(No attempt has been made so far to deal with the problem after the occurrence of such a cusp, but something could certainly be done about it.). Under these circumstances, two different grinding wheels are required for the concave and convex sides (Fixed-Setting method). Hence, The expansion of the angle characteristic up to the fourth order for a refracting surface of revolution was derived in § 4.1. Hu, in Mechanics of Sheet Metal Forming … A surface of revolution is an area generated by revolving a segment about an axis (see figure). With a large number of blade groups, the lengthwise tooth curvature at the toe is significantly larger than that at the heel (but its values on the concave side and those on the convex side are comparable). Subsequently, having nearly reached the local angular velocity, the liquid moves outwards as a thinning/diverging film under the prevailing centrifugal acceleration as will be shown below. Let P = (xo,x1,…, xn) be a partition of [a, b] and for each r = 0, 1, …, n, let Xr be the point (xr, f (xr)) on the curve. [Morgan and Johnson, Theorem 2.2] show that in any smooth compact Riemannian manifold, minimizers for small volume are nearly round spheres. Added May 1, 2019 by mkemp314 in Astronomy. If greater accuracy is required, the full system is solved iteratively using this solution as an initial value. (My use of the word "approximate" will be explained shortly, and until then I'll just keep saying disk and I'll also stop specifying that we only want the surface areas of the boundaries.) The circles in M generated under revolution by each point of C are called the parallels of M; the different positions of C as it is rotated are called the meridians of M.This terminology derives from the geography of the sphere; however, a sphere is not a surface of revolution as defined above. Therefore, parameters RpCNV and φCNV must be selected accordingly. For objects such as cubes or bricks, the surface area of the object is … A curve in. We define the area of such a surface by first approximating the curve with line segments. The following results are fundamental for Riemannian immersions: Let f(M, g) → (N, h) be a pseudo-Riemannian immersion. The axis of revolution is taken as x-axis, and the surface is defined initially in cylindrical coordinates (x, r) by giving x and r as functions of the arc length s along a meridian; for subsequent times s is retained as a Lagrangean parameter. Show that the covariant surface base vectors, with u = u1 and v = u2, are, in background cartesian co-ordinates and that the covariant metric tensor has components, which are functions of u but not v, while the contravariant metric tensor is, A surface vector A has covariant and contravariant components with respect to the surface base vectors given, respectively, by, it follows by comparison with eqn (3.26) that duα/ds = λα represents the contravariant components of a unit surface vector. To be determined are the cylindrical coordinates x(s, t), r(s, t) of the deformed surface. Round balls about the origin are known to be minimizing in certain two-dimensional surfaces of revolution (see the survey by Howards et al. Also acting on the element are the principal tensions, Tθ = σθt and Tϕ = σϕt. (12.18) has to be modified to take into account the vertical component of the forces due to self-weight. Surface area is the total area of the outer layer of an object. Let f(x) be a nonnegative smooth function over the interval [a, b]. is a differentiable map X : I —> R3. For a straight blade tool, the corresponding grinding wheel geometry is specified by the four parameters in Fig. Such short diffusion/conduction path lengths stimulate excellent heat, mass and momentum transfer between the gas phase and the liquid, and between the rotating surface and the liquid. On the other hand, when the grinding wheel is finishing the convex side at the heel (minimum curvature), its lengthwise curvature must be smaller than or comparable with that of the tooth. This result may be compared with the general equations for a scalar product in eqn (1.54). Proof The proof is omitted. We shall make use of these results in Section 12. 2.1 What Is a Curve. If it were 1, that piece of surface would not be separating any regions. Surface of Revolution Description Calculate the surface area of a surface of revolution generated by rotating a univariate function about the horizontal or vertical axis. Therefore r = R cos β gives the extreme lines of latitude on the shell reached by the geodesic. (mathematics) A surface formed when a given curve is revolved around a given axis. With the aid of Hamilton's principle the equations of motion are found to be: The subscripts refer to differentiations. The use of the coordinate system associated with trajectories is not always the most effective method of geometrization. 