hyperbola, with its two unbounded branches, intersects the line at infinity in two distinct real points, representing the directions of its asymptotes.^[76] Circles, as a special case of ellipses, have no real points at infinity but intersect the line at infinity at two complex conjugate points known as the circular points I and J, with homogeneous coordinates [1 : i : 0] and [1 : -i : 0], respectively.^[77] These points lie on every circle in the projective plane, highlighting the projective equivalence of all circles.^[77] For hyperbolas, the asymptotes can be understood projectively as the tangent lines to the curve at its two points at infinity.^[78] This perspective resolves the Euclidean notion of lines approached but never touched, treating the asymptotes instead as actual tangents in the extended plane.^[78]

Projective Definitions and Equivalence

In projective geometry, conics can be characterized purely in terms of incidence and cross-ratio properties, independent of any metric structure. One such definition, due to Jakob Steiner, describes a conic as the envelope of lines obtained by joining corresponding rays from two projective pencils of lines based at distinct points.^[79] This construction ensures that the resulting curve is a non-degenerate conic, as the projectivity between the pencils defines the correspondence without reference to distances or angles.^[80] Another metric-free characterization is provided by Karl Georg Christian von Staudt, who defined a conic as the locus of points P in the projective plane such that, for every complete quadrangle, the pencil of lines through P harmonically divides the diagonal triangle of the quadrangle.^[81] This definition relies solely on the harmonic cross-ratio, a fundamental invariant in projective geometry, allowing conics to be identified through their harmonic properties relative to quadrangles.^[82] Von Staudt's approach underscores the synthetic nature of projective geometry, where conics emerge from incidence relations alone.^[83] A key consequence of these projective definitions is the equivalence of all non-degenerate conics under projective transformations. Specifically, any non-degenerate conic in the real projective plane can be mapped to the unit circle via a suitable projective transformation, as the projective group acts transitively on the set of non-degenerate conics.^[84] This equivalence highlights that distinctions between ellipses, parabolas, and hyperbolas are affine or Euclidean artifacts, not projective ones.^[85] Among conics, the circle holds a special projective role as the unique non-degenerate conic that intersects the line at infinity precisely at the circular points I and J.^[86] These imaginary points at infinity distinguish circles projectively from other conics, as any conic passing through I and J is a circle in the Euclidean sense.^[86]

Key Theorems and Constructions

Pascal's theorem, a fundamental result in projective geometry, asserts that if a hexagon is inscribed in a conic section, then the intersection points of the three pairs of opposite sides are collinear.^[87] This collinearity holds regardless of the specific positions of the vertices on the conic, provided the hexagon is simple and the intersections are well-defined in the projective plane.^[88] The theorem provides a powerful tool for verifying whether points lie on a conic or for constructing additional points on a given conic using five known points.^[88] The dual of Pascal's theorem is Brianchon's theorem, which states that if a hexagon is circumscribed about a conic section—meaning its sides are tangent to the conic—then the three main diagonals connecting opposite vertices are concurrent at a single point.^[89] This concurrency point is invariant under projective transformations and characterizes the tangential hexagon uniquely for the conic.^[90] Brianchon's theorem complements Pascal's by shifting focus from inscribed figures to circumscribed ones, enabling symmetric applications in pole-polar relations.^[91] Key geometric constructions involving conics include drawing tangents from an external point to the conic. To construct these tangents projectively, one identifies the polar line of the external point with respect to the conic; the points of intersection between this polar and the conic serve as the points of tangency, from which the tangent lines are drawn to the external point.^[92] This method relies solely on straightedge operations once the conic and point are given, preserving projective properties without metric assumptions.^[92] Central to these constructions is the pole-polar duality, a projective involution that associates each point (the pole) with a unique line (the polar) relative to the conic, such that the polar of a point is the locus of points harmonic to it with respect to the conic's intersections.^[89] If a point lies on the polar of another, their roles are interchanged, establishing a reciprocal relation that underlies theorems like Pascal's and Brianchon's.^[89] This duality facilitates the construction of tangents, polars, and envelopes, as the tangent at a point on the conic is precisely its own polar.^[92] Poncelet's porism extends these ideas to periodic figures, stating that if there exists a closed n-sided polygon inscribed in one conic and circumscribed about another (interlocking) conic, then infinitely many such n-gons exist, generated by successive tangent and secant steps around the conics.^[93] This closure property holds under projective mappings between the conics and implies that starting from any vertex on the outer conic, the polygonal path closes after n steps if and only if the initial condition is satisfied.^[94] The porism highlights the rich interplay of projective equivalences in generating families of polygons tangent to or inscribed in conics.^[95]

