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42-576: The first Anastigmat was designed by Paul Rudolph for the German firm Carl Zeiss AG in 1890 and marketed as the Protar ; it consisted of four elements in two groups, as an asymmetric arrangement of two cemented achromatic lens doublets and was improved to a five-element, two-group design in 1891, substituting a cemented triplet for the rear group. In 1892, the Swiss mathematician Emil von Höegh designed

84-416: A = b , with radius r = a = b . For the parabola, the standard form has the focus on the x -axis at the point ( a , 0) and the directrix the line with equation x = − a . In standard form the parabola will always pass through the origin. For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are

126-401: A . (Here a is the semi-major axis defined below.) A parabola may also be defined in terms of its focus and latus rectum line (parallel to the directrix and passing through the focus): it is the locus of points whose distance to the focus plus or minus the distance to the line is equal to 2 a ; plus if the point is between the directrix and the latus rectum, minus otherwise. In addition to

168-515: A quadratic equation in two variables which can be written in the form A x 2 + B x y + C y 2 + D x + E y + F = 0. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.} The geometric properties of the conic can be deduced from its equation. In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending

210-412: A change of coordinates ( rotation and translation of axes ) these equations can be put into standard forms . For ellipses and hyperbolas a standard form has the x -axis as principal axis and the origin (0,0) as center. The vertices are (± a , 0) and the foci (± c , 0) . Define b by the equations c = a − b for an ellipse and c = a + b for a hyperbola. For a circle, c = 0 so

252-406: A focus; its half-length is the semi-latus rectum ( ℓ ). The focal parameter ( p ) is the distance from a focus to the corresponding directrix. The major axis is the chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the semi-major axis ( a ). When an ellipse or hyperbola are in standard position as in

294-418: A line, five points determine a conic . Formally, given any five points in the plane in general linear position , meaning no three collinear , there is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear

336-506: A measure of how far the ellipse deviates from being circular. If the angle between the surface of the cone and its axis is β {\displaystyle \beta } and the angle between the cutting plane and the axis is α , {\displaystyle \alpha ,} the eccentricity is cos ⁡ α cos ⁡ β . {\displaystyle {\frac {\cos \alpha }{\cos \beta }}.} A proof that

378-425: A non-circular conic to be the set of those points whose distances to some particular point, called a focus , and some particular line, called a directrix , are in a fixed ratio, called the eccentricity . The type of conic is determined by the value of the eccentricity. In analytic geometry , a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy

420-400: A non-degenerate conic. There are three types of conics: the ellipse , parabola , and hyperbola . The circle is a special kind of ellipse, although historically Apollonius considered it a fourth type. Ellipses arise when the intersection of the cone and plane is a closed curve . The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone; for

462-432: A right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, "conic" in this article will refer to

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504-411: A right cone, this means the cutting plane is perpendicular to the axis. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola . In the remaining case, the figure is a hyperbola : the plane intersects both halves of the cone, producing two separate unbounded curves. Compare also spheric section (intersection of a plane with

546-488: A separate family of anastigmat lens designs, including the Voigtländer Heliar (designed by Hans Harting, 1900), Ludwig Bertele 's Ernostar (1919), and the later Zeiss Sonnar (Bertele, 1929). All modern photographic lenses are close to being anastigmatic, meaning that they can create extremely sharp images for all objects across their field of view ; the underlying limitation is that the lens can deliver

588-575: A series of anastigmatic lenses consisting of multiple cemented achromats in 1895, designed by Hugh L. Aldis , marketed as the Stigmatic . The first Stigmatic was a six-element, three-group design. Aldis simplified the lens to a three-element, two-group design after leaving Dallmeyer in 1901. Zeiss would withdraw the Anastigmat from the market in favor of the Unar and Tessar types, developed in

630-410: A sphere, producing a circle or point), and spherical conic (intersection of an elliptic cone with a concentric sphere). Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points P whose distance to a fixed point F (called the focus ) is a constant multiple e (called the eccentricity ) of the distance from P to a fixed line L (called