4) is due to the fact that, unlike in the Semi-Completing process, two different grinding wheels are used here for the concave (CNV) and the convex (CVX) sides. The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution. R.J. Lewandowski, W.F. When the grinding wheel is finishing the concave side at the toe (maximum curvature), its lengthwise curvature must be larger than or comparable with that of the tooth, otherwise it would interfere with other tooth parts. Corollary 16.7.3 Let C be the curve given by the polar equation, where r has a continuous derivative on [α, β]. (b) Principal radii of curvature at the point P. (c) Geometric relations at P. A. Artoni, ... M. Guiggiani, in International Gear Conference 2014: 26th–28th August 2014, Lyon, 2014. The same rolling argument implies that the root of the tree has just one branch. The approximate solution given by Shewmon (1964) has the same form as (7.6), but a different coefficient. A smooth map f : M → N is a pseudo-Riemannian immersion if it satisfies f*h = g. In this case we may consider the tangent bundle TM as a sub-bundle of the induced vector bundle f*(TN) to which we give the pseudo-Riemannian structure induced from h and the linear connection Then, using the addition theorem of § 5.4 it follows from (12) on comparison with § 5.3 (3) that, These are the Seidel formulae for the primary aberration coefficients of a general centred system of refracting surfaces. Surfaces of revolution. For that reason we summarise the main results of immersion theory. Below is a sketch of a function and the solid of revolution we get by rotating the function about the x x -axis. If for simplicity the arbitrary length λ0 in the plane of the entrance pupil is taken equal to unity, and the relations Di = t′i - s′i = ti+1 - si+1 are used, it is seen from (9) that hi and Hi may be calculated in succession from the relations*w. From (9) and from the Abbe relations (4) and (5) we obtain the following relation, which may be used as check on the calculations and which will be needed later: Colin McGregor, ... Wilson Stothers, in Fundamentals of University Mathematics (Third Edition), 2010, Let f be a real function with a continuous derivative on [a, b], and consider the surface of revolution formed by revolving, once about the x-axis, the curve. Then the argument above shows that the resulting surface of revolution is exactly M: g(x, y, z) = c. Using the chain rule, it is not hard to show that dg is never zero on M, so M is a surface. The structure theorem now follows, since the only possible structures are bubbles of one region in the boundary of the other. The resulting surface therefore always has azimuthal symmetry. For a spherical inclusion of radius R,∫y2dA=8πR4/3, so that. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. R3. (5.225) formulated for a basic surface that is not necessarily a surface of revolution. We use cookies to help provide and enhance our service and tailor content and ads. We can derive a formula for the surface area much as we derived the formula for arc length. The angle characteristic of a reflecting surface of revolution. In drawing processes (along the left-hand diagonal) the material does not change thickness and it is preferable to use a non-strain-hardening sheet as there is no danger of necking; strain-hardening would only increase the forming loads and make the process more difficult to perform. MAX BORN M.A., Dr.Phil., F.R.S., EMIL WOLF Ph.D., D.Sc., in Principles of Optics (Sixth Edition), 1980. This example is from Wikipedia and may be reused under a CC BY-SA license. Not mine but couldnt figure out how to use my subscription fee to see steps Solid of Revolution - Visual. Viewed from E3 this vector λ has cartesian components, (which may be regarded as a set of direction cosines) and background contravariant curvilinear components, The angle θ between directions specified by unit surface vectors λ and μ each satisfying aαβλαλβ = 1and aαβμαμβ = 1is given by. 5.9. D¯ which is induced from the Levi-Civita connection from h. Let NM be the orthogonal complement of TM in f*(TN). After eliminating h in the preceding relation: The surface energy of the spherical cap with surface tension γ is. What happened was that the membrane began to move toward the axis of revolution, eventually reaching it at some point. Surface Area = ∫b a(2πf(x)√1 + (f′ (x))2)dx. A surface of revolution is the surface that you get when you rotate a two dimensional curve around a specific axis. With reference to Figure 23, the interface is a surface of revolution. Regularity, including the 120-degree angles, comes from applying planar regularity theory [Morgan 19] to the generating curves; also the curves must intersect the axis perpendicularly. Parameter s is the arc length along the profile direction: s = 0 at the beginning of the root fillet, and it increases going upwards. The surface element is at a radius r and subtends an angle dθ. Hu, in Mechanics of Sheet Metal Forming (Second Edition), 2002. 