Complex and Degenerate Cases

Conics over Complex Numbers

In the complex projective plane \mathbb{CP}^2, conic sections are defined by homogeneous quadratic equations with coefficients in \mathbb{C}, extending the real case to allow complex coordinates and transformations. Using homogeneous coordinates over \mathbb{C}, points are represented as [x : y : z] where x, y, z \in \mathbb{C} are not all zero, up to multiplication by nonzero complex scalars. This setting unifies the classification of conics, as the algebraically closed nature of \mathbb{C} enables broader equivalence under the action of the projective linear group \mathrm{PGL}(3, \mathbb{C}). Non-degenerate conics, those with full rank quadratic forms, are smooth curves of genus zero in \mathbb{CP}^2.^[96] All non-degenerate conics in \mathbb{CP}^2 are projectively equivalent over \mathbb{C}, meaning any such conic can be mapped to any other via an invertible projective transformation. In particular, every non-degenerate conic is equivalent to the standard circle defined by the equation x^2 + y^2 - z^2 = 0 in homogeneous coordinates. This equivalence implies that traditional real distinctions—such as ellipses, parabolas, and hyperbolas—dissolve, as an affine transformation over \mathbb{C} can map any ellipse to a circle, and projective extensions handle the remaining types. The transitive action of \mathrm{PGL}(3, \mathbb{C}) on the space of non-degenerate conics ensures this uniformity, contrasting with the real projective plane where multiple orbits exist.^[96]^[97] The circular points at infinity, denoted I = [1 : i : 0] and J = [1 : -i : 0], are key to understanding this unification. These complex points lie on the line at infinity z = 0 and are the intersection points of all circles in the real affine plane when extended projectively. Over \mathbb{C}, I and J are ordinary points in \mathbb{CP}^2, and a conic is a circle precisely if it passes through both. Since projective transformations over \mathbb{C} can map any pair of distinct points on a conic to I and J (preserving the conic's non-degeneracy), every non-degenerate conic can be transformed into one passing through these points, rendering it a "circle" in the complex projective sense. This perspective reveals all non-degenerate conics as complex analogs of circles.^[86]^[98] Conics over \mathbb{C} also connect to the Riemann sphere via stereographic projection, providing a geometric visualization. The Riemann sphere compactifies the complex plane \mathbb{C} to \mathbb{CP}^1, and stereographic projection from the north pole maps the sphere minus that point bijectively to \mathbb{C}. In this framework, non-degenerate conics in the affine complex plane project to closed curves on the Riemann sphere that are images of great circles or small circles under the inverse projection; however, the projective equivalence over \mathbb{C} ensures these correspond uniformly to circular sections, emphasizing the "circleness" of all conics in the complex domain. This projection aids in studying intersections and transformations, as lines in \mathbb{C} (degenerate conics) map to circles through the projection pole. In applications, the matrix representation of a conic facilitates classification using complex eigenvalues. The general conic equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 corresponds to a symmetric $3 \times 3 matrix M whose entries encode the coefficients, with the conic as the set where \mathbf{X}^T M \mathbf{X} = 0 for \mathbf{X} = [x, y, z]^T. Over \mathbb{C}, M is diagonalizable, and for non-degenerate conics (\det M \neq 0), the eigenvalues can be transformed via congruence to those of the circle matrix \operatorname{diag}(1, 1, -1), confirming the equivalence. Complex eigenvalues arise naturally in this setting, distinguishing non-degenerate cases from degenerates and enabling computational classification in algebraic geometry software.^[99]

