Columnar body manufacturing method and columnar body manufactured using said manufacturing method

Abstract

Provided is a columnar body manufacturing method capable of manufacturing a columnar body more easily and at lower cost. This method comprises: a step for, regarding a columnar surface having a desired curved shape, calculating a curvature radius along a direction of a generatrix thereof; a step for manufacturing a plate-shaped body having a cross-sectional area defined along the direction of the generatrix so as to be proportional to the curvature radius; and a step for curving the plate-shaped body along the direction of the generatrix. The generatrix may be any differentiable curve, is preferably a parabola, an ellipse, or a hyperbola, and also may be a curve having an inflection point.

Description

Method for manufacturing a cylindrical body and a cylindrical body manufactured using the method

[0001] The present invention relates to a technique for producing a cylindrical body, and more specifically to a method for producing a cylindrical body by calculating the radius of curvature of a cylindrical surface having a desired curved shape and curving a plate-like body having a cross-sectional area determined to be proportional to the radius of curvature.

[0002] Many improved reflecting telescopes have been developed, including Newtonian reflecting telescopes that use parabolic reflectors. However, the high level of precision required for polishing the parabolic reflectors used as primary mirrors and the complexity of the manufacturing process remain unchanged to this day. As a result, medium- to large-sized reflecting telescopes are expensive, and from both technical and cost perspectives, they are difficult to manufacture without a large-scale organization. Therefore, if it were possible to manufacture a high-light-gathering reflecting telescope that requires enormous investment, but is cheaper, easier, and lighter than conventional telescopes, even ordinary citizens would be able to own or build their own reflecting telescopes.

[0003] Techniques proposed to solve these problems in reflecting telescopes include those disclosed in, for example, Patent Documents 1 to 3. These techniques use a concave reflector that is curved in only one direction, i.e., a cylindrical reflector.

[0004] JP 2005-164881 A Japanese Patent No. 6602942 A

[0005] Kanazawa Institute of Technology, KIT Mathematics Navigation, URL https: / / w3e.kanazawa-it.ac.jp / math / category / kika / heimenkika / henkan-tex.cgi?target= / math / category / kika / heimenkika / radius_of_curvature.html&pcview=2 Shozo Koshi, Introduction to Differential and Integral Calculus, Academic Press, January 1979

[0006] The techniques proposed in Patent Documents 1 to 3 require the fabrication of a cylindrical reflector that is curved in only one direction. While fabricating such a cylindrical reflector is not as difficult as fabricating the primary mirror of a parabolic reflector, it still requires expensive machine tools. On the other hand, cylindrical reflectors used in simple telescopes can be fabricated using, for example, a 3D printer, but this requires the surface to be mirror-finished or a mirror body to be attached to the surface. Furthermore, 3D CAD or the like must be created using data from the 3D printer, making it difficult to fine-tune the resulting product.

[0007] Therefore, an object of the present invention is to provide a method for producing a prism more easily and inexpensively. Another object of the present invention is to provide a prism that can be produced by this method and has a structure different from that of a prism that is formed by bending a rectangular plate in only one direction.

[0008] When a force F is applied to a rectangular plate with a cross-sectional area A parallel to its surface (cross-sectional area A), the distortion θ of the rectangle is proportional to F and inversely proportional to the cross-sectional area A. In other words, 1 / θ is proportional to the cross-sectional area A. Based on this relationship, the inventors discovered that if the cross-sectional area A is changed in advance along the curve of a certain differentiable function f so that θ is proportional to the radius of curvature r obtained from the function f when a force is applied to the plate with a cross-sectional area A, then θ will change inversely proportional to r in the direction along the curve, and therefore it is possible to create a cylindrical body using the function f, which is the basis of the radius of curvature r, as a conductor.

[0009] In one aspect, the present invention provides a cylindrical body manufacturing method for manufacturing a cylindrical body having a desired curved shape from a rectangular plate. The method includes the steps of calculating the radius of curvature of the cylindrical surface along the direction of a conductor line, fabricating a plate having a cross-sectional area along the direction of the conductor line that is proportional to the radius of curvature, and curving the plate along the direction of the conductor line. The conductor line can be any differentiable curve, preferably a parabola, ellipse, or hyperbola, and can also be a curve with an inflection point.

[0010] In one embodiment, the step of producing a plate-shaped body with a defined cross-sectional area includes producing a plate-shaped body with a uniform thickness and a length of a generatrix perpendicular to the conductor that is proportional to the radius of curvature. In another embodiment, the step of producing a plate-shaped body with a defined cross-sectional area includes producing a plate-shaped body with a uniform length of a generatrix perpendicular to the conductor and a thickness along the conductor that is proportional to the radius of curvature.