12.7 subjected to internal pressure. Elementary Differential Geometry (Second Edition), Handbook of Computer Aided Geometric Design, Theory of Intense Beams of Charged Particles, The expansion up to fourth degree for the angle characteristic associated with a reflecting, The expansion of the angle characteristic up to the fourth order for a refracting, Fundamentals of University Mathematics (Third Edition), is either the standard double bubble or another. This eliminates the first problem, but produces the opposite of the second problem, giving higher weighting to errors in position of points nearer the axis. Let us consider the spatial flows with no symmetry and define the coordinate system xi by the relation, The presence of the new unknown function v3 allows implementation of a coordinate system with g13 ≡ 0. The quantity ρ is the initial surface density per unit area, and r0(s) is the radial coordinate of the initial surface. Figure 14.10.2. Proof sketch. This makes an angle ϕ with the axis. The bubble mustbe connected, or moving components could create illegal singularities (or alternatively an asymmetric minimizer). I = [a, b] be an interval on the real line. 2. However, to do so requires a knowledge of appropriate techniques of numerical analysis (which are in turn based on the mathematical theory of the partial differential equations involved), and the availability of a high speed digital computer. Hence, using (16.7.1), the area of revolution is. Example 16.7.5 Find the surface area of a sphere, radius R. Solution We can think of the required area A as the area of revolution generated by the upper half of the circle x2 + y2 = R2 which has the polar equation, Frank Morgan, in Geometric Measure Theory (Third Edition), 2000. Generalization to a centred system consisting of any number of refracting surfaces is now straightforward. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Micro-Drops and Digital Microfluidics (Second Edition), Mechanics of Sheet Metal Forming (Second Edition), Grinding face-hobbed hypoid gears through full exploitation of 6-axis hypoid generators, International Gear Conference 2014: 26th–28th August 2014, Lyon, Motions of Microscopic Surfaces in Materials. A surface of revolution is formed when a curve is rotated about a line. The rotation of a curve (called generatrix ) around a fixed line generates a surface of revolution. Round balls are known to be minimizing also in Sn and Hn [Schmidt]. when x and r are assumed independent of time, the equilibrium positions are obtained by the rotation of catenaries to yield the classical form given by the calculus of variations when the problem of minimizing the area of surfaces of revolution is studied (since the surface of minimum area yields the configuration having minimum potential energy). If N (β) sin βdβ is the number of fibers per unit length of the equator with inclinations to it lying between β and β + dβ, it can be shown that for a sphere, The fiber distribution is independent of the angle β. The other principal radius of curvature of the surface is ρ1, as shown. The stresses set up on any element are thus only the so-called "membrane stresses" σ1 and σ2 mentioned above, no additional bending stresses being required. Strictly, all three of these stresses will vary in magnitude through the thickness of the shell wall but provided that the thickness is less than approximately one-tenth of the major, i.e. Surface Area of Revolution . We define a tensor B: TM ⊕ NM → TM such that for vectors U, V in TM and X in NM. 55. He considered various types of materials, such as rubber-like (Mooney) materials, metallic materials, and the soap film. Because of this limitation on thickness, which makes the system statically determinate, the shell can be considered as a membrane with little or no resistance to bending. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. 12.7. The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution. In the simplest application, i.e. It is, therefore, an effective enabler for the desktop strategy noted above. using eqn (3.17). For small A, the solution is a disc, for large A, the solution is an annular band. If it were 0, an argument given by [Foisy, Theorem 3.6] shows that the bubble could be improved by a volume-preserving contraction toward the axis (r → (rn−1 − ε)1/(n−1)). Using the same notation as in the preceding section (cf. for (da, d¯a) under the constraints ‖da‖ = 1, 〈da, d¯a 〉 = 0. In general, you can skip parentheses, but be … Then, the surface area of the surface of revolution formed by revolving the graph of f(x) around the x-axis is given by. See Figure 16.7.1. Because of (4) we have, Using this relation, (2) may be written as, In (6), the arguments may be replaced by their Gaussian approximations; in particular, the Seidel variables referring to points on the incident and the refracted ray may be interchanged. ), in certain n-dimensional cones [Morgan and Ritoré], and in Schwarzschild-like spaces by Bray