Degenerate Conics

Degenerate conics arise when the general equation of a conic section factors into linear terms, resulting in geometric objects of lower dimension such as points or lines rather than smooth curves. These cases are characterized by the singularity of the associated conic matrix, where the determinant of the 3×3 symmetric matrix representing the quadratic form is zero, indicating that the conic is reducible over the field.^[100] In the real affine plane, the general Cartesian form reduces to special cases that can be classified by the nature of these factors. The primary types of degenerate conics include a pair of intersecting lines, which represents a degenerate hyperbola; a pair of parallel lines, corresponding to a degenerate parabola; two coincident lines forming a double line; or a single point, which is a degenerate ellipse.^[101] A double line occurs when the conic equation is the square of a linear equation, effectively representing the same line with multiplicity two, while a single point results from an equation where the only real solution is a isolated vertex-like configuration. These degenerations typically emerge when the slicing plane passes through the vertex of the cone in the classical geometric definition.^[5] Classification of these degenerates relies on the discriminant of the quadratic terms in the general equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, given by B^2 - 4AC, combined with the rank of the conic matrix. When B^2 - 4AC > 0, the form suggests a hyperbola that degenerates into two intersecting lines if the overall matrix rank is 2; for B^2 - 4AC = 0, a parabolic form degenerates into two parallel lines or a double line when the rank is less than 3, with further rank conditions (such as rank 1 for the double line) distinguishing subtypes; and for B^2 - 4AC < 0, an elliptic form degenerates to a point when the matrix rank is 2 and the constant term aligns to restrict solutions to a single location. The full degeneracy requires the determinant of the conic matrix to be zero, with rank less than 3 overall, and sub-rank analysis (e.g., rank < 2 for certain line degenerations) provides finer categorization.^[102] In projective geometry, degenerate conics are viewed through the lens of the conic matrix's singularity, where a zero determinant implies the quadratic form factors into linear factors, yielding either two distinct lines (rank 2), a double line (rank 1), or a point (rank 2 with isotropic properties over the reals).^[73] This perspective unifies the affine types, as parallel lines intersect at a point at infinity, and points represent collapsed conics tangent to the line at infinity. Representative examples illustrate these cases. The equation x^2 - y^2 = 0 factors as (x - y)(x + y) = 0, yielding two intersecting lines and exemplifying a degenerate hyperbola.^[101] Similarly, x^2 = 0 represents a double line along the y-axis, a case of coincident lines with rank 1 in the conic matrix.^[102]

Advanced Topics

Pencils of Conics

A pencil of conics is a one-dimensional linear family of conic sections defined algebraically in the projective plane as the set of all curves satisfying the equation \lambda C_1 + \mu C_2 = 0, where C_1 and C_2 are two distinct conics represented by quadratic forms X^T A_1 X = 0 and X^T A_2 X = 0, with \lambda, \mu scalars not both zero, and the combined form X^T (\lambda A_1 + \mu A_2) X = 0. This parametrization arises naturally when considering all conics passing through the four intersection points of C_1 and C_2, as Bézout's theorem guarantees that two distinct conics intersect at exactly four points (counting multiplicity and points at infinity).^[103] Consequently, every member of the pencil shares these four base points, ensuring fixed intersections among non-degenerate members.^[104] In such a pencil, degenerate members occur when the matrix \lambda A_1 + \mu A_2 has rank less than 3, resulting in pairs of lines rather than irreducible conics. A general pencil contains exactly three degenerate conics, corresponding to the roots of the cubic determinant equation \det(\lambda A_1 + \mu A_2) = 0, unless the entire pencil is degenerate (e.g., all members are pairs of lines through two fixed points). These degenerate cases divide the pencil into components, with the pairs of lines typically connecting the base points in different pairings, such as the complete quadrangle formed by the four points.^[104] The fixed intersection properties of pencils relate to broader intersection theorems, such as the Cayley-Bacharach theorem, which in the conic case underscores that the four base points remain invariant across the family, analogous to how the theorem predicts an eighth intersection point for cubics sharing nine points.^[103] In projective geometry, pencils of conics facilitate the generation of conics through dual constructions, notably Steiner's definition, where a conic emerges as the locus of intersections between two projectively related (but not perspectively) pencils of lines from distinct vertices.^[105] This approach highlights the projective invariance of conics, enabling constructions independent of metric properties.

