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Longer titles found: Quartic plane curve (view), Real plane curve (view), Cubic plane curve (view)

searching for Plane curve 91 found (227 total)

alternate case: plane curve

Curvature (6,488 words) [view diff] exact match in snippet view article find links to article

curve can be parametrized with respect to arc length. In the case of a plane curve, this means the existence of a parametrization γ(s) = (x(s), y(s)), where

curves with the same turning number are regular homotopic. Every immersed plane curve lifts to an embedded space curve via separating the intersection points

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at

differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the

geometry, an equichordal point is a point defined relative to a convex plane curve such that all chords passing through the point are equal in length. Two

affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine

by saying that P, in the worst case, will be of high degree and the plane curve it defines will have singular points; and the coefficients of P may be

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 {\displaystyle

another circle Epitrochoid – Plane curve formed by rolling a circle on the outside of another Epicycloid – Plane curve traced by a point on a circle

plane sextic curve, studied by Wirtinger in 1892, is a degree 6 genus 4 plane curve with double points at the 6 vertices of a complete quadrilateral. Coble

3}}\approx 1.8928} . Sierpiński demonstrated that his carpet is a universal plane curve. That is: the Sierpiński carpet is a compact subset of the plane with

Neumann, Walter (1985). Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematical Studies. Vol. 110. Princeton, N

{\displaystyle \gamma \cdot v\equiv 0} , so α {\displaystyle \alpha } is a plane curve. Consider a point γ ( T ) {\displaystyle \gamma (T)} such that curves

an open interval and let γ : I → R2 be a smooth parametrisation of a plane curve. We assume that γ(I) is a non-degenerate curve (in the sense of Nomizu

See Salmon (1879, p. 127). 2.  A certain plane curve of degree (n–1)(n–2) constructed from a plane curve of degree n and a generic point. (Coolidge

endpoints of a corresponding plane curve C {\displaystyle C} of less curvature. Suppose C ( s ) {\displaystyle C(s)} is a plane curve with curvature κ ( s )

finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents

Campillo, F. Delgado, and S. M. Gusein-Zade. "The Alexander polynomial of a plane curve singularity via the ring of functions on it". Duke Mathematical Journal

no vertical or oblique asymptotes. Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)). Suppose that the curve tends to infinity

Rotation of a celestial body's orbital line of apsides Cardioid – Type of plane curve Cyclograph – Instrument for drawing arcs of large diameter with no clear

Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus g = ( d − 1 ) ( d − 2 ) 2 − s , {\displaystyle

French mathematician Louis Poinsot. Mathematics portal Cotes's spiral – Plane curve Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves. New York:

follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line.

invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom

Every non-degenerate triangle is strictly convex. Convex curve – Type of plane curve Concave polygon – Simple polygon which is not convex Convex polytope –

Many of these elements can be classified according to form (straight, plane / curve) and dimensionality (one-dimensional / two-dimensional): Columns are

previous example may be generalized to any algebraic plane curve in the following way. Let C be a plane curve defined by the equation f(x,y) = 0, where f is

Plane curve formed by rolling a circle on the outside of another

rocks, or debris Tell – Ancient settlement mound Witch of Agnesi – Cubic plane curve Whittow, John (1984). Dictionary of Physical Geography. London: Penguin

function field of the projective line. Consider the affine algebraic plane curve defined by the equation y 2 = x 5 + 1 {\displaystyle y^{2}=x^{5}+1}

function's epigraph in terms of its supporting hyperplanes. Convex curve - a plane curve that lies entirely on one side of each of its supporting lines. The interior

given rational point on it. By the Riemann–Roch theorem, an irreducible plane curve of degree d {\displaystyle d} given by the vanishing locus of a section

generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the generatrix

points of a plane algebraic curve meeting a given line. If C is a complex plane curve given by the zeros of a polynomial f(x,y) of two variables, and L is

Cohen–Macaulay curves can be classified completely by looking at the plane curve case. Using the criterion, there are easy examples of non-Cohen–Macaulay

contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures

t ) {\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)} are the plane curve given by h {\displaystyle h} , but the fibers over V = V ( s t ) {\displaystyle

derivatives of the curvature are equal. For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal

{\mathbb {C} \{z_{1},\ldots ,z_{n}\}}{(y^{2}-x^{n})}}} representing a plane-curve singularity. A germ of analytic algebras is then an object in the opposite

the ruling on the surface. The envelope of this family of lines is a plane curve whose inverse image under the development is the edge of regression.

can use the arithmetic genus of the curve. Noether's method takes a plane curve and repeatedly applies quadratic transformations (determined by a singular

surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation: A x 2 + B x y + C y 2 + D x z + E y z +

injective on the set of the solutions and that its image is an irreducible plane curve. For more difficult examples, a similar proof may always be given, but

the normal line to S {\displaystyle S} cuts S {\displaystyle S} in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve

example, consider the analytification X {\displaystyle X} a complex plane curve Proj ( C [ x , y , z ] x n + y n + z n ) {\displaystyle {\text{Proj}}\left({\frac

or y=x. Let I ⊂ R be an open interval and let γ : I → R2 be a smooth plane curve parametrised by arc length. Consider the one-parameter family of normal

Algebraic plane curve

chern classes. For example, if we have a degree d {\displaystyle d} plane curve in P 2 {\displaystyle \mathbb {P} ^{2}} , we find the total chern class

are not determined by X(k) in any reasonable sense. For example, the plane curve X over the real numbers defined by x2 + y2 = −1 has X(R) empty, but X(C)