and Morgan, with applications to the Penrose Inequality in general relativity. The numerical integration of the dynamical equations was carried out by R. W. Dickey in the vicinity of the unstable equilibrium position predicted by the variational method after disturbing the system in various ways. in which α is the inclination of the geodesic to the line of latitude that has a radial distance r from the axis, and β is the inclination of the geodesic to the line of latitude of radius R. Attention here is restricted to shells of revolution in which r decreases with increasing z2. The stress system set up will be three-dimensional with stresses σ1 (hoop) and σ2 (meridional) in the plane of the surface and σ3 (radial) normal to that plane. (For a development and discussion of this theory, see [10].) When a liquid flow is supplied to, or near, the centre of a rotating surface of revolution an outwardly flowing liquid film is generated. The formulas we use to find surface area of revolution are different depending on the form of the original function and the a We can use integrals to find the surface area of the three-dimensional figure that’s created when we take a function and rotate it around an axis and over a certain interval. Date: 1840 a surface formed by the revolution of a plane curve about a line in its plane New Collegiate Dictionary. Find the volume of the solid of revolution formed. Now, suitable values of RpCVX and φCVX should be determined, but they would be different from those selected for the concave side: in particular, we would end up with RpCVX > RpCNV. Z. Marciniak, ... S.J. However, when m0 and m1 are eliminated from (39) with the help of the two identities connecting the ray components, different expressions for T (as a function of four ray components) are obtained in the two cases. Hence, if (4) is also used, where (8) and § 5.2 (7) was used, (7) becomes, If as before, r2, ρ2 and κ2 denote the three rotational invariants, the terms in the curly brackets of (6) become. We also have to determine the quantities hi and Hi. Examples of how to use “surface of revolution” in a sentence from the Cambridge Dictionary Labs Its profile curve must twice meet the axis of revolution, so two “parallels” reduce to single points. Thus for a dome of subtended arc 2θ with a force per unit area q due to self-weight, eqn. If the minimizer were continuous in A, it would have to become singular to change type. The manufacturing equipment used to filament wind is more expensive than that required for hand lay-up but production is much faster and less hand labor is required. In a later section we wish to consider surfaces of revolution obtained by rotation of special curves. Area of a Surface of Revolution. See the proof of Corollary 16.6.3. R-, R, R∇ are the curvature operators of This is the normal bundle of the immersion. When the region is rotated about the z-axis, the resulting volume is given by V=2piint_a^bx[f(x)-g(x)]dx. Calculate the surface area generated by rotating the curve around the x-axis.. Rotate the line. where smallest, radius of curvature of the shell surface, this variation can be neglected as can the radial stress (which becomes very small in comparison with the hoop and meridional stresses). The coordinate r is the radius from the origin to the point P (or the distance to the origin) and θ … (1.109) appear as, For an equipotential emitter, we have at U = const, The current conservation equation in (1.109) takes the form, The Poisson equation in (1.109) remains unchanged. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. The Gaussian lateral magnification between the object and the image plane (l1/l0) and between the planes of the entrance and the exit pupil (λ1/λ0) may be obtained from § 4.4 (14) and § 4.4 (10), or more simply by noting that imaging by a spherical surface is a projection from the centre of the sphere. Such a surface is We want to define the area of a surface of revolution in such a way that it corresponds The necessity of the properness condition on the patches in Definition 1.2 is shown by the following example. BrittJr., in Comprehensive Composite Materials, 2000. The point of this example is that one can, even in such a highly nonlinear problem involving a continuous system nevertheless calculate the motion successfully, starting from an unstable equilibrium position, when the parameters are varied in different ways. Nevertheless Hsiang (1993) announced an example of a singular bubble in the Cartesian product H7 × S7 of hyperbolic space with the round sphere. Then the area of revolution generated by C is. The co-ordinate curves form an orthogonal network if a12 = F = 0 everywhere. Copyright © 2021 Elsevier B.V. or its licensors or contributors. The calculator will find the area of the surface of revolution (around the given axis) of the explicit, polar or parametric curve on the given interval, with steps shown. In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.. The