Intersections of Conics

In algebraic geometry, the intersection of two conic sections in the plane is governed by Bézout's theorem, which states that two plane algebraic curves of degrees m and n intersect in exactly m n points, counted with multiplicity and including points at infinity in the projective plane.^[106] For two conics, each of degree 2, this yields precisely four intersection points.^[107] These points may coincide or lie at infinity, affecting the geometric configuration, such as when parallel branches of hyperbolas meet on the line at infinity.^[108] To compute the intersection points algebraically, one approach involves solving the system of two quadratic equations defining the conics, often by eliminating one variable to obtain a quartic equation whose roots correspond to the points.^[109] This elimination can be performed using the resultant of the two quadratics with respect to one variable, yielding a degree-4 polynomial in the other variable.^[109] An alternative numerical method parameterizes the problem via a linear combination of the conic matrices and solves an eigenvalue problem to find the ratios determining the intersection points.^[109] Special cases arise based on multiplicity and degeneracy. Tangency occurs when two conics touch at a point with multiplicity 2, reducing the distinct real intersections while still satisfying the total count of four (with the remaining points possibly complex or at infinity).^[110] In degenerate scenarios, if both conics reduce to pairs of straight lines—representing the opposite sides of a complete quadrilateral—their intersections yield the six vertices of that quadrilateral, with multiplicities accounting for the Bézout total.^[111] Dually, the problem of finding common tangents to two conics transforms under the pole-polar relation: the common tangents correspond to the polars of the intersection points of the dual conics.^[112] This duality preserves the fourfold intersection count, where each intersection point in the dual space defines a common tangent line in the primal.^[112]

Generalizations

Conic sections generalize to higher dimensions through quadric surfaces, which are the three-dimensional analogs defined by quadratic equations in three variables. These surfaces include ellipsoids, hyperboloids of one and two sheets, elliptic and hyperbolic paraboloids, cones, and cylinders, each arising as the intersection of a plane with a quadratic cone in four-dimensional space. For instance, an ellipsoid results from a plane cutting through all four nappes of such a cone, while a hyperboloid emerges from a plane intersecting three nappes, and a paraboloid from a plane parallel to a generator of the cone.^[113]^[114] In abstract algebraic geometry, conic sections are viewed as smooth projective curves of genus zero over an algebraically closed field, birationally equivalent to the projective line \mathbb{P}^1. Any irreducible curve of genus zero admits a rational parametrization and can be embedded as a conic in the projective plane \mathbb{P}^2_k, where it is defined by a homogeneous quadratic equation. Over non-algebraically closed fields, such curves may be nontrivial, but they remain genus zero and are classified up to projective equivalence by their behavior under field extensions. This perspective unifies conics with rational curves, emphasizing their role as the simplest non-trivial algebraic curves.^[115]^[116] Conic sections extend to non-Euclidean geometries, where they are defined via quadratic forms adapted to the underlying metric. In the hyperbolic plane, conics are classified based on their intersection properties with the absolute conic, yielding analogs of ellipses (bounded regions inside the absolute), hyperbolas (regions crossing the absolute), and parabolas (tangent to the absolute), with eccentricity determining the type. Similarly, in elliptic geometry, conics appear as closed curves on the projective plane modulo antipodes, often visualized on the sphere, where they correspond to great or small circles generalized by quadratic intersections. These definitions preserve key properties like foci and directrices but adjust for the constant curvature.^[117] In higher-dimensional algebraic geometry, conic bundles represent a multivariable generalization, consisting of a proper flat morphism \pi: X \to S from a variety X to a base S, where the generic fiber is a smooth conic (genus-zero curve) over the function field of S. Defined over schemes where 2 is invertible, these bundles are relatively minimal if no -1-curve fibers exist, and they often feature a discriminant curve controlling singular fibers. Conic bundles over surfaces or threefolds are central to rationality problems, as their geometry influences whether X is rational or stably rational.^[118]^[119]