[0011] In one embodiment, the step of fabricating a plate-like body having a cross-sectional area determined along the direction of the conductor so as to be proportional to the radius of curvature includes fabricating a plate-like body having one straight side and the other curved side along the direction of the conductor, while in another embodiment, the step of fabricating a plate-like body having a cross-sectional area determined along the direction of the conductor so as to be proportional to the radius of curvature includes fabricating a plate-like body having two curved sides along the direction of the conductor.

[0012] In one embodiment, the step of bending the plate-like body along the conductor direction includes fixing at least two opposing points on the plate-like body across the center of the plate-like body in the conductor direction while maintaining the curve. In another embodiment, the step of bending the plate-like body along the conductor direction includes fixing at least two opposing points on the plate-like body across the center of the plate-like body in the conductor direction and a point corresponding to the inflection point.

[0013] In another aspect, the present invention provides a cylindrical body having a desired curved shape. The cylindrical body has a cross-sectional area determined along the direction of the conductor line so as to be proportional to a radius of curvature calculated along the direction of the conductor line of the cylindrical body having the desired curved shape. The cylindrical body may have a uniform thickness and the length of a generatrix perpendicular to the conductor line is determined to be proportional to the radius of curvature. Alternatively, the length of the generatrix perpendicular to the conductor line is constant and the thickness along the direction of the conductor line is determined to be proportional to the radius of curvature. The conductor line may be any differentiable curve, preferably a parabola, ellipse, or hyperbola, and may also be a curve having an inflection point.

[0014] In yet another aspect, the present invention provides a cylindrical body design method for designing a cylindrical body having a desired curved shape from a rectangular plate-like body, the method comprising the steps of calculating a radius of curvature along the direction of a conductor for the cylindrical surface having the desired curved shape, and determining a cross-sectional area of ​​the plate-like body along the direction of the conductor so as to be proportional to the radius of curvature.

[0015] According to the present invention, a cylindrical body can be produced more easily and inexpensively without using expensive machine tools or sophisticated mirror finishing.

[0016] 1 is a schematic diagram showing an example of the configuration of an optical system having a reflecting mirror that can be fabricated using a method for fabricating a cylinder according to an embodiment of the present invention; FIG. 2 is a schematic diagram of a parabolic cylinder, which is an example of a cylinder that can be fabricated using a method for fabricating a cylinder according to an embodiment of the present invention, where (a) shows a plan view of a plate before bending, and (b) is a perspective view after bending. This is a parabolic cylinder in which one side along the direction of the conductor is curved and the other side is straight. FIG. 3 is a flow diagram showing a method for fabricating a cylinder according to an embodiment of the present invention; FIG. 4 is a graph of the radius of curvature calculated in fabricating a parabolic cylinder with a focal length of 50 mm; and FIG. 5 is a partial graph of the length of the parabola calculated in fabricating a parabolic cylinder with a focal length of 50 mm. FIG. 6 is a schematic diagram of a parabolic cylinder, which is another example of a cylinder that can be fabricated using a method for fabricating a cylinder according to an embodiment of the present invention, where (a) shows a plan view of a plate before bending, and (b) is a perspective view after bending. This is a parabolic cylinder in which both sides along the direction of the conductor are curved. 10 is a graph comparing the curvature of a cylinder whose width is determined according to the cylinder manufacturing method of the present invention and a cylinder curved without removing the widthwise ends. It is a photograph of a prototype elliptical cylinder manufactured according to the cylinder manufacturing method of the present invention. It is a diagram of an actual rectangular plate traced to show the positions where the widthwise ends are removed to create a parabolic cylinder with a focal length of 500 mm. It shows an optical system constructed using a parabolic cylinder created by removing the ends from the plate of FIG. 9, whose back surface has been traced, and then bending it. (a) is an image taken by the optical system of FIG. 10, and (b) is a photograph of an actual landscape including (a).

[0017] An embodiment of the present invention will be described below with reference to the drawings. The following describes the method for manufacturing a cylindrical body according to the present invention, taking as an example a cylindrical reflector that can be used in optical systems such as telescopes. However, the method for manufacturing a cylindrical body according to the present invention can be used not only for designing and manufacturing reflectors for optical systems, but also for designing and manufacturing cylindrical bodies for various other applications. The method for manufacturing a cylindrical body according to the present invention can also be used, for example, for designing and manufacturing the shapes of exterior walls and roofs of buildings, designing and manufacturing automobile bodies, designing and manufacturing antenna shapes for receiving radio waves, and designing two cylindrical antennas arranged with their concave surfaces facing each other that can be used as an alternative to radar antennas.

[0018] (Configuration example of optical system) Figure 1 shows a configuration example of an optical system that can use a cylindrical reflecting mirror manufactured by the manufacturing method according to the present invention (see Patent Document 2). Figure 1 shows an example in which a cylindrical reflecting mirror is used as a primary mirror of an optical system.