presence of the subscript “CNV” next to the symbols Rp and φ (Fig. If the resulting surface is a closed one, it also defines a solid of revolution. R1. Revolve each line segment Xr–1Xr (r = 1, …, n), once about the x-axis, to produce a surface. For an arbitrary vortex beam, the motion Eqs. More generally, any surface obtained by rotating the curve y=f(x) about the x-axis has the following expression for area [4], In our case, with the same notations as in Figure 3.6, the curve is defined by r=f(z) and is rotated about the z-axis, so that the preceding formula becomes, Upon integration between the two limits R—h and R—one obtains, Other forms of the expression of the surface of the spherical cap are useful. We use a solution suggested by Pottmann and Randrup [63], and define the error to be the product of the distance and the sine of the angle between the normal line, andthe plane of the axis and the data point. The normals to a surface of revolution intersect the axis of revolution (in a projective sense, i.e. It will be useful to make one further modification. The expansion up to fourth degree for the angle characteristic associated with a reflecting surface of revolution can be derived in a similar manner. It follows that in a region in which the thickness is uniform, the tensions will also satisfy a similar condition, and this is illustrated for plane stress, in Figure 7.2. The curve generating the shell, C, is illustrated in Figure 7.3(b) and the outward normal to the curve (and the surface) at P is N P→. D¯ induces a connection on TM and NM. Since the relations between the Seidel variables and the ray components are linear, the order of the terms does not change by transition from the one set of variables to the other. The grinding wheel surface is obtained by rotating the profile curve around the grinding wheel axis by an angle χ. In RP3, the least-area way to enclose a given volume V is: forsmall V, a round ball; for large V, its complement; and for middle-sized V, a solid torus centered on an equatorial RP1. One way to discuss such surfaces is in terms of polar coordinates ( r, θ). For objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces. (12.18) becomes: Combining this equation with one obtained from vertical equilibrium considerations yields the required values of σ1 and σ2. The image below shows a function f(x) over an interval [a,b], and the surface of revolution you get when you rotate it around the x axis. We have seen that using the surface of revolution as a basic stream tube, based on the assumption that Vl, Vψ depend only on l, reduces the problem under consideration to the integration of an ordinary differential equation and, possibly, to the calculation of a quadrature for η. We revolve around the x-axis an element of arc length ds. Mass conservation relates the flux J to the velocity v, and the virtual mass displacement δI to the virtual translation δr: The integral extends over the area of the interface. The associated Abbe invariants (§ 4.4 (7)) will be denoted by K and L respectively: Before substituting into (1) the expressions for the ray components in terms of the Seidel variables, it will be useful to re-write (1) in a slightly different form. We’ll start by dividing the interval into n n equal subintervals of width Δx Δ x. where (Xi), i = 1, …, n, is an orthonormal basis at x. The grinding wheel is still a surface of revolution whose axial profile curve coincides with (or, is very similar to) the cutting edge, whose geometry depends on the tool type (straight blade, curved blade, with Toprem, etc.). Surface Area of a Surface of Revolution. This may be considered as a tension yield locus and following an approach similar to that in Section 3.7, we identify an effective or representative tension function T¯. Find the equation z=f(x,y) describing a surface of revolution. Figure 7.3. Generally only 3 or 4 iterations are needed. Then. By continuing you agree to the use of cookies. It will be useful to summarize the relevant Gaussian formulae. The Hutchings Basic Estimate 14.9 also has the following corollary. To simplify the statements of later theorems, we use a slightly different terminology in this case; see Exercise 12. Let C be a curve in a plane P ⊂ R3, and let A be a line in P that does not meet C. When this profile curve C is revolved around the axis A, it sweeps out a surface of revolution M in R3. The force f, defined by (7.3), is in the direction of the axis of revolution, the x-axis; y is the radius of the surface of revolution. The equation for H from the system of Eqs. For these problems, you divide the surface into narrow circular bands, figure the surface area of a representative band, and then just add up the areas of all the bands to get the […] If the revolved figure is a circle, then the object is called a torus. Elastic surfaces in motion are to be considered, with attention confined to surfaces of