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Precessing conic sections - Villanova University
Einstein's general relativity adds an additional element to these conic section orbits in the field of a rotating mass or black hole, namely a precession ( ...
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Lissajous Figures
A more complicated curve called a Lissajous figure. In this demonstration we look at curves of the formMissing: oscillations sources
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Tied-arch bridges - SteelConstruction.info
[top]Global design A parabolic arch is the best shape for structural efficiency because, under uniform load there should just be axial forces in the arch ...
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[PDF] the importance of the shape of arches
The uneconomical nature of the circular arch as a bridge arch is highlighted. Parabolic arch shape increases the stresses up to |9| MPa and catenary shape to | ...<|separator|>
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Sound Reflections in Auditoriums - HyperPhysics
Locations where the rotunda effect is experienced are sometimes called "whispering galleries". The dome of St. Paul's Cathedral in London is a famous ...
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Wide Field-of-View Imaging Using a Combined Hyperbolic Mirror
A wide field-of-view (FOV) image contains more visual information than a conventional image. This study proposes a new type of hyperbolic mirror for wide FOV ...
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Construction and design of cycloidal gears - tec-science
Dec 21, 2018 · A cycloid is constructed by rolling a rolling circle on a base circle. A fixed point on the rolling circle describes the cycloid as a trajectory curve.Missing: conic | Show results with:conic
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[PDF] Unit – 1(a) - mechanical
Engineering Applications: In cycloidal teeth gears, the faces are of Epicyloidal profile and the flanks are of Hypocycloidal profile to ensure correct meshing.
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Galileo's Discovery of the Parabolic Trajectory - jstor
When Galileo published his discus sion of the parabolic trajectory in 1638, he did not refer to any experiments. All he could derive was an ideal law that.
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[PDF] The Fourth Day from Galileo's Two New Sciences (1638)
A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola.
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Parabolic Dish Reflector - Antenna Theory
The basic structure of a parabolic dish antenna is shown in Figure 3. It consists of a feed antenna pointed towards a parabolic reflector. The feed antenna is ...
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[PDF] Exploring Parabolas: The shape of a satellite dish
The top figure to the left shows a satellite dish with a radio receiver located at the focus of the parabola. The radio rays are reflected from the parabolic ...
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Robust optical design of high-contrast vehicle headlamps with ...
Apr 25, 2025 · Traditionally, reflective optics have been employed to collect and direct light from the source, with conic shapes (e.g., parabolic, elliptical) ...
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[PDF] Basics of Projective Geometry - UPenn CIS
In terms of coordinates, this corresponds to “homogenizing.” For example, the homogeneous equation of a conic is ax2 +by2 +cxy+dxz+eyz+ fz2 = 0.
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[PDF] An Introduction to Projective Geometry for computer vision
Mar 12, 1998 · In homogeneous coordinates the line becomes Y = 0 which yields the solution X;0;0 , the ideal point associated with the horizontal direction.
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[PDF] PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin - People
The techniques of projective geometry, in particular homogeneous coordinates, ... The different types of conic sections – ellipses, hyperbolas and ...
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Conic 1.2
A hyperbola is a conic which has two points in common with the line at infinity; these are the points in the directions of the two asymptotes. A parabola is a ...
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[PDF] Basic Notions
... circular points at infinity (1,±i,0). Thus all circles have the two points (1,±i,0) at infinity in common. Taken together with the two finite points of ...
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[PDF] GEOMETRY REVISITED H. S. M. Coxeter S. L. Greitzer - Aproged
fore one of the points at infinity on the hyperbola is the point of contact of the tangent u, and of course the other is the point of contact of v. 6. Since ...
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Conic 1.3
The Conic as a Locus of Lines. We defined a conic as the set of points of intersection of corresponding lines in two projective pencils.
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[PDF] Foundations of Projective Geometry
Sep 3, 2012 · 5 we used homogeneous coordinates to parameterize points on a line. We proved the following: Page 18. 188 Exploring Geometry - Web Chapters.