[0019] The optical system shown in FIG. 1 includes a primary mirror M that reflects light from an object K, and a cylindrical lens L (concentrator) that is disposed on the optical axis of the light reflected from the primary mirror M and transmits the light reflected by the primary mirror M. The primary mirror M has a concave reflective surface 12 that is curved in only one direction (i.e., only in the y-axis direction in the figure), and this reflective surface 12 is a cylindrical reflector that can reflect light from the object K. An optical system in which the cylindrical lens L used in this optical system is replaced with a secondary mirror that is a cylindrical reflector whose reflective surface faces the direction of the primary mirror M, as disclosed in Patent Document 1, for example, is also possible. These primary mirrors and secondary mirrors can be fabricated using the fabrication method of the present invention.

[0020] (Definition of a Cylinder) In this specification, a cylinder is an object with a curved surface (cylinder face) that is created when a straight line that intersects a curve at a single point moves along the curve while maintaining a constant direction. The original curve is called the conductor, and the individual lines that form the cylinder face are called the generatrix.

[0021] (Manufacturing a parabolic cylinder) [Method of changing the width in proportion to the radius of curvature] Figure 2 is a schematic diagram of a parabolic cylinder, which is an example of a cylinder that can be manufactured by a cylinder manufacturing method according to one embodiment of the present invention, where (a) shows a plan view of the plate-like body before bending, and (b) is a perspective view after bending. Figure 3 is a flow chart showing a cylinder manufacturing method according to one embodiment of the present invention. A method for manufacturing the cylinder of Figure 2 will now be described.

[0022] (1) Calculation of the radius of curvature of a parabolic cylinder First, for a cylinder having a desired curved shape, the radius of curvature along the direction of the conductor is calculated (step S1 in FIG. 3). In this case, since the cylinder having the desired curved shape is a parabolic cylinder, the conductor is a parabola. A parabola is expressed by the following equation (1): y = f(x) = ax ２ (1)

[0023] The radius of curvature r of a function y=f(x) that represents an arbitrary differentiable curve can be calculated from the following equation (2), as shown in, for example, Non-Patent Document 1 and Non-Patent Document 2. Here, in the case of the parabola in (1), y' = 2ax and y" = 2a, so the radius of curvature r (parabola) of the parabola is Using equation (3), when the origin of the x-axis is set to the midpoint in the length direction of a certain plate-like body, the radius of curvature of the parabola corresponding to the value of x can be calculated.

[0024] (2) Fabrication of Plate-Like Object Next, a plate-like object is fabricated whose cross-sectional area is determined along the direction of the conductor so as to be proportional to the radius of curvature calculated by the above method (step S2 in FIG. 3). The inventors have found that by calculating the radius of curvature r obtained from a function f representing an arbitrary differentiable curve, and then changing the cross-sectional area of ​​a plate-like object having a length direction along the curve of function f so as to be proportional to the calculated radius of curvature r, a cylindrical object can be fabricated whose generating line is perpendicular to the curve of function f (which becomes the conductor of the cylindrical object). In one embodiment, a method can be considered for determining the cross-sectional area of ​​a plate-like object along the conductor so as to be proportional to the radius of curvature, in which the thickness is uniform and the length of the generating line perpendicular to the conductor (i.e., the width of the plate-like object) is determined to be proportional to the radius of curvature.

[0025] As a specific example, a rectangular plate measuring 450 mm in length, 150 mm in width, and 1 mm in thickness is prepared, and the width of this rectangular plate is changed along the length to produce a plate forming a parabolic cylinder with a focal length f = 50 mm. For a parabola with a focal length f = 50 mm, a in equation (1) is 0.005, based on the relationship 1 / (4f) = a (4). Substituting this a into equation (3) yields the radius of curvature of the parabola at the x position. Figure 4 shows a graph of the radius of curvature calculated in this manner, plotting the radius of curvature r (parabola) corresponding to the x value when the center of the rectangular plate in the length direction is set to 0 on the x axis.

[0026] Next, if necessary, the position along the length of the rectangular plate corresponding to the position on the x-axis, i.e., the length of the plate from the center of the length to the position corresponding to the position on the x-axis, is calculated. The radius of curvature r calculated using the above formula (3) is the radius of curvature of the parabola at the position corresponding to the value on the x-axis. However, since the cylinder is curved, the length along the plate from the center of the length of the plate (i.e., point 0 on the x-axis) toward the end to the position corresponding to the x value is different from the length along the x-axis from point 0 to that x value, and the length along the plate is longer than the length along the x-axis. Therefore, in order to determine the width along the length of the rectangular plate so that it is proportional to the radius of curvature, it may be necessary to calculate the relationship between the value p on the x-axis from the center of the length and the length L of the plate to the position corresponding to p. However, for cylinders with a large radius of curvature (cylinders with a large focal length), for example, there may be little difference between the length on the x-axis and the length along the cylinder within the range of the length in the conductor direction required as a reflector, so this step of calculating the relationship between p and L is not necessary.