672-619: Is a specialization of the homogeneous form used in the more general setting of projective geometry (see below ). The conic sections described by this equation can be classified in terms of the value B 2 − 4 A C {\displaystyle B^{2}-4AC} , called the discriminant of the equation. Thus, the discriminant is − 4Δ where Δ is the matrix determinant | A B / 2 B / 2 C | . {\displaystyle \left|{\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right|.} If

714-399: Is again the determinant of the 2 × 2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form. In polar coordinates , a conic section with one focus at the origin and, if any, the other at a negative value (for an ellipse) or a positive value (for a hyperbola) on the x -axis, is given by the equation where e is

756-401: Is also called the semi-minor axis. The following relations hold: For conics in standard position, these parameters have the following values, taking a , b > 0 {\displaystyle a,b>0} . After introducing Cartesian coordinates , the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of

798-1545: The Cartesian coordinate system , the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate ), and all conic sections arise in this way. The most general equation is of the form with all coefficients real numbers and A, B, C not all zero. The above equation can be written in matrix notation as ( x y ) ( A B / 2 B / 2 C ) ( x y ) + ( D E ) ( x y ) + F = 0. {\displaystyle {\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}D&E\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+F=0.} The general equation can also be written as ( x y 1 ) ( A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ) ( x y 1 ) = 0. {\displaystyle {\begin{pmatrix}x&y&1\end{pmatrix}}{\begin{pmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}=0.} This form

840-602: The Dagor (aka Double Anastigmatic Goerz ) for Goerz , a symmetric lens with six elements in two groups, made of two cemented triplets. The Orthostigmat (1893) and Collinear (1895) were developed at around the same time by Steinheil and Voigtländer, respectively, and had a similar symmetric construction with six elements in two groups. At about the same time, Rudolph created the Double Protar (1894/1895), which consisted of eight elements in two groups. The Cooke Triplet

882-531: The Tessar design can clearly be traced from the Protar through the Unar . At about the same time the Unar was released by Zeiss, von Höegh modified the Dagor as a symmetric lens with four elements in four groups, released by Goertz as the Type B in 1899 and later renamed Celor and Syntor . The so-called dialyte-type lens consists of a pair of air-spaced two-element achromats arranged back-to-back, and later

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924-449: The directrix ). For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane. The eccentricity of an ellipse can be seen as

966-521: The hyperbola , the parabola , and the ellipse ; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga 's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines

1008-429: The 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, as is the determinant of the 3 × 3 matrix above . The constant term F and the sum D + E are invariant under rotation only. When the conic section is written algebraically as the eccentricity can be written as a function of the coefficients of the quadratic equation. If 4 AC = B

1050-400: The Euclidean plane to include a line at infinity, obtaining a projective plane , the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides

1092-508: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.150 via cp1114 cp1114, Varnish XID 923839984 Upstream caches: cp1114 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 08:00:24 GMT Conic section A conic section , conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane . The three types of conic section are

1134-402: The above curves defined by the focus-directrix property are the same as those obtained by planes intersecting a cone is facilitated by the use of Dandelin spheres . Alternatively, an ellipse can be defined in terms of two focus points, as the locus of points for which the sum of the distances to the two foci is 2 a ; while a hyperbola is the locus for which the difference of distances is 2

1176-484: The anastigmatic performance only up to a maximum aperture (i.e., it has a minimum F-number ) and only within a given working distance (focusing range). Note that all optical aberrations (except spherical aberration) become more pronounced towards the edges of the field of view, even with high-grade anastigmatic lenses. Anastigmatic performance is accomplished by a proper combination of multiple lenses (optical surfaces), usually three or more. Aspheric lenses can minimize

1218-401: The conic is non-degenerate , then: In the notation used here, A and B are polynomial coefficients, in contrast to some sources that denote the semimajor and semiminor axes as A and B . The discriminant B – 4 AC of the conic section's quadratic equation (or equivalently the determinant AC – B /4 of the 2 × 2 matrix) and the quantity A + C (the trace of