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[PDF] an overview of definitions and their relationships.
Mar 22, 2021 · Now we are ready to give Von Staudt's definition of a conic: a Von Staudt conic in a pappian projective plane PG(2,F) with F a field (char F ...Missing: harmonic | Show results with:harmonic
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[PDF] Conic curves revisited via harmonicity
Aug 30, 2023 · The definition of conic curves as the locus of points in the projective plane that see a quadrangle as a harmonic set, is introduced.
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Harmonic curves and the beauty of projective geometry
Nov 16, 2024 · We now prove that von Staudt's definition of conic curves with mild extra hypothesis gives harmonic curves. Lemma 3. Given a polarity in the ...
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Projective Geometry and Transformations of 2D
in 2D projective geometry all non-degenerate conics are equivalent under projective transformations. The equation of a conic in inhomogeneous coordinates is.
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[PDF] Algorithms for Computing a Planar Homography from Conics in ...
All full rank indefinite conics are projectively equivalent to a circle, i.e., every such conic can be transformed to a circle with a homography [3].
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Circular Point at Infinity -- from Wolfram MathWorld
All conics passing through the circular points at infinity are circles. The circular points at infinity are the fixed points of the orthogonal involution.
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Dandelin's three-dimensional proof of Pascal's Theorem ... - UBC Math
Pascal's Theorem asserts: If one is given six points on a conic section and makes a hexagon out of them in an arbitrary order, then the points of intersection ...<|control11|><|separator|>
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[PDF] Module 1: Projective Geometry Constructions
If the lines are parallel, that point exists but is the infinite point on the line. A new pair of lines with different orientation meet at a different point at ...
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Chapter II: Polarities and Conic Sections - ScienceDirect
This chapter discusses the polarities and conic sections. It also aims to study the mapping of one conic on a second, or on itself that are induced by ...
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[PDF] A selfdual generalization of the Theorems of Pascal and Brianchon
Feb 4, 2025 · Note that Theorem 2.6 contains Brianchon's Theorem 2.5, namely if the conics E and C coincide. Also note that by Theorem 2.3 the polar line of ...
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[PDF] Scholarly Commons - University of the Pacific
The construction of conic sections by means of Pascal's and. Brianchon's theorems. Benjamin Lee Welker Jr. University of the Pacific. Follow this and ...
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[PDF] Introduction to Projective Geometry
The conic itself can be described as the locus of self-conjugate points (and the envelope of self-conjugate lines) in the polarity (ABC)(Pp), where p = PD. ...
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[PDF] Poncelet's porism: a long story of renewed discoveries, I - Oliver Nash
Jul 8, 2018 · Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then ...
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[PDF] Poncelet porism in singular cases - arXiv
The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane ...<|control11|><|separator|>
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[PDF] A Simple Proof of Poncelet's Theorem (on the occasion of its ... - UZH
Poncelet's treatise was a milestone in the development of projective geometry, and his theorem is widely considered the deepest and most beautiful result ...
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[PDF] Conics on the Cubic Surface - Naval Academy
interested in conics in the projective plane P. C. 2. In this setting, all conics are projectively equivalent. Moreover, each nonsingular conic is a rational ...
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[PDF] arXiv:2012.14883v1 [math.HO] 29 Dec 2020
Dec 29, 2020 · Any non-degenerate conic is projectively equivalent to a circle, while the statements for degenerate conics can be obtained by a limiting ...
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[PDF] Projective Geometry - CS@Purdue
The true setting for algebraic geometry is complex projective space. Example: The circle x2 + y2 = 1 homogenizes to x2 + y2 = z2 with points at infinity (±1,i) ...
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[PDF] Additional notes on Quadratic forms - Manuela Girotti
The eigenvalues are: λ1 = 2 > 0 and λ2 = 4 > 0 (guess: this could be an ellipse or 2 complex lines intersecting in one point). P is a 2 × 2 orthogonal matrix ...
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Conic_Equation
2. Degenerate conics. The conic is called degenerate or singular or reducible, when its discriminant which is the determinant |M| of the matrix is zero.
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[PDF] MATH431: Real Projective 2-Space - UMD MATH
