| 单词 | quadric |
| 例句 | “We were all trying to solve this quadric the same way, and then I realized we just had to invert the way we were looking at it.” Tradition 2018-05-01T00:00:00Z When a quadric surface intersects a coordinate plane, the trace is a conic section. Calculus, Volume 3 2016-03-30T00:00:00Z A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z If the boundary is a closed quadric, there is one possible congruence-group of the hyperbolic type. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" 2011-02-27T03:00:31.973Z However, this does not have to be the case for all quadric surfaces. Calculus, Volume 3 2016-03-30T00:00:00Z Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z ELLIPSOID, a quadric surface whose sections are ellipses. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 3 "Electrostatics" to "Engis" 2011-02-06T03:00:53.093Z Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Consider first a numerical quadric equation with imaginary coefficients. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" 2011-02-27T03:00:31.973Z We also note there are x terms and y terms that are not squared, so this quadric surface is not centered at the origin. Calculus, Volume 3 2016-03-30T00:00:00Z The diameter of a quadric surface is a line at the extremities of which the tangent planes are parallel. Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4 "Diameter" to "Dinarchus" "Tell this wise gentleman your solution of that pretty question relating to the concomitants of a system of ternary quadrics." Punch Among the Planets A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Evidently the method gives for L� a quadric equation, which is the “resolvent” equation in this particular case. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" 2011-02-27T03:00:31.973Z Use the graph of the given quadric surface to answer the questions. Calculus, Volume 3 2016-03-30T00:00:00Z Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Painvin’s complex generated by the lines of intersection of perpendicular tangent planes of a quadric, the singular surface now being Fresnel’s wave surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z By projecting this quadric stereographically into space of four dimensions we obtain Klein’s analogy. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Hence:— If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Use b. to write the equation of the quadric surface in standard form. Calculus, Volume 3 2016-03-30T00:00:00Z If through one point on a quadric surface, two, and only two, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is ruled quadric surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Using the definitions at the end of � 95, we may also say:— On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Instead of a circle or sphere we may take any conic or quadric. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z If a line moves so that it always cuts three given lines of which no two meet, then it generates a ruled quadric surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, the equation of a quadric surface is given. Calculus, Volume 3 2016-03-30T00:00:00Z From what has been said the theorem follows: A ruled quadric surface contains two sets of straight lines. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The tangent plane to a quadric surface either cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z This configuration of sixteen points and planes has many interesting properties; thus each plane contains six points which lie on a conic, while through each point there pass six planes which touch a quadric cone. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The tangents drawn from a point P to a quadric surface form a cone of the second order, for the polar plane of P cuts it in a conic. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface. Calculus, Volume 3 2016-03-30T00:00:00Z If, however, the rays a1 and a2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z It begins with Arthur Cayley, who showed that metrical properties are projective properties relative to a certain fundamental quadric, and that different geometries arise according as this quadric is real, imaginary or degenerate. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Every plane section of a quadric surface is a conic or a line-pair. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, the graph of a quadric surface is given. Calculus, Volume 3 2016-03-30T00:00:00Z The following propositions follow:— A quadric surface has at every point a tangent plane. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The surface itself is therefore called a quadric surface, or a surface of the second order. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z These are called chief-tangent curves; on a quadric surface they are the above straight lines. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a node, it must therefore consist of two straight lines. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Determine the axis of symmetry of the quadric surface. Calculus, Volume 3 2016-03-30T00:00:00Z In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions; when two lines intersect the representative points are on the same generator of this quadric. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Every conic which has five points in common with a quadric surface lies on the surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The harmonic complex, first studied by Battaglini, is generated in an infinite number of ways by the lines cutting two quadrics harmonically. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Through two conics which lie in different planes, but have two points in common, and through one external point always one quadric surface may be drawn. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, rewrite the given equation of the quadric surface in standard form. Calculus, Volume 3 2016-03-30T00:00:00Z Every plane which cuts a quadric surface in a line-pair is a tangent plane. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in �� 89-93, but with one important exception. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Every line which has three points in common with a quadric surface lies on the surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Poles and Polar Planes.—The theory of poles and polars with regard to a conic is easily extended to quadric surfaces. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z This plane is called the polar plane of the point P, with regard to the quadric surface. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The Species of Quadric Surfaces.—Surfaces represented by equations of the second degree are called quadric surfaces. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometry 733 at this stage The theory of distance will be considered after the principles of descriptive geometry have been developed. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z There is one congruence-group corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z |
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