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Type of mathematical curve
In mathematics, a plane cubic curve, often called simply a cubic, is a plane algebraic curve defined by a homogeneous polynomial of degree 3 in three variables
Plane_cubic_curve
Cubic plane curve
mathematics, the Tschirnhausen cubic is a cubic plane curve defined in Cartesian coordinates ( x , y ) {\displaystyle (x,y)} by the cubic equation 27 a y 2 = (
Tschirnhausen_cubic
Mathematical concept
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases
Plane_curve
Curve defined as zeros of polynomials
algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous
Algebraic_curve
Curve used in computer graphics and related fields
Four points P0, P1, P2 and P3 in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at P0 going toward P1 and arrives
Bézier_curve
Plane curve unique to a given triangle
In Euclidean geometry, the McCay cubic (also called M'Cay cubic or Griffiths cubic) is a cubic plane curve in the plane of a reference triangle and associated
McCay_cubic
Cubic function used for interpolation
they were synonymous. Cubic polynomial splines are extensively used in computer graphics and geometric modeling to obtain curves or motion trajectories
Cubic_Hermite_spline
Otto Hesse, is a pencil (one-dimensional family) of cubic plane curves in the complex projective plane, defined by the equation x 3 + y 3 + z 3 − λ x y z
Hesse_pencil
Serpentine curve Trident curve Trisectrix of Maclaurin Tschirnhausen cubic Witch of Agnesi Quartic plane curves include Ampersand curve Bean curve Bicorn
List_of_curves
Plane curve associated with any triangle
In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is
Neuberg_cubic
Fastest curve descent without friction
brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), or curve of fastest descent, is the one lying on the plane between
Brachistochrone_curve
Mathematical idealization of the trace left by a moving point
century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real
Curve
Topics referred to by the same term
Cubic form, a homogeneous polynomial of degree 3 Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 Cubic plane curve
Cubic
Algebraic curve in projective 3-space
mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially
Twisted_cubic
gallery of curves used in mathematics, by Wikipedia page. See also list of curves. Line Circle Ellipse Parabola Hyperbola Cubic curve Cubic polynomial
Gallery_of_curves
Online mathematics resource for cubic plane curves
Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle
Catalogue_of_Triangle_Cubics
Figure-eight-shaped curve
the wool from which ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth
Lemniscate
28 lines which touch a general quartic plane curve in two places
a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane. The
Bitangents_of_a_quartic
Homogeneous polynomial of degree 3
variables, the zero set is a cubic plane curve. In (Delone & Faddeev 1964), Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients
Cubic_form
Mathematical function defined piecewise by polynomials
knot and the last are the same) in the plane is just a polygon. A common spline is the natural cubic spline. A cubic spline has degree 3 with continuity
Spline_(mathematics)
Set of points associated with a triangle's circumconic centers
Thomson cubic of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent. Cubic plane curve § Thomson cubic Weisstein
Thomson_cubic
Cubic plane curve
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form y 2 − a 2 x 3 = 0 y^{2}-a^{2}x^{3}=0
Semicubical_parabola
In mathematics, straight line touching a plane curve without crossing it
line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined
Tangent
Algebraic curve in mathematics
enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian variety – that is, it
Elliptic_curve
Following is a list of shapes studied in mathematics. Cubic plane curve Quartic plane curve Fractal Conic sections Unit circle Unit hyperbola Folium of
List_of_mathematical_shapes
Cubic plane curve
Greek κισσοειδής (kissoeidēs) 'ivy-shaped'; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean
Cissoid_of_Diocles
Set on which a group acts freely and transitively
number field (the theory of the Selmer group). In fact a typical plane cubic curve C over Q has no particular reason to have a rational point; the standard
Principal_homogeneous_space
Curve from a cone intersecting a plane
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola
Conic_section
In geometry, a Hessian curve is a cubic plane curve similar to the folium of Descartes that belongs to the Hesse pencil. It is named after the German mathematician
Hessian form of an elliptic curve
Hessian_form_of_an_elliptic_curve
Polynomial equation of degree 3
approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have
Cubic_equation
Type of plane curve
convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going
Convex_curve
Cubic plane curve
[aɲˈɲeːzi, -eːsi; -ɛːzi]) is a cubic plane curve defined from two diametrically opposite points of a circle. The curve was studied as early as 1653 by
Witch_of_Agnesi
Cubic plane curve
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle
Trisectrix_of_Maclaurin
Statement about cubic curves in the projective plane
statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in the
Cayley–Bacharach_theorem
Type of spline curve
Pythagorean hodograph curves are cubic curves. Not every cubic curve can be parameterized in this way. The cubic Pythagorean hodograph curves can be described
Pythagorean_hodograph_curve