Oct 6, 2021 · Other examples of degenerate conics are intersecting lines (a degenerate hyperbola) and parallel lines or a single line (a degenerate parabola).Missing: coincident | Show results with:coincident
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[PDF] UCSD CSE - University of California San Diego
Degenerate conics. If the matrix C is not of full rank, then the conic is termed degen- erate. Degenerate point conics include two lines (rank 2), and a ...
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[PDF] CHAPTER III: CONICS AND QUADRICS - Moodle UPM
Proposition. A pencil of conics in P2 contains three degenerate conics or less, unless the pencil is entirely composed by degenerate conics. λC1 + µC2 ≡ (x0,x1 ...
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[PDF] Cayley-Bacharach Formulas arXiv:1405.6438v2 [math.AG] 24 Dec ...
Dec 24, 2014 · Let Cλ,µ = λC1 + µC2 denote the pencil of conics through the points P1,P2,P3,P4, and. 4. Page 5. let Lλ,µ = λL1 +µL2 be the pencil of lines ...
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[PDF] February 17th: The Intersection of Conics and a Pencil of Conics
Feb 17, 2020 · If K = R, then the pencil has at least one degenerate conic. Proof: A cubic form has at least 3 roots by Section 1.8. In addition, over R, it ...
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[PDF] Projective Geometry in a Plane Fundamental Concepts - Earlham CS
Theorem 4 (Steiner's Definition). A conic is the set of intersections of two pencils of lines that are projectively, but not perspectively, related. Note.
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[PDF] 3264 & All That Intersection Theory in Algebraic Geometry
... ellipse moves away from the real points of the line, and the same for the point of intersection of two lines as the lines become parallel.) Over the course ...<|control11|><|separator|>
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[PDF] Bézout's Theorem - Math (Princeton)
Nov 30, 2016 · The degree of a polynomial f (x,y) is the largest sum of powers of x and y. Jennifer Li (University of Massachusetts). Bézout's Theorem.
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[PDF] Counting Conics - Naval Academy
Nov 18, 2005 · The intersection of the corresponding hypersurfaces in P5 consists of. (1)3(2)2 = 4 points by Bézout's Theorem. These points correspond to.
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[2403.08953] On the Intersection of Two Conics - arXiv
Mar 13, 2024 · Once a linear combination of the two conic matrices has been constructed, the solution of an eigenvalue problem provides four possible ...
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[PDF] Tangency of Conics and Quadrics - WSEAS US
To find the relationship between two conics, we use the pole-polar relationship. For example, if two con- ics are tangent to each other, they should share a.
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ConicsFamily
... degenerate conics represented by the pairs of opposite sides of the complete quadrilateral ABCD. w1, w2, w3 coincide with the intersection points of the ...
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Wilson Stothers' Cabri Pages - Geometric Proofs
Then P lies on the polar of Q if and only if Q lies on the polar of P. Suppose that we can draw two tangents from the point P to the conic C, and that these ...
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Quadratic Surface -- from Wolfram MathWorld
A second-order algebraic surface given by the general equation (1) Quadratic surfaces are also called quadrics, and there are 17 standard-form types.
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A.8 Conic Sections and Quadric Surfaces
A conic section is the curve of intersection of a cone and a plane that does not pass through the vertex of the cone. This is illustrated in the figures ...Missing: center | Show results with:center<|control11|><|separator|>
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Section 53.10 (0C6L): Curves of genus zero—The Stacks project
Nov 14, 2017 · A Gorenstein proper genus zero curve is a plane curve of degree 2, i.e., a conic. A general proper genus zero curve is obtained from a ...
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[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41
All genus 0 curves can be described as conics in P2 k. Proof. Any genus 0 curve has a degree −2 line bundle — the canonical ...
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[PDF] Elementary Constructions for Conics in Hyperbolic and Elliptic Planes
For visualizing the hyperbolic plane we use F. Klein's projective geometric model, elliptic geome- try will be visualized on the sphere. In a Euclidean plane a ...
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[1901.07616] Conic Representations of Topological Groups - arXiv
Jan 22, 2019 · Then we inspect embeddings of irreducible conic representations of semi-simple Lie groups in some "regular" conic representation they possess.
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[1712.05564] The rationality problem for conic bundles - arXiv
Dec 15, 2017 · This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure.
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[PDF] Conic bundles and iterated root stacks - arXiv
Conic bundles. Definition 1. Let S be a regular scheme such that 2 is invertible in its local rings. A regular conic bundle over S is ...