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Point on a curve where motion must move backwards
such a singularity is in the same differential class as the cusp of equation x 2 − y 5 = 0 , {\displaystyle x^{2}-y^{5}=0,} which is a singularity of type
Cusp_(singularity)
Point where a mathematical object behaves irregularly
coordinate system has a singularity (called a cusp) at ( 0 , 0 ) {\displaystyle (0,0)} . For singularities in algebraic geometry, see singular point of an algebraic
Singularity_(mathematics)
Topics referred to by the same term
a pointed structure on a tooth. Cusp or CUSP may also refer to: Cusp (singularity), a singular point of a curve Cusp catastrophe, a branch of bifurcation
Cusp
Mathematical theory
mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable
Singularity_theory
Neighborhood of a singularity of cusp type
In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity. The cusp neighborhood for a hyperbolic Riemann surface can
Cusp_neighborhood
Area of mathematics
dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena
Catastrophe_theory
Point on a curve not given by a smooth embedding of a parameter
singular point at the origin. However, a node such as that of y 2 − x 3 − x 2 = 0 {\displaystyle y^{2}-x^{3}-x^{2}=0} at the origin is a singularity of
Singular_point_of_a_curve
Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius
astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside
Astroid
Envelope of light rays reflected or refracted by a curved surface/object
as patches of light or their bright edges, shapes which often have cusp singularities. Concentration of light, especially sunlight, can burn. The word caustic
Caustic_(optics)
Concept in algebraic geometry
C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)} with the cusp singularity at the origin. Its normalization can be given by the map Spec ( k [
Normal_scheme
In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. Hirzebruch (1973)
Signature_defect
Concept in algebraic geometry
the rhamphoid cusp y2 = x5 has a singularity of order 2 at the origin. After blowing up at its singular point it becomes the ordinary cusp y2 = x3, which
Resolution_of_singularities
Concept in commutative algebra
Noetherian domain that is not a J-1 ring as S has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring
Excellent_ring
Compact astronomical body
hole would create a so-called naked singularity, a singularity outside of a black hole. Because these singularities make the universe inherently unpredictable
Black_hole
Mathematical curve with two cusps
cusps. This curve was further studied by Arthur Cayley in 1867. The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities
Bicorn
S} -module. Also S {\displaystyle S} has a cusp singularity at every closed point, so the set of singular points is not closed. (Danilov 2001) Akizuki
Nagata_ring
Topics referred to by the same term
Cuspidal point can refer to: Cuspidal point of a curve, see Cusp (singularity) Cuspidal point of a surface, see Pinch point (mathematics) This disambiguation
Cuspidal_point
Algebraic surface in mathematics
corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of X, but also singularities at its cusps. The surface Y
Hilbert_modular_variety
Physical theory of the cosmos
measurements of the expansion rate of the universe place the initial singularity at an estimated 13.787±0.02 billion years ago, which is considered the
Big_Bang
Mathematical knot
link of the cusp singularity z 2 + w 3 {\displaystyle z^{2}+w^{3}} ; the Hopf link (oriented correctly) is the link of the node singularity z 2 + w 2 {\displaystyle
Fibered_knot
Point on a curve at which two or more osculating circles are tangent
geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more)
Tacnode
is not a J-0 ring. More precisely S has a cusp singularity at every closed point, so the set of non-singular points consists of just the ideal (0) and
J-2_ring
Shape with same width in all directions
of constant width from involutes of curves with an odd number of cusp singularities, having only one tangent line in each direction (that is, projective
Curve_of_constant_width
Curve defined as zeros of polynomials
equations of the branches. For describing a singularity, it is worth to translate the curve for having the singularity at the origin. This consists of a change
Algebraic_curve
Analytic function on the upper half-plane with a certain behavior under the modular group
and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C-vector spaces of modular and cusp forms of weight
Modular_form
Italian-British applied mathematician
transition of a soap film surface by the emergence of a twisted fold (cusp) singularity. His current work aims to establish connections between isophase minimal
Renzo_L._Ricca