is the projective line. For n = 2 it is the plane conic Z0Z2 = Z2 1, and for n = 3 it is the twisted cubic. The term "normal" refers to projective normality
Rational_normal_curve
Class of quartic plane curves
In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points
Cassini_oval
Mathematically-calculated curve in which a straight section changes into a curve
and curves, or between two different curves. Centripetal force on vehicles on roads without and with a transition curve In the horizontal plane, the
Track_transition_curve
Curve of a specific cubic implicit function
xy+ax^{3}+bx^{2}+cx=d} . Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0 {\displaystyle x=0} ,
Trident_curve
Study of geometry using a coordinate system
coordinate method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, by which every point
Analytic_geometry
Concept in algebraic geometry
P3, branched over a nonsingular genus 4 curve cut out by a cubic surface. Degree 2: they have 56 (−1)-curves corresponding to the minuscule vectors of
Del_Pezzo_surface
Generalization of the concept of parallel lines
A parallel curve of a given (progenitor) curve is the envelope of a family of congruent (equal-radius) circles centered on the curve. It generalises the
Parallel_curve
Formulas in differential geometry
is a cuspidal cubic to order o(s3). The rectifying plane is the plane containing T and B. The projection of the curve onto this plane is: r ( 0 ) + (
Frenet–Serret_formulas
Family of algebraic curves of the form r = sec(θ) + a*cos(θ)
acnode (0,0) not present in polar form. They are rational, circular, cubic plane curves. These expressions have an asymptote x = 1 (for a ≠ 0). The point
Conchoid_of_de_Sluze
timelike curve concavity Conchoid (mathematics) Confocal Contact (mathematics) Contour line Crunode Cubic Hermite curve Curvature Curve orientation Curve fitting
List_of_curves_topics
Family of closed mathematical curves
superellipse is a plane algebraic curve of order p/q. In particular, when a = b = 1 and n is an even integer, then it is a Fermat curve of degree n. In
Superellipse
Plane curve constructed from two other curves and a fixed point
Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves C1, C2 and a point O (the pole). Let L be a variable
Cissoid
polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point
Polar_curve
Plane curve: conic section
In mathematics, a parabola (/pəˈræbələ/ pə-RA-bə-lə) is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially
Parabola
Any finite subset of an algebraic curve has a linkage which traces it
self-intersecting cubic, smooth elliptic cubic and the trifolium curves Y. Liu's mechanical computation for drawing algebraic plane curves M. Gallet et al
Kempe's_universality_theorem
dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C* called the dual curve. In the correspondence
Plücker_formula
In algebraic geometry, a point with rational coordinates
rational over some finite extension of k (unless it is the cone over a plane cubic curve). Campana's conjecture would also imply that a K3 surface X (such
Rational_point
Compact Riemann surface of genus 3
conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane H2 by a certain cocompact group G that
Klein_quartic
Serpent-like curve
Newton and classified as a cubic curve in 1701. "Serpentine". Maths History. Retrieved 2025-09-20. Weisstein, Eric. "Serpentine Curve". Wolfram MathWorld. Retrieved
Serpentine_curve
Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique, 2e Série, 55 (3): 235–273, arXiv:math/0611590
Hessian_group
Quantity of a three-dimensional space
derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of
Volume
Curve whose curvature changes linearly
curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve
Euler_spiral
Algebraic surface defined by a cubic polynomial
irreducible cubic surface (possibly singular) over an algebraically closed field is rational unless it is the projective cone over a cubic curve. In this
Cubic_surface
Mathematical object studied in the field of algebraic geometry
curve; it can be viewed as the curve in the projective plane P2 = {[x, y, z]} defined by x = 0. For another example, first consider the affine cubic curve
Algebraic_variety
Point on a curve where motion must move backwards
the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation x = f ( t ) y = g ( t )
Cusp_(singularity)
Theorem in classical algebraic geometry
genus–degree formula relates the degree d {\displaystyle d} of an irreducible plane curve C {\displaystyle C} with its arithmetic genus g {\displaystyle g} via
Genus–degree_formula
Limit of the tangent line at a point that tends to infinity
books.google.com Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane", Mathematics Magazine, 72 (3): 183–192, doi:10.2307/2690881
Asymptote
x^{d}\right).} In the Euclidean plane, the moment curve is a parabola, and in three-dimensional space it is a twisted cubic. Its closure in projective space
Moment_curve
Curve created by a geometric operation
considered as a curve in the complex projective plane. In general, inversion with respect to an arbitrary curve may produce an algebraic curve with proportionally
Inverse_curve
Quartic plane curve
quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly
Limaçon_trisectrix
in 3 tangent points, such as a tritangent conic to a cubic curve or a tritangent plane of a cubic surface. trope A trope is a singular (meaning special)
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Mathematical paradox
two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after
Cramer's_paradox
