In geometry, the foci (singular focus) are a pair of special points used in describing conic sections. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface The four types of conic sections are the circle, parabola, ellipse, and hyperbola. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions
The focus has two equivalent defining properties; and they always fall on the major axis of symmetry of the conic. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect The simpler depends on the type of conic:
- In an ellipse, the sum of the distances from any point on the ellipse to the two foci is a constant (which is always the length of the major axis of the ellipse). In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae
- In a circle, there is only one focus, the center of the circle, and all the points of the circle are equidistant from it. (This can be viewed a special case of the above, with a circle being an ellipse with two foci at the same point; the sum of the distances is the diameter. )
- In a hyperbola, the difference of the distances is always constant.
- A parabola also only has one focus (although it is sometimes useful to speak of a focus at infinity); but there is a line called the directrix such that the distance from any point of the parabola to the focus is equal to the (perpendicular) distance from the point to the directrix. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface
The rule for the parabola can be generalized to other conics, and this is the other defining property: A conic section can be defined as the set of points such that the ratio of distance to its focus to the distance to the corresponding directrix is a constant, called the eccentricity. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section.
The circle has eccentricity 0, and the directrix is a line at infinity. "Ideal line" redirects here For the ideal line in racing see Racing line. The focus-directrix property is thus true of the circle, but it is also true of every other point on the plane.
Conics in projective geometry
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface
For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle.
For the parabola, the center of the directrix moves to the point at infinity (see projective geometry). Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. The directrix 'circle' becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and 'at infinity' become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).
To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the 'right-hand' arm of one branch of a hyperbola meets the 'left-hand' arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity.
In projective geometry, all conics are equivalent in the sense that every theorem that can be proved for one conic section applies to all the others.
Astronomical significance
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). Gravitation is a natural Phenomenon by which objects with Mass attract one another In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other
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