British mathematician and educator
Bifurcations of the global stable set of a planar endomorphism near a cusp singularity. International Journal of Bifurcation and Chaos in Applied Sciences
Catherine_Hobbs
Mathematical theory of knots
however cusps can appear. Generically, the front diagram of a knot as no tangency point, no triple intersection and standard cusp singularities. In this
Thurston–Bennequin_number
Singularities of algebraic varieties
(1985) and Reid. In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Canonical_singularity
2009 American film
DuMond. It is based on the ideas of transhumanism and the technological singularity. The film was released on October 2, 2009 at the Woodstock Film Festival
2B_(film)
French mathematician (1661–1704)
by James Gregory in letters to Collins (1670–1671), ibid. This singularity is a cusp of the second kind, in which both branches are concave from the
Guillaume_de_l'Hôpital
Classical theory of gravitation
singularity at the beginning of the universe, such as in the black hole cosmology, quantum cosmology, static universe, and cyclic model. Singularity theorems
Einstein–Cartan_theory
Point where the derivative of a function is zero or undefined (in certain cases)
the degrees of the polynomials that define the variety. Singular point of a curve Singularity theory Nullcline Milnor, John (1963). Morse Theory. Princeton
Critical_point_(mathematics)
Cubic plane curve
(0,0)} . At point ( 0 , 0 ) {\displaystyle (0,0)} the curve has a singularity (cusp). The proof follows from the tangent vector ( 2 t , 3 t 2 ) {\displaystyle
Semicubical_parabola
Two functions having equal values and derivatives at a given point
generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation
Contact_(mathematics)
Model for the origin of the universe
avoid a singularity. However, research in loop quantum cosmology purported to show that a previously existing universe collapses not to a singularity, but
Big_Bounce
Isolated point in the solution set of a polynomial equation in two real variables
must have a local minimum or a local maximum at the singularity. Singular point of a curve Crunode Cusp Tacnode Hazewinkel, M. (2001) [1994], "Acnode", Encyclopedia
Acnode
August 2023. Staff (2021–2025). "Energy Singularity: Faster Path to Commercial Fusion Energy". Energy Singularity. Shanghai, China. Retrieved 11 December
List of nuclear fusion companies
List_of_nuclear_fusion_companies
Type of mathematical curve
4a^{3}+27b^{2}=0} , the cubic is singular. If 4 a 3 + 27 b 2 = a = 0 {\displaystyle 4a^{3}+27b^{2}=a=0} , the singular point is a cusp. If 4 a 3 + 27 b 2 =
Cubic_plane_curve
British mathematician (1925–2016)
a British mathematician known for his work in geometric topology and singularity theory. Zeeman's main contributions to mathematics were in topology,
Christopher_Zeeman
Solution of Einstein field equations
regular (singularity-free) solution of the Einstein field equations. Gödel's original chart is geodesically complete and free of singularities. Therefore
Gödel_metric
Pn-1 is singular. double curve A 1-dimensional singularity, usually of a surface, of multiplicity 2 double point 1. A 0-dimensional singularity of multiplicity
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Centers of curvature of a curve
of singularity theory, evolutes are envelopes of smooth families of lines and can exhibit typical singularities such as cusps. These singularities correspond
Evolute
Belizean-American mathematical physicist
high energy physics, differential geometry, singularities, and probability theory. His monograph "Singularity Theory and Gravitational Lensing" developed
Arlie_Petters
Point of extreme curvature on a curve
a curve will generically have a cusp when the curve has a vertex; other, more degenerate and non-stable singularities may occur at higher-order vertices
Vertex_(curve)
points converge to the singular point and only 3 inflection remain along the singular curve. If the cubic degenerates and gets a cusp then only one inflection
Plücker_formula
1983 painting by Salvador Dalí
therefore seven possible discontinuities, or "elementary catastrophes": fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic
The_Swallow's_Tail
Subset of a function's domain on which its value is equal
example at a local extremum of f ) or may have a singularity such as a self-intersection point or a cusp. A set of the form L c − ( f ) = { ( x 1 , … ,
Level_set
Viewing certain time-periods more favorably
defined as "the egotism that one's own generation is poised on the very cusp of history". The term had been used earlier in a study about attitudes to
Chronocentrism
Branch of mathematics
function fields, and p-adic fields. A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry
Algebraic_geometry
Type of algebraic equation
