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Conjugate focus

Focus Fo"cus, n.; pl. E. Focuses, L. Foci. [L. focus hearth, fireplace; perh. akin to E. bake. Cf. Curfew, Fuel, Fusil the firearm.] 1. (Opt.) A point in which the rays of light meet, after being reflected or refrcted, and at which the image is formed; as, the focus of a lens or mirror. 2. (Geom.) A point so related to a conic section and certain straight line called the directrix that the ratio of the distace between any point of the curve and the focus to the distance of the same point from the directrix is constant. Note: Thus, in the ellipse FGHKLM, A is the focus and CD the directrix, when the ratios FA:FE, GA:GD, MA:MC, etc., are all equal. So in the hyperbola, A is the focus and CD the directrix when the ratio HA:HK is constant for all points of the curve; and in the parabola, A is the focus and CD the directrix when the ratio BA:BC is constant. In the ellipse this ratio is less than unity, in the parabola equal to unity, and in the hyperbola greater than unity. The ellipse and hyperbola have each two foci, and two corresponding directrixes, and the parabola has one focus and one directrix. In the ellipse the sum of the two lines from any point of the curve to the two foci is constant; that is: AG+GB=AH+HB; and in the hyperbola the difference of the corresponding lines is constant. The diameter which passes through the foci of the ellipse is the major axis. The diameter which being produced passes through the foci of the hyperbola is the transverse axis. The middle point of the major or the transverse axis is the center of the curve. Certain other curves, as the lemniscate and the Cartesian ovals, have points called foci, possessing properties similar to those of the foci of conic sections. In an ellipse, rays of light coming from one focus, and reflected from the curve, proceed in lines directed toward the other; in an hyperbola, in lines directed from the other; in a parabola, rays from the focus, after reflection at the curve, proceed in lines parallel to the axis. Thus rays from A in the ellipse are reflected to B; rays from A in the hyperbola are reflected toward L and M away from B. 3. A central point; a point of concentration. Aplanatic focus. (Opt.) See under Aplanatic. Conjugate focus (Opt.), the focus for rays which have a sensible divergence, as from a near object; -- so called because the positions of the object and its image are interchangeable. Focus tube (Phys.), a vacuum tube for R[oe]ntgen rays in which the cathode rays are focused upon the anticathode, for intensifying the effect. Principal, or Solar, focus (Opt.), the focus for parallel rays.

Focus Fo"cus, n.; pl. E. Focuses, L. Foci. [L. focus hearth, fireplace; perh. akin to E. bake. Cf. Curfew, Fuel, Fusil the firearm.] 1. (Opt.) A point in which the rays of light meet, after being reflected or refrcted, and at which the image is formed; as, the focus of a lens or mirror. 2. (Geom.) A point so related to a conic section and certain straight line called the directrix that the ratio of the distace between any point of the curve and the focus to the distance of the same point from the directrix is constant. Note: Thus, in the ellipse FGHKLM, A is the focus and CD the directrix, when the ratios FA:FE, GA:GD, MA:MC, etc., are all equal. So in the hyperbola, A is the focus and CD the directrix when the ratio HA:HK is constant for all points of the curve; and in the parabola, A is the focus and CD the directrix when the ratio BA:BC is constant. In the ellipse this ratio is less than unity, in the parabola equal to unity, and in the hyperbola greater than unity. The ellipse and hyperbola have each two foci, and two corresponding directrixes, and the parabola has one focus and one directrix. In the ellipse the sum of the two lines from any point of the curve to the two foci is constant; that is: AG+GB=AH+HB; and in the hyperbola the difference of the corresponding lines is constant. The diameter which passes through the foci of the ellipse is the major axis. The diameter which being produced passes through the foci of the hyperbola is the transverse axis. The middle point of the major or the transverse axis is the center of the curve. Certain other curves, as the lemniscate and the Cartesian ovals, have points called foci, possessing properties similar to those of the foci of conic sections. In an ellipse, rays of light coming from one focus, and reflected from the curve, proceed in lines directed toward the other; in an hyperbola, in lines directed from the other; in a parabola, rays from the focus, after reflection at the curve, proceed in lines parallel to the axis. Thus rays from A in the ellipse are reflected to B; rays from A in the hyperbola are reflected toward L and M away from B. 3. A central point; a point of concentration. Aplanatic focus. (Opt.) See under Aplanatic. Conjugate focus (Opt.), the focus for rays which have a sensible divergence, as from a near object; -- so called because the positions of the object and its image are interchangeable. Focus tube (Phys.), a vacuum tube for R[oe]ntgen rays in which the cathode rays are focused upon the anticathode, for intensifying the effect. Principal, or Solar, focus (Opt.), the focus for parallel rays.

