| 单词 | quadric surface |
| 例句 | When a quadric surface intersects a coordinate plane, the trace is a conic section. Calculus, Volume 3 2016-03-30T00:00:00Z 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 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 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" Determine the axis of symmetry of the quadric surface. Calculus, Volume 3 2016-03-30T00:00:00Z 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 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 For the following exercises, rewrite the given equation of the quadric surface in standard form. 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 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 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 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 Use b. to write the equation of the quadric surface in standard form. Calculus, Volume 3 2016-03-30T00:00:00Z 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 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 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 For the following exercises, the equation of a quadric surface is given. 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 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 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 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 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 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 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 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 For the following exercises, the graph of a quadric surface is given. Calculus, Volume 3 2016-03-30T00:00:00Z 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 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 |
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