plane curve it defines will have singular points; and the coefficients of P may be very large numbers. Further, the 'cusps' of the moduli problem, which
Modular_equation
Extinct genus of Cartilaginous fish
a singular species, A. harmsenae. It is named for sedimentologist Dr. Fraka Harmsen. It has unique teeth which are broad with widely divergent cusps separated
Aztecodus
Algebraic variety of dimension two
like algebraic curves, may possess singularities, which are points where there is no tangent plane. A singularity may be a self-crossing point or a point
Algebraic_surface
Proposed era of humanity after the Information Age
the Imagination Age concept through speeches at the O'Reilly Media, TED, Cusp, and Business Innovation Factory conferences. The term Imagination Age was
Imagination_Age
mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is
Hyperelliptic_surface
{x} \}\times M\to \mathbb {R} } has degenerate singularity at some p. A function has degenerate singularity if both the Jacobian matrix of first order partial
Affine_focal_set
Four-letter name of God in the Hebrew Bible
Name, in the mystery of ten and the mystery of four." Namely, the upper cusp of the Yod is Arich Anpin and the main body of Yod is and Abba; the first
Tetragrammaton
Roulette curve made from circles with radii that differ by factors of 3 or 1.5
known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference
Deltoid_curve
Point where a curve intersects itself at an angle
{\partial ^{2}f}{\partial x~\partial y}}\right)^{2}<0.} Singular point of a curve Acnode Cusp Tacnode Saddle point Salmon, George (1879). A treatise on
Crunode
"rubbish of the Dark Ages." Yet just as Petrarch, seeing himself at the cusp of a "new age," was criticising the centuries before his own time, so too
List of common misconceptions about the Middle Ages
List_of_common_misconceptions_about_the_Middle_Ages
amounts to saying that the singular point is a double point, rather than a cusp. Deciding whether this condition holds is effectively computable by Tate's
Semistable_abelian_variety
Theorem in geometric topology
run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching
Poincaré_conjecture
Theory of rapid universe expansion
densities. Such an interaction averts the unphysical Big Bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which
Cosmic_inflation
Reference frame
that is, the metric has a singularity. The Landau group have found that the synchronous frame necessarily forms a time singularity, that is, the time lines
Synchronous_frame
Mathematical concept
either an elliptic curve (type I0), or have a double point (type I1), or a cusp (type II). If the intersection matrix is affine A1, there are 2 components
Elliptic_surface
projections of a family of curves on the surface. The apparent contour can have cusps when the ray has higher order contact with the surface. This occurs when
Apparent_contour
Term for the Early Middle Ages
F. Oakley, The medieval experience: foundations of Western cultural singularity (University of Toronto Press, 1988), pp. 1-4. Daileader, Philip (2001)
Dark_Ages_(historiography)
American indie pop band
F<ck F^ck F>ck on July 29, 2022. The band's most recent album, Lady on the Cusp, was released on May 17, 2024. On April 2, 2026, the band released a single
Of_Montreal
Creature in Greek mythology
siren, who usually took a half-human, half-animal form somewhere on the cusp between nature and culture. Leonardo da Vinci wrote of them in his notebooks
Siren_(mythology)
English musician and actor (1947–2016)
2023. Retrieved 9 April 2023. Watts, Michael (22 January 2006). "On the cusp of fame, Bowie tells Melody Maker he's gay – and changes pop for ever". The
David_Bowie
Geometric space
adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb
Moduli_of_algebraic_curves
Electricity generation by nuclear fusion
Variations included the Tandem Mirror, magnetic bottle and the biconic cusp. A series of mirror machines were built by the US government in the 1970s
Fusion_power
Algebraic curve in mathematics
coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the
Elliptic_curve
Method for representing the local symmetries of a curve
repeated singularity for some u _ ∈ U . {\displaystyle {\underline {u}}\in U.} By a repeated singularity, we mean that the jacobian matrix is singular. Since
Symmetry_set
Generalized manifold
have conical singularities, because Rn/Zk has such a singularity at the fixed point of Zk. In string theory, gravitational singularities are usually a
Orbifold
Curve in the dual projective plane made from all lines tangent to a given curve
curve has three singularities – a node in the center, and two cusps at the lower right and lower left. The black curve has no singularities but has four
Dual_curve
Number, approximately 3.14