revolution. Both types occur for a critical value of A, when the minimizer jumps from one type to the other. The sum of the areas of these surfaces is. Surface of revolution definition is - a surface formed by the revolution of a plane curve about a line in its plane. If, for example, S1 were not spherical, replacing it by a spherical piece enclosing the same volume (possibly extending a different distance horizontally) would decrease area, as follows from the area-minimizing property of the sphere. Fig. A surface of revolution is a surface globally invariant under the action of any rotation around a fixed line called axis of revolution. So far I have not discussed anything resembling a structure, but the time for that has now arrived. (This theory is a dynamical counterpart to the static theory called the membrane theory of shells.) The differential equations of motion are, in that case: In the static case, i.e. (b) x = t – sin t, y = 1 – cos t (0 ≤ t ≤ 2π). There are results on R × Hn by Hsiang and Hsiang, on RP3, S1 × R2, and T2 × R by Ritoré and Ros ([2]; [1], [Ritoré]), on R × Sn by Pedrosa, and on S1 × Rn, S1 × Sn, and S1 × Hn by Pedrosa and Ritoré. An area-minimizing double bubble in Rn is either the standard double bubble or another surface of revolution about some line, consisting of a topological sphere with a tree of annular bands (smoothly) attached, as in Figure 14.10.1. Definition 16.7.1 Let f be a real function with a continuous derivative on [a, b]. 12.7(b) where r1 is the radius of curvature of the element in the horizontal plane and r2 is the radius of curvature in the vertical plane. Since everything else can be rolled around S1 or S2 without creating any illegal singularities, they must be spheres and the bubble must be the standard double bubble. A surface generated by revolving a plane curve about an axis in its plane. (3.9), we find. Tamas Varady, Ralph Martin, in Handbook of Computer Aided Geometric Design, 2002. Definition 2.1. Where C can be expressed in the form y = f(x) (a ≤ x ≤ b), f having a continuous derivative on [a, b] and x: [α, β] → [a, b] bijective, the proof is similar to that of Theorem 16.6.2 under the same restrictions. Fig. Substituting from these relations into (6) and recalling (1), we finally obtain the required expression for ψ(4): This formula gives, on comparison with the general expression § 5.3 (3), the fourth-order coefficients A, B, … F of the perturbation eikonal of a refracting surface of revolution. 4.5. As an error measure for least squares minimisation, we would ideally like to use the distance of these two lines, but this has two problems: (i) a normal parallel to the axis does not have zero error, and (ii) for a given angular deviation in normal, a greater error will result for the normal through a point further from the axis than a point nearer the axis. Strength is derived from the glass orientation, pretensioning of the glass roving, and the high glass to resin content. Valeriy A. Syrovoy, in Advances in Imaging and Electron Physics, 2011. Tom Willmore, in Handbook of Differential Geometry, 2000. *, Equations (13) express the primary aberration coefficients in terms of data specifying the passage of two paraxial rays through the system, namely a ray from the axial object point and a ray from the centre of the entrance pupil. Filament winding is a popular method of fabricating but it is applicable only to surfaces of revolution. By rotating the line around the x-axis, we generate. Miles, in Basic Structured Grid Generation, 2003, A surface of revolution may be generated in E3 by rotating the curve in the cartesian plane Oxz given in parametric form by x = f(u), z = g(u) about the axis Oz. concrete domes or dishes, the self-weight of the vessel can produce significant stresses which contribute to the overall failure consideration of the vessel and to the decision on the need for, and amount of, reinforcing required. Solid of Revolution--Washers. An axisymmetric shell, or surface of revolution, is illustrated in Figure 7.3(a). Figure 14.10.1. What does surface-of-revolution mean? The glass to resin ratio can be as high as 0.75 by weight, but the low resin content means that this laminate is not as corrosion resistant as the HLU laminate. Surface of revolution definition, a surface formed by revolving a plane curve about a given line. Parameters specifying the grinding wheel geometry for the CNV side. As C is revolved, each of its points (q1, q2, 0) gives rise to a whole circle of points, Thus a point p = (p1, p2, p3) is in M if and only if the point, If the profile curve is C: f(x, y) = c, we define a function g on R3 by. (1.89). 4.4 and 4.5) all the equations of § 4.1.4 up to and including (39) apply without change in the present case; hence (39) is also the angle characteristic of a reflecting surface of revolution, when regarded as a function of all the six ray components.

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