Conjugate focus

Conjugate Con"ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con- + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See Join.] 1. United in pairs; yoked together; coupled. 2. (Bot.) In single pairs; coupled. 3. (Chem.) Containing two or more radicals supposed to act the part of a single one. [R.] 4. (Gram.) Agreeing in derivation and radical signification; -- said of words. 5. (Math.) Presenting themselves simultaneously and having reciprocal properties; -- frequently used in pure and applied mathematics with reference to two quantities, points, lines, axes, curves, etc. Conjugate axis of a hyperbola (Math.), the line through the center of the curve, perpendicular to the line through the two foci. Conjugate diameters (Conic Sections), two diameters of an ellipse or hyperbola such that each bisects all chords drawn parallel to the other. Conjugate focus (Opt.) See under Focus. Conjugate mirrors (Optics), two mirrors so placed that rays from the focus of one are received at the focus of the other, especially two concave mirrors so placed that rays proceeding from the principal focus of one and reflected in a parallel beam are received upon the other and brought to the principal focus. Conjugate point (Geom.), an acnode. See Acnode, and Double point. Self-conjugate triangle (Conic Sections), a triangle each of whose vertices is the pole of the opposite side with reference to a conic.

Conjugate Con"ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con- + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See Join.] 1. United in pairs; yoked together; coupled. 2. (Bot.) In single pairs; coupled. 3. (Chem.) Containing two or more radicals supposed to act the part of a single one. [R.] 4. (Gram.) Agreeing in derivation and radical signification; -- said of words. 5. (Math.) Presenting themselves simultaneously and having reciprocal properties; -- frequently used in pure and applied mathematics with reference to two quantities, points, lines, axes, curves, etc. Conjugate axis of a hyperbola (Math.), the line through the center of the curve, perpendicular to the line through the two foci. Conjugate diameters (Conic Sections), two diameters of an ellipse or hyperbola such that each bisects all chords drawn parallel to the other. Conjugate focus (Opt.) See under Focus. Conjugate mirrors (Optics), two mirrors so placed that rays from the focus of one are received at the focus of the other, especially two concave mirrors so placed that rays proceeding from the principal focus of one and reflected in a parallel beam are received upon the other and brought to the principal focus. Conjugate point (Geom.), an acnode. See Acnode, and Double point. Self-conjugate triangle (Conic Sections), a triangle each of whose vertices is the pole of the opposite side with reference to a conic.

- perception of the moving object, and focus on the object rather than the rest of the visual world. When changing focus, conjugate eye movements allow for the perception...

- two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy...

- phase-conjugate beam, Ξ4. Its direction is given by k4 = k1 + k2 − k3, and so if the two pump beams are counterpropagating (k1 = −k2), then the conjugate and...

- the points lie on two conjugate diameters (see below). (If a = b {\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If...

- axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate axis or minor axis of length 2 b {\displaystyle 2b} , corresponding to...

- muscle of the upper eyelid. The conjugate gaze is the motion of both eyes in the same direction at the same time, and conjugate gaze palsy refers to an impairment...

- June 1960 The focus distance to have the DOF extend between given near and far object distances is the harmonic mean of the object conjugates. Most helicoid-focused...

- other words, a point P is a focus if both PI and PJ are tangent to C. When C is a real curve, only the intersections of conjugate pairs are real, so there...

- {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1\ .} A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, such diameters...

- In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following...

- two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy...

- phase-conjugate beam, Ξ4. Its direction is given by k4 = k1 + k2 − k3, and so if the two pump beams are counterpropagating (k1 = −k2), then the conjugate and...

- the points lie on two conjugate diameters (see below). (If a = b {\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If...

- axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate axis or minor axis of length 2 b {\displaystyle 2b} , corresponding to...

- muscle of the upper eyelid. The conjugate gaze is the motion of both eyes in the same direction at the same time, and conjugate gaze palsy refers to an impairment...

- June 1960 The focus distance to have the DOF extend between given near and far object distances is the harmonic mean of the object conjugates. Most helicoid-focused...

- other words, a point P is a focus if both PI and PJ are tangent to C. When C is a real curve, only the intersections of conjugate pairs are real, so there...

- {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1\ .} A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, such diameters...

- In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following...

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