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.

The standard projection model for the atom probe is an emitter geometry that is based upon a revolution of a conic section, such as a sphere, hyperboloid or paraboloid.

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 ).

As a parabola is a conic section, some sources refer to quadratic Béziers as " conic arcs ".

* Conic constant, a quantity describing conic sections, and is represented by the letter K

* Lambert conformal conic projection ( LCC ) is a conic map projection, which is often used for aeronautical charts

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 ).

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points ( the foci ) is constant.

Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

However, in projective geometry every conic section is equivalent to an ellipse.

* Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.

190 BCE ) is mainly known for his investigation of conic sections.

The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone.

The other conic sections are the parabola, the ellipse, and the circle ( the circle is a special case of the ellipse ).

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.

For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for " deficient " and " comparable "; these terms may refer to the eccentricity of these curves, which is greater than one ( hyperbola ), less than one ( ellipse ) and exactly one ( parabola ), respectively.

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

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.

A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis.

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

A general conic is defined by five points: given five points in general position, there is a unique conic passing through them.

Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci.

The Braikenridge – Maclaurin theorem may be applied in the Braikenridge – Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point.

A cissoid of Zahradnik ( name after Karel Zahradnik ) is defined as the cissoid of a conic section and a line relative to any point on 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.

Any conic section can be defined as the locus of points whose distances to a point ( the focus ) and a line ( the directrix ) are in a constant ratio.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section.

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

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 trisectrix of Maclaurin can be defined from conic sections in three ways.

This is a result of Feuerbach's conic theorem that states that for all circumconics of a reference triangle that also passes through its orthocenter, the locus of the center of such circumconics form the nine-point circle and that the circumconics can only be rectangular hyperbolas.

# 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 first theorem is that a closed conic section ( i. e. an ellipse ) is the locus of points such that the sum of the distances to two fixed points ( the foci ) is constant.

Using the Dandelin spheres, it can be proved that any conic section is the locus of points for which the distance from a point ( focus ) is proportional to the distance from the directrix.

This " polarity " can then be used to define the conic, in a manner that is perfectly symmetrical and immediately self-dual: a conic is simply the locus of points which lie on their polars, or the envelope of lines which pass through their poles.

A conic, C, being a ( 1, 1 ) curve on Q, meets the image of any line of Af, which we have already found to be a Af curve on Q, in Af points.

* Two conic sections generally intersect in four points, some of which may coincide.

:* Any conic should meet the line at infinity at two points according to the theorem.

Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles.

Furthermore because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular circumconic, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.

Any conic that passes through the four orthocentric points can only be a rectangular hyperbola.

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

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.

A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position ( no three are collinear ).

This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations ( e. g. conic sections ).

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.