1260-411: The conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by where η = 1 if the determinant of the 3 × 3 matrix above is negative and η = −1 if that determinant is positive. It can also be shown that the eccentricity is a positive solution of

1302-476: The coordinate axes and the line x = y is the principal axis. The foci then have coordinates ( c , c ) and (− c , − c ) . The first four of these forms are symmetric about both the x -axis and y -axis (for the circle, ellipse and hyperbola), or about the x -axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines y = x and y = − x . These standard forms can be written parametrically as, In

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1344-428: The early 1900s. Rudolph's Unar (1899) was derived from the earlier Protar but used four elements in four groups, eliminating the cemented interfaces. This in turn was improved by returning to the cemented rear group, resulting in the enduring four-element, three-group Tessar design (1902). Although some have speculated the Tessar was influenced by the earlier Cooke Triplet , Rudolf Kingslake emphatically declared

1386-432: The eccentricity ( e ), foci, and directrix, various geometric features and lengths are associated with a conic section. The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center . A parabola has no center. The linear eccentricity ( c ) is the distance between the center and a focus. The latus rectum is the chord parallel to the directrix and passing through

1428-449: The eccentricity and l is the semi-latus rectum. As above, for e = 0 , the graph is a circle, for 0 < e < 1 the graph is an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. The polar form of the equation of a conic is often used in dynamics ; for instance, determining the orbits of objects revolving about the Sun. Just as two (distinct) points determine

1470-419: The equation can be converted to canonical form in transformed variables x ~ , y ~ {\displaystyle {\tilde {x}},{\tilde {y}}} as or equivalently where λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} are the eigenvalues of

1512-410: The equation where again Δ = A C − B 2 4 . {\displaystyle \Delta =AC-{\frac {B^{2}}{4}}.} This has precisely one positive solution—the eccentricity— in the case of a parabola or ellipse, while in the case of a hyperbola it has two positive solutions, one of which is the eccentricity. In the case of an ellipse or hyperbola,

1554-402: The equations below, with foci on the x -axis and center at the origin, the vertices of the conic have coordinates (− a , 0) and ( a , 0) , with a non-negative. The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis ( b ), the same value b as in the standard equation below. By analogy, for a hyperbola the parameter b in the standard equation

1596-498: The matrix ( A B / 2 B / 2 C ) {\displaystyle \left({\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right)} — that is, the solutions of the equation — and S {\displaystyle S} is the determinant of the 3 × 3 matrix above , and Δ = λ 1 λ 2 {\displaystyle \Delta =\lambda _{1}\lambda _{2}}

1638-401: The means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry . A conic is the curve obtained as the intersection of a plane , called the cutting plane , with the surface of a double cone (a cone with two nappes ). It is usually assumed that the cone is

1680-498: The number of surfaces required and thus the bulk and weight of the composite lens; however, aspheric surfaces are more costly to manufacture than spherical and other conic section (hyperbolic, parabolic) ones. Many high-end catoptric telescopes are three-mirror anastigmat , while the corresponding catadioptric telescopes use two mirrors (reflector) and one lens (refractor) to accomplish the same result. Paul Rudolph (physicist) Too Many Requests If you report this error to

1722-587: Was developed by H. Dennis Taylor for T. Cooke & Sons in York and patented in 1893. Cooke was not interested in manufacturing the lens, so a smaller workshop in Leicester , Taylor, Taylor and Hobson (no relation), was contracted to build the lens, bearing the Cooke brand. Its relatively simple three-element, three-group construction gave it a cost advantage over prior designs. J H Dallmeyer Ltd first released

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1764-618: Was developed into the Goertz Artar by W. Zschokke. The Dagor also was modified by E. Arbeit who removed one cemented surface, leaving it as a six-element, four-group design. The Schulz and Billerbeck company of Potsdam released Arbeit's modification as the Euryplan in 1903, generically known as the air-spaced Dagor . Paul Rudolph would go on to release a similar design for Hugo Meyer as the Plasmat in 1918. The Cooke Triplet spawned

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