For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

* Case g = 0: no points or infinitely many ; C is handled as a conic section.

These are ten in number ; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position ; compare to two points determining a line and how five points determine a conic.

A degenerate case of Pascal's Theorem ( four points ) is interesting ; given points ABCD on a conic Γ, the intersection of alternate sides, AB ∩ CD, BC ∩ DA, together with the intersection of tangents at opposite vertices ( A, C ) and ( B, D ) are collinear in four points ; the tangents being degenerate ' sides ', taken at two possible positions on the ' hexagon ' and the corresponding Pascal Line sharing either degenerate intersection.

Pick a generic point P on the conic and choose λ so that the cubic h = f + λg vanishes on P. Then h = 0 is a cubic that has 7 points A, B, C, D, E, F, P in common with the conic.

When C is a conic, the real foci defined this way are exactly the foci which can be used in the geometric construction of C.

* The projection from a point on C yields a conic section.

In mathematics, a Severi – Brauer variety over a field K is an algebraic variety V which becomes isomorphic to projective space over an algebraic closure of K. Examples are conic sections C: provided C is non-singular, it becomes isomorphic to the projective line over any extension field L over which C has a point defined.

The key tool is the curve X given by the set of pairs ( p, ℓ ) where p is on the conic C and ℓ is tangent to the conic D. Then X is smooth ; more specifically X is an elliptic curve.

For example, five points determine a conic, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.

The second theorem is that for any conic section, the distance from a fixed point ( the focus ) is proportional to the distance from a fixed line ( the directrix ), the constant of proportionality being called the eccentricity.

In the gravitational two-body problem, the orbits of the two bodies are described by two overlapping conic sections each with one of their foci being coincident at the center of mass ( barycenter ).

It can be thought of as a measure of how much the conic section deviates from being circular.

This can be shown to result in the trajectory being ideally a conic section ( circle, ellipse, parabola or hyperbola ) with the central body located at one focus.

A parabola may also be characterized as a conic section with an eccentricity of 1.

Scale 1: 1, 250 000 ; Lambert conformal conic proj.

2b, M < sub > 1 </ sub > and M '< sub > 2 </ sub > are tilted with respect to each other, the interference fringes will generally take the shape of conic sections ( hyperbolas ), but if M < sub > 1 </ sub > and M '< sub > 2 </ sub > overlap, the fringes near the axis will be straight, parallel, and equally spaced.

For example, eliminating z between the two equations x < sup > 2 </ sup > + y < sup > 2 </ sup > − z < sup > 2 </ sup > = 0 and x + 2y + 3z − 1 = 0, which defines an intersection of a cone and a plane in three dimensions, we obtain the conic section 8x < sup > 2 </ sup > + 5y < sup > 2 </ sup > − 4xy + 2x + 4y − 1 = 0, which in this case is an ellipse.

The male ( pollen ) cones are slender conic, 5 – 11 cm long and 1 – 2 cm broad and reddish-brown in colour and are lower on the tree than the seed cones.

If four points are collinear, however, then there is not a unique conic passing through them – one line passing through the four points, and the remaining line passes through the other point, but the angle is undefined, leaving 1 parameter free.

The Hilbert-Hurwitz result from 1890 reducing the diophantine geometry of curves of genus 0 to degrees 1 and 2 ( conic sections ) occurs in Chapter 17, as does Mordell's conjecture.

# a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than 1 ( the first recorded proofs of the properties, which do not appear in Apollonius ).

The cones are conic, cylindrical or ovoid ( egg-shaped ), and small to very large, from 2 – 60 cm long and 1 – 20 cm broad.

The theorem was generalized by Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points.

This means genus 0 for the case n = 2 ( a conic ) and genus 1 only for n = 3 ( an elliptic curve ).

A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If e is between zero and one the conic is an ellipse ; if e = 1 the conic is a parabola ; and if e > 1 the conic is a hyperbola.

" The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves ( or surfaces ) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form

A cross between a spiral and a helix, such as the curve shown in red, is known as a conic helix.

Orbits are conic sections, so, naturally, the formulas for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:

Any conic section defined over F with a rational point in F is a rational curve.

Apollonius's genius reaches its highest heights in Book v. Here he treats of normals as minimum and maximum straight lines drawn from given points to the curve ( independently of tangent properties ); discusses how many normals can be drawn from particular points ; finds their feet by construction ; and gives propositions that both determine the center of curvature at any point and lead at once to the Cartesian equation of the evolute of any conic.

In mathematics, a degenerate conic is a conic ( degree-2 plane curve, the zeros of a degree-2 polynomial equation, a quadratic ) that fails to be an irreducible curve.

The solution is elementary ( as we would now say, it computes a local zeta-function, for a curve that is a conic ).

The conic is isomorphic to the canonical system via the isomorphism, and each of the six lines is naturally isomorphic to the dual canonical system via the identification of theta divisors and translates of the curve.

Hirsch extends this argument to any surface of revolution generated by a conic, and shows that intersection with a bitangent plane must produce two conics of the same type as the generator when the intersection curve is real.

A conic is a degree 2 plane curve, thus defined by an equation:

He developed a new conic curve based on the unit square.

He described the string construction of the ellipse and he wrote a book on conic sections, which was excellent preparation for designing the elaborate vaulting of Hagia Sophia.

Particularly of interest to Pascal was a work of Desargues on conic sections.

It states that if a hexagon is inscribed in a circle ( or conic ) then the three intersection points of opposite sides lie on a line ( called the Pascal line ).

Gauss's method involved determining a conic section in space, given one focus ( the Sun ) and the conic's intersection with three given lines ( lines of sight from the Earth, which is itself moving on an ellipse, to the planet ) and given the time it takes the planet to traverse the arcs determined by these lines ( from which the lengths of the arcs can be calculated by Kepler's Second Law ).

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum ( Theory of motion of the celestial bodies moving in conic sections around the Sun ).

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections.

Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

* Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject.

Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the cone.

If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola.

190 BC ) in his definitive work on the conic sections, the Conics.

Aeroplane pilots use aeronautical charts based on a Lambert conformal conic projection, in which a cone is laid over the section of the earth to be mapped.

The earliest known work on conic sections was by Menaechmus in the fourth century BC.

** Conon of Samos, Greek mathematician and astronomer whose work on conic sections ( curves of the intersections of a right circular cone with a plane ) serves as the basis for the fourth book of the Conics of Apollonius of Perga ( b. c. 280 BC )

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant, the same can be done for a second-order PDE at a given point.

* Conon of Samos, Greek mathematician and astronomer whose work on conic sections ( curves of the intersections of a right circular cone with a plane ) serves as the basis for the fourth book of the Conics of Apollonius of Perga ( b. c. 280 BC )

For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

* Apollonius of Perga ( Pergaeus ), Greek astronomer and mathematician specialising in geometry and noted for his writings on conic sections ( d. c. 190 BC )