What is "cusp"

Cusp \Cusp\ (k[u^]sp), n. [L. cuspis, -idis, point, pointed end.]

1. (Arch.) A triangular protection from the intrados of an arch, or from an inner curve of tracery.

2. (Astrol.) The beginning or first entrance of any house in the calculations of nativities, etc.

3. (Astron) The point or horn of the crescent moon or other crescent-shaped luminary.

4. (Math.) A multiple point of a curve at which two or more branches of the curve have a common tangent.

5. (Anat.) A prominence or point, especially on the crown of a tooth.

6. (Bot.) A sharp and rigid point.

Cusp \Cusp\, v. t. [imp. & p. p. Cusped (k[u^]spt); p. pr. & vb. n. Cusping.] To furnish with a cusp or cusps.

1580s, from Latin cuspis "point, spear, pointed end, head," which is of unknown origin. Astrological use is earliest.

n. 1 A sharp point or pointed end. 2 (context figuratively English) An important moment when a decision is made that will determine future events.

1. n. point formed by two intersecting arcs (as from the intrados of a Gothic arch)

2. a thin triangular flap of a heart valve [syn: leaflet]

3. small elevation on the grinding surface of a tooth

Cusp may refer to:

• Cusp (anatomy), a pointed structure on a tooth
• Cusp (astrology), a boundary between zodiac signs and houses
• Cusp (novel), a 2005 science-fiction story by Robert A. Metzger
• Cusp (singularity) (geometry and astrophysics), a singular point of a curve
• Beach cusps, a pointed and regular arc pattern of the shoreline at the beach
• Behavioral cusp (psychology), an important behavior-change with far-reaching consequences
• Cusp catastrophe, a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry
• Cusp form in modular form theory
• Cusp generation, a name given to those born during the transitional years of two generations.
• Cusps of heart valves, leaflets of a heart valve
• Cuspidal representation, a generalization of cusp forms in the theory of automorphic representations
• Caltech-USGS Seismic Processing, software for analyzing earthquake data
• Center for Urban Science and Progress, a research institute planned for New York
• CubeSat for Solar Particles (CuSP) a planned NASA satellite

In astrology, a cusp (from the Latin for spear or point) is the imaginary line that separates a pair of consecutive signs in the zodiac or houses in the horoscope.

Because the solar disc has a diameter of approximately half a degree, it is possible for the Sun to straddle the cusp as it moves across the sky. When this occurs at the moment of birth such a person is said to be "born on the cusp" and some believe that their life is influenced by the characteristics of both signs. For example, if an individual was born when the Sun (by convention the point at the centre of the Solar disc) was located at 29 degrees, 50 minutes Gemini, then one might say that he was born on the cusp of Gemini and Cancer. Much of the Solar disc was actually in Cancer even though the centre was in Gemini.

Although the term "cusp" is universally used for the boundaries of signs, not all astrologers agree that an object can ever be included in more than one sign. Many consider relevant only the location of the Sun's centre, which must be entirely in one sign, and would describe the natal Sun in the example as simply being in Gemini. If late degrees of Gemini have a Cancer-like character, they would describe that as simply the nature of that part of Gemini rather than some influence spilling over from the next sign.

On the other hand, astrologers who consider objects "on the cusp" to be meaningfully different from objects entirely in one sign may apply such a description even when no part of the object crosses the boundary. That point of view may consider the Sun to be "on the cusp" even when its centre is as much as two degrees away from the sign boundary. They may also call other objects (much less than half a degree in diameter) "on the cusp" despite no part of the object being in the adjacent sign. Their claim is that the influence of the cusp gets weaker but does not suddenly disappear as the object gets further from the cusp.

A similar debate applies to cusps between houses. The majority of astrologers treat the Sun as always being in exactly one sign, and consider the "Sun between signs, or in two at once" interpretation of cusps to be a misinterpretation.

Cusp is a science fiction novel written by Robert A. Metzger, in the category of hard science fiction. It deals with two perpendicular rings running along the Earth's surface, that act as cosmic jets, using ionized hydrogen. In this universe, the fusion of organic and non organic material is an everyday thing. Multiple characters are portrayed in the story, and it is told from a third-person narrative.

A cusp is a pointed, projecting, or elevated feature. In animals, it is usually used to refer to raised points on the crowns of teeth.

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

For a plane curve defined by a differentiable parametric equation

$$\begin{align} x&=f(t)\\ y&=g(t), \end{align}$$
a cusp is a point where both derivatives of and are zero, and at least one of them changes sign. Cusps are local singularities in the sense that they involve only one value of the parameter , contrarily to self-intersection points that involve several values.

For a curve defined by an implicit equation

F(x, y) = 0, 
cusps are points where the terms of lowest degree of the Taylor expansion of are a power of a linear polynomial; however not all singular points that have this property are cusps. In some contexts, and in the remainder of this article, one restricts the definition of a cusp to the case where the non-zero part of lowest degree of the Taylor expansion of has degree two.

A plane curve cusp may be put in one of the following forms by a diffeomorphism of the plane: x − y = 0, where k ≥ 1 is an integer.