{\displaystyle \gamma :I\to \mathbb {R} ^{2}} be a parametrisation of a smooth plane curve. The symmetry set of γ ( I ) ⊂ R 2 {\displaystyle \gamma (I)\subset \mathbb

formula. This can also be derived from multivariable integration. If a plane curve is given by ⟨ x ( t ) , y ( t ) ⟩ {\displaystyle \langle x(t),y(t)\rangle

Eisenbud, D. Neumann, W. Three-dimensional link theory and invariants of plane curve singularities. Ann. of Math. Stud. 110 Budney, R. JSJ-decompositions

361–369. MR1500754 1917: "The maximum number of cusps of an algebraic plane curve, and enumeration of self-dual curves", Bulletin of the American Mathematical

Algebraic plane curve

\end{aligned}}} This can also be derived from multivariable integration. If a plane curve is given by ⟨ x ( t ) , y ( t ) ⟩ {\displaystyle \langle x(t),y(t)\rangle

line intersects it. Suppose γ {\displaystyle \gamma } is a rectifiable plane curve. Given an oriented line ℓ, let n γ {\displaystyle n_{\gamma }} (ℓ) be

circumscribing square (whose side is of length d). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle

straightness measures, defined by ISO 12780 on a cross-section (the plane curve resulting from the intersection of the surface of interest and a plane

and the discovery of a turning point singularity on the involute of a plane curve near an inflection point. L'Hôpital exchanged ideas with Pierre Varignon

smooth compactification. For example, on a genus 0 {\displaystyle 0} plane curve C {\displaystyle C} logarithmic cohomology of C {\displaystyle C} with

Quadratrix as a plane curve for side length a = 1 {\displaystyle a=1} , as given by the parametric formula for −∞ < t < ∞, with singularities when t is

{\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ {\displaystyle \gamma } divides

topological generalization of the classical four-vertex theorem, that a plane curve has at least four extrema of curvature, specifically, at least two local

first appeared in Roquette (1970). The Tate curve is the projective plane curve over the ring Z[[q]] of formal power series with integer coefficients

of p in K to itself. The ramification locus of the double cover is a plane curve C of degree 6, and all the nodes of K which are not p map to nodes of C

of a function. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at

centre Hypersurface, a generalization of the concepts of hyperplane, plane curve, and surface Hyperstructure, an algebraic structure equipped with at

{\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ {\displaystyle \gamma } divides

Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial

in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at

equals a, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric

stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal. A formal power series version of Zariski's

dimension less than n − 1 {\displaystyle n-1} . For example, a complex plane curve near every singular point z 0 {\displaystyle z_{0}} has its Milnor fiber

Soc. 45 (1939) 158–163. doi:10.1090/S0002-9904-1939-06929-2 Anomalous plane curve systems associated with singular surfaces. Bull. Amer. Math. Soc. 46

identifying solutions to an equation of two variables with points on a plane curve. To achieve geometric solutions without using coordinates, Bolzano introduced

is zero-dimensional, it may be non-transverse, for example, if V is a plane curve and W is one of its tangent lines. The first problem requires the machinery

to 2 (that is, no element is its own additive inverse) is an affine plane curve defined by the equation: E E , a , d : a x 2 + y 2 = 1 + d x 2 y 2 {\displaystyle

the differential geometry of curves, the total curvature of a smooth plane curve is the amount it turns anticlockwise, in radians, from start to finish

+d_{r}-n-1)|_{X},} by the adjunction formula. For example, the dualizing sheaf of a plane curve X of degree d is O ( d − 3 ) | X {\displaystyle O(d-3)|_{X}} . In particular

proof contained. It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving. In fact, even though Gauss redid

i / 3 {\displaystyle \tau =e^{2\pi i/3},e^{\pi i/3}} pg 78. Given a plane curve by its Weierstrass equation y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b}

{\displaystyle {\mathcal {F}}} a quasi-coherent sheaf. Given a smooth projective plane curve C {\displaystyle C} of degree d {\displaystyle d} , the sheaf cohomology

previous isomorphism to verify the genus of a degree d {\displaystyle d} plane curve. Since x d + y d + z d {\displaystyle x^{d}+y^{d}+z^{d}} is a smooth

intersection points and of the common horizontal asymptotes. A rational plane curve may be defined by a parametric equation x = P ( t ) R ( t ) , y = Q (

Conic plane curve associated with a given triangle

sufficiently large set of points in general position, we may ensure that no plane curve of degree N (and hence any lower degree) contains all the points of Z

lines. Usually a surface of revolution is determined by a generating plane curve (meridian) m 0 {\displaystyle m_{0}} . Rotating m 0 {\displaystyle m_{0}}

these generalized conics. For constants r0 ≥ 0, λ ≥ 0 and real k, a plane curve described by the polar equation r = r 0 1 + λ sin ⁡ ( k θ ) {\displaystyle

t_{b_{2}({\mathfrak {X}})}} . Its mirror manifold is singular over a complex plane curve Σ {\displaystyle \Sigma } given by a polynomial equation P ( e x , e

x n + 1 = r x ( 1 − x n ) {\displaystyle x_{n+1}=rx(1-x_{n})} is the plane curve that plots the relationship between x n {\displaystyle x_{n}} and x n

Plane curve drawn by an odd number of circular arcs.