| 单词 | tangent plane |
| 例句 | In this case, a surface is considered to be smooth at point P if a tangent plane to the surface exists at that point. Calculus, Volume 3 2016-03-30T00:00:00Z The characteristic property of all cylindrical surfaces is that the tangent planes are parallel to the axis. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. The Teaching of Geometry 2011-10-12T02:00:52.133Z The singular surface of the general quadratic complex is the famous quartic, with sixteen nodes and sixteen singular tangent planes, first discovered by E.E. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The direction of frictional resisting forces is in the common tangent plane of the two surfaces, and directly opposed to their relative motion. Friction, Lubrication and the Lubricants in Horology 2011-01-20T03:00:07.073Z If a function is differentiable at a point, then a tangent plane to the surface exists at that point. Calculus, Volume 3 2016-03-30T00:00:00Z Considered as a curved surface, concentric with the earth, a tangent plane to such a surface is the plane of the horizon. Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 7 "Arundel, Thomas" to "Athens" 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" The nodes of the singular surface are points whose complex cones are coincident planes, and the complex conic in a singular tangent plane consists of two coincident points. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Now the plane, which is tangent to the three spheres, is also evidently tangent to the three conical surfaces; and therefore the vertices of those conical surfaces are all in the tangent plane. Scientific American, Volume XXXVI., No. 8, February 24, 1877 A Weekly Journal of Practical Information, Art, Science, Mechanics, Chemistry, and Manufactures. For a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable at that point. Calculus, Volume 3 2016-03-30T00:00:00Z The curve is similar to the intersection of the surface with a parallel to the tangent plane and indefinitely near it. The Gutenberg Webster's Unabridged Dictionary Section I, J, K, and L If these lines lie in the same plane, they determine the tangent plane at that point. Calculus, Volume 3 2016-03-30T00:00:00Z 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 A tangent plane at a regular point contains all of the lines tangent to that point. Calculus, Volume 3 2016-03-30T00:00:00Z We define the term tangent plane here and then explore the idea intuitively. Calculus, Volume 3 2016-03-30T00:00:00Z This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. Calculus, Volume 3 2016-03-30T00:00:00Z A tangent plane to a surface does not always exist at every point on the surface. Calculus, Volume 3 2016-03-30T00:00:00Z Also the previous definition of an angle can be adapted to this case, by making t1 and t2 to be the tangent planes through the line p1p2 to the imaginary conic. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Figure 4.29 Graph of a function that does not have a tangent plane at the origin. Calculus, Volume 3 2016-03-30T00:00:00Z Figure 4.28 Calculating the equation of a tangent plane to a given surface at a given point. Calculus, Volume 3 2016-03-30T00:00:00Z When working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same. Calculus, Volume 3 2016-03-30T00:00:00Z Figure 4.31 Using a tangent plane for linear approximation at a point. Calculus, Volume 3 2016-03-30T00:00:00Z 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 If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z To develop a method that makes surface integrals easier to compute, we approximate surface areas ΔSij with small pieces of a tangent plane, just as we did in the previous subsection. 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 If P lies on the surface we take the tangent plane of P as its polar. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables. 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 |
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