Anglo-Irish mathematician and Vice-Provost of Trinity College Dublin
Contact of Cubic Curves, The Transactions of the Royal Irish Academy 25, Science (1875), 559–565.' 'On the Intersections of Plane Curves of the Third
Andrew_Searle_Hart
Mathematical concept
elliptic curves over K {\displaystyle K} (up to isomorphism). Suppose E {\displaystyle E} is given as a cubic curve in the projective plane by a homogeneous
Supersingular_elliptic_curve
Branch of mathematics
hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies
Algebraic_geometry
Algebraic curve
Retrieved 5 September 2013. Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves. New York: Dover Publications. pp. 106–8. ISBN 978-0-486-60288-2. Wikimedia
Folium_of_Descartes
Spline function
parametric curve does not interpolate the control points. Usually the curve does not pass through the control points. A cubic B-spline curve C ( t ) {\displaystyle
B-spline
Mathematical plane curve
negative pedal curve is a plane curve that can be constructed from another plane curve C and a fixed point P. For each point X ≠ P on the curve C, the negative
Negative_pedal_curve
Point where the curvature of a curve changes sign
inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the
Inflection_point
Cubic graph with 10 vertices and 15 edges
Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited
Petersen_graph
Geometric configuration of 9 points and 9 lines
any nonsingular cubic plane curve in the Euclidean plane, three real inflection points of the curve, and a fourth point on the curve, there is a unique
Pappus_configuration
Type of surface in algebraic geometry
a product of two curves and are not a complete intersection of divisors in an Abelian variety. The Fano surface S of a smooth cubic threefold F into P4
Fano_surface
curve Conics, Pascal's theorem, Brianchon's theorem Twisted cubic Elliptic curve, cubic curve Elliptic function, Jacobi's elliptic functions, Weierstrass's
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Curve that winds around a central point
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns
Spiral
Concept in algebraic geometry
information for the case g = 4, when a canonical curve is an intersection of a quadric and a cubic surface; and for g = 5 when it is an intersection
Canonical_bundle
Zero of the derivative of a function
they correspond to the points on the graph where the tangent plane is parallel to the xy plane. The notion of a stationary point allows the mathematical
Stationary_point
Theorem in projective geometry
elliptic curves by way of continuity. Suppose f is the cubic polynomial vanishing on the three lines through AB, CD, EF and g is the cubic vanishing
Pascal's_theorem
Geometry with 7 points and 7 lines
In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points
Fano_plane
Coordinates used to specify position of a line
projective plane, the dual of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane. For a curve f(x, y) = 0 in the plane, the
Line_coordinates
Concept in algebraic geometry
any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree. For points in the plane or on
General_position
geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew
Laguerre–Forsyth_invariant
Line tangent to a curve at two locations
algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with a
Bitangent
Simplest non-trivial closed knot with three crossings
the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic). If one end of a tape or belt is turned
Trefoil_knot
Plane algebraic curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(x, y) = 0, where F is a polynomial with real coefficients
Circular_algebraic_curve
Principle in geometry
points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve). There are additional subtleties
Five_points_determine_a_conic
Type of algebraic curve
hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There
Imaginary_hyperelliptic_curve
Straight figure with zero width and depth
ISBN 978-0-86720-093-5 Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane", Mathematics Magazine, 72 (3): 183–192, doi:10.2307/2690881
Line_(geometry)
Geometric configuration of 9 points and 12 lines
the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by
Hesse_configuration
Number of points needed to determine an algebraic curve
algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-degenerate
Cramer's theorem (algebraic curves)
Cramer's_theorem_(algebraic_curves)
variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the algebra is called
Elliptic_algebra
Model for thermodynamic phase transitions
set of geometrical instructions that modify a given constant temperature curve (isotherm) to produce its experimentally observed vapor-liquid phase transition
Maxwell_construction
Curve external to a family of curves in geometry
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency
Envelope_(mathematics)
which moves in a plane such that the sum of its distances from two fixed points – the foci – in the plane is a constant. The curve obtained when the
Generalized_conic
Stress intensity factor at which a crack's propagation increases drastically
thickness alone dictates the slope of R-curve. There are cases where even plane strain fracture ensues in rising R-curve due to "microvoid coalescence" being
Fracture_toughness
Scottish mathematician
He also proved the correctness of Isaac Newton's classification of cubic plane curves. Stirling was born on 11 May 1692 O.S. at Garden House near Stirling
James Stirling (mathematician)
James_Stirling_(mathematician)
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