result approaches π as ε approaches zero. The point (0.25 + ε, 0) at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly:
Pi
Connects non-singular algebraic curves with compact Riemann surfaces
and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion
Belyi's_theorem
Attitudes and behaviors towards sex in ancient Rome
became androgynous, emphasizing that although the handsome youth was on the cusp of sexual adulthood, he rejected love as Narcissus had and likewise at the
Sexuality_in_ancient_Rome
almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the
Topological_modular_forms
Asymptotically stable in the sense of geometric invariant theory
0,1} are smooth and the degenerate points only have one double-point singularity. This example can be generalized to the case of a one-parameter family
Stable_curve
Mathematics award
three-manifolds, including proofs of Bers' conjecture on the density of cusp points in the boundary of the Teichmüller space, and Kra's theta-function
Fields_Medal
Indian mathematician (1887–1920)
has a generating function as the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a
Srinivasa_Ramanujan
Fundamental object of geometry
values 0 and 1. Accumulation point Affine space Boundary point Critical point Cusp Event (relativity) Foundations of geometry Locus (mathematics) Position (geometry)
Point_(geometry)
Plane algebraic curve defined by a 4th-degree polynomial
determines the size of the curve. The bicuspid has only the two cusps as singularities, and hence is a curve of genus one. The bow curve is a quartic plane
Quartic_plane_curve
In mathematics, straight line touching a plane curve without crossing it
vertical. The graph y = x2/3 illustrates another possibility: this graph has a cusp at the origin. This means that, when h approaches 0, the difference quotient
Tangent
'leaf', folium, is neuter. In descriptions of a single leaf, the neuter singular ending of the adjective is used, e.g. folium lanceolatum 'lanceolate leaf'
Glossary_of_leaf_morphology
Embedding a topological space into a compact space as a dense subset
single points for each cusp, making them Riemann surfaces (and so, since they are compact, algebraic curves). Here the cusps are there for a good reason:
Compactification (mathematics)
Compactification_(mathematics)
Medical condition
Caviar tongue is also referred to as sublingual varices (plural) and varix (singular) and look like varicose veins in the tongue. It is a benign, asymptomatic
Caviar_tongue
Fourth season of the V-League revival
October 2025. Retrieved 6 October 2025. "Adamson rallies past FEU, moves on cusp of historic title". V-League.ph. 6 October 2025. Retrieved 9 October 2025
2025 V-League Collegiate Challenge
2025_V-League_Collegiate_Challenge
for "boyfriend". culm /ˈ-ʌlm/ rhymes with stulm, another word for an adit. cusp /ˈ-ʌsp/ rhymes with dusp, an acronym for "dual-specificity phosphatase enzyme"
List of English words without rhymes
List_of_English_words_without_rhymes
American actor and filmmaker (born 1937)
stretch his acting skill: "I like to play people that haven't existed yet, a 'cusp character'", he said, "I have that creative yearning. Much in the way Chagall
Jack_Nicholson
Egyptian actress (born 1990)
Elbay's performance as "charming", and Esquire called her an actress "on the cusp of her big break." In October 2022, Elbay starred as Diana, Princess of Wales
Rosaline_Elbay
Family of closed mathematical curves
n = 2/3 and a = b, is a hypocycloid with four cusps. Deltoid curve, the hypocycloid of three cusps. Squircle, the superellipse with n = 4 and a = b
Superellipse
Curve of constant width Curve of pursuit Curves in differential geometry Cusp Cyclogon De Boor algorithm Differential geometry of curves Eccentricity (mathematics)
List_of_curves_topics
Tangent spaces in algebraic geometry
are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases
Zariski_tangent_space
Cohort born from 1946 to 1964
about 37,818,000 people. Others use the term Generation Jones to refer to a cusp generation, which includes those born in the latter half of the Baby Boomers
Baby_boomers
Mathematical idealization of the trace left by a moving point
onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points. A plane curve may also be completed to a curve
Curve
Locally spherical point on a mathematical surface
and outer deltoid - monstar pattern Outside outer deltoid - lemon pattern Cusps of the inner deltoid - cubic (symbolic) umbilics On the diagonals and the
Umbilical_point
Three dimensional analogue of uniformization conjecture
Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology
Geometrization_conjecture
CUSP SINGULARITY
CUSP SINGULARITY
Surname or Lastname
English
English : possibly a variant of Copp.Possibly an Americanized spelling of German Kopp.
Boy/Male
Biblical
Ethiopians, blackness.
Girl/Female
Gujarati, Indian
Cup
Boy/Male
Biblical
Threshold, silver cup.
Biblical
a hill; cup
Biblical
cup-bearer of the prince
Girl/Female
Arabic, Muslim
Wine Cup
Girl/Female
Biblical, Dutch, German
A Hill; Cup
Girl/Female
Arabic, Muslim
Wine Cup
Male
English
Anglicized form of Hebrew Kuwsh, CUSH means "black," i.e. "Ethiopian." In the bible, this is the name of a land and its people. It is also the name of a Benjamite and the son of Ham and grandson of Noah.
Boy/Male
Greek Latin
Cup bearer to the gods.
Biblical
threshold; silver cup
Girl/Female
Biblical
Hill, cup, thing lifted up.
Boy/Male
English
Cup bearer.
Boy/Male
Biblical
Cup-bearer of the prince.
Girl/Female
British, English
Happy
Boy/Male
English American
Forest; cup bearer.
Biblical
Cushan, Cushi, Ethiopians; blackness
Surname or Lastname
English
English : variant of Kiss.Americanized spelling of German and Jewish Kusch.
Boy/Male
Arthurian Legend
Name of a king.
CUSP SINGULARITY
CUSP SINGULARITY
Boy/Male
Hindu
Armed with the curved kodhanda bow
Surname or Lastname
English
English : variant of Dole or of Doll.Dutch : nickname for a stupid person.Americanized spelling of German Dollmann (see Dollman).Hungarian Dolmán : variant of Dolmány, metonymic occupational name or nickname from dolmány ‘embroidered coat’, named after a Szekler village in Transylvania called Dolmán. In some cases this may be an Americanized spelling of Dolmáni, habitational name for someone from the village itself.
Male
Egyptian
, a sacred scribe.
Girl/Female
Tamil
Mother
Girl/Female
Indian
Boy/Male
American, Australian
From the Lake
Boy/Male
American, Australian, British, Celtic, Christian, English, French, German
Honourable; Servant Spear; Derived from Combining an Old Name; Spear
Girl/Female
Hindu, Indian, Traditional
Time; Season
Boy/Male
Tamil
Shikhandin | ஷிகாநà¯à®¤à¯€à®¨
Lord Shiva, Lord Vishnu
Boy/Male
Hindu, Indian
Worship of God; One who Sings God's Praises
CUSP SINGULARITY
CUSP SINGULARITY
CUSP SINGULARITY
CUSP SINGULARITY
CUSP SINGULARITY
n.
A small vessel, used commonly to drink from; as, a tin cup, a silver cup, a wine cup; especially, in modern times, the pottery or porcelain vessel, commonly with a handle, used with a saucer in drinking tea, coffee, and the like.
n.
Anything shaped like a cup; as, the cup of an acorn, or of a flower.
n.
The point or horn of the crescent moon or other crescent-shaped luminary.
p. pr. & vb. n.
of Cusp
n.
A sharp and rigid point.
n.
A cup used for holding an egg, at table.
a.
Cup-shaped.
n.
The beginning or first entrance of any house in the calculations of nativities, etc.
imp. & p. p.
of Cusp
v. t.
To furnish with a cusp or cusps.
n.
The cusk. See Cusk.
a.
Having a calyx or cup; cup-shaped.
v. t.
To make concave or in the form of a cup; as, to cup the end of a screw.
v. t.
To supply with cups of wine.
p. pr. & vb. n.
of Cup
n.
A triangular protection from the intrados of an arch, or from an inner curve of tracery.
n.
A multiple point of a curve at which two or more branches of the curve have a common tangent.
n.
One who sneaks from his cups; one who balks his glass.
n.
A prominence or point, especially on the crown of a tooth.