[0027] In the parabola of equation (1), the length L of the parabola from the origin to the position x = p is If t=2ax in equation (5), equation (5) becomes as follows. where: Using the above relationship, L is expressed by the following equation (8). FIG. 5 shows a portion of a graph of the length of the plate calculated in the fabrication of a parabolic cylinder with a focal length of 50 mm.

[0028] Based on the above concept, a width is calculated along the conductor direction so as to be proportional to the radius of curvature, and a plate-like body having the calculated width is produced. Here, a method for producing a plate-like body having a width determined along the conductor direction so as to be proportional to the radius of curvature by physically removing the end portion from the original rectangular plate-like body is described. However, this is not limited to this, and a plate-like body having a width calculated along the conductor direction in advance may also be produced.

[0029] There are two methods for changing the width of a rectangular plate along the conductor direction by removing its ends: removing only one widthwise end of the rectangular plate, and removing both widthwise ends. Using the method of removing only one widthwise end results in a plate with one straight edge and the other curved edge along the conductor direction. Using the method of removing both widthwise ends results in a plate with both curved edges along the conductor direction. The parabolic cylinder shown in Figure 2 is a cylinder made using the former method, i.e., removing only one widthwise end of the rectangular plate.

[0030] Table 1 shows some of the values ​​used to fabricate the parabolic cylinder. In Table 1, p is the length (mm) from point 0 on the x-axis, L is the length (mm) of the conducting wire (parabola) to the position corresponding to p on the x-axis, and r is the radius of curvature (mm) of the conducting wire (parabola) at position p. The length of the conducting wire corresponds to the length of the plate. In the table, r(p) is the radius of curvature of the plate at the position x = p, and r(pe) is the radius of curvature of the plate at the end of the length of the plate, i.e., at the position L = 450 / 2 = 225 mm, when the center of the plate in the longitudinal direction is set as point 0 on the x-axis. r(p) / r(pe) indicates the ratio of the radius of curvature at position p to the radius of curvature at the end position. Therefore, the width of the plate can be determined based on this ratio (step S2-1 in Figure 3). Note that, in order to simplify the table, Table 1 does not list all p's and their corresponding values; only representative points are listed. The granularity of p may be determined depending on the size of the parabolic cylinder, the focal length, and other conditions.

[0031]

[0032] FIG. 2(a) is a schematic plan view of a rectangular plate produced by removing only one widthwise end of the rectangular plate according to Table 1, and FIG. 2(b) is a schematic perspective view of the parabolic cylinder obtained by bending the plate shown in FIG. 2(a). Alternatively, a parabolic cylinder can be produced by removing both widthwise ends of the rectangular plate. In this case, a plate with both curved sides along the conductor direction can be obtained by removing both widthwise ends of the rectangular plate equally or one widthwise end more than the other according to the ratios shown in Table 1. FIG. 6 is a schematic diagram of a parabolic cylinder obtained in this manner. FIG. 6(a) is a schematic plan view of a rectangular plate produced by removing both widthwise ends of the rectangular plate before bending, and FIG. 6(b) is a schematic perspective view of the cylinder obtained by bending the plate shown in FIG. 6(a).

[0033] [Method of Changing Thickness Proportional to the Radius of Curvature] Up to this point, we have described a method for fabricating a plate-like body whose width is determined to be proportional to the radius of curvature along the conductor direction. However, as another method for achieving a plate-like body whose cross-sectional area is proportional to the radius of curvature, the thickness of the plate-like body can be changed along the conductor direction (step S2-2 in Figure 3). The method for changing the thickness is not limited. For example, a plate-like body whose cross-sectional area is proportional to the radius of curvature can be fabricated by grinding one or both surfaces of the plate-like body. When fabricating a reflecting mirror, it is preferable to grind the surface opposite the reflecting surface. Alternatively, the thickness can be changed by applying, for example, putty to one or the other surface of the plate-like body. Alternatively, a plate-like body having a predetermined thickness along the conductor direction may be fabricated.

[0034] Furthermore, as a means for varying the thickness of the plate along the conductor direction, grooves extending perpendicular to the conductor can be formed with a density inversely proportional to the radius of curvature. Specifically, grooves extending perpendicular to the conductor from one end of the plate to the other can be formed with varying density along the length of the plate. Multiple grooves can be formed so that the spacing between adjacent grooves is wide near the ends of the plate and narrows toward the center. This groove density per unit length along the length of the plate gradually increases from the ends to the center. Within a certain range, areas with high groove density are equivalent to reducing the thickness of the plate, while areas with low groove density are equivalent to increasing the thickness of the plate.

[0035] (3) Curving of Plate-Shaped Body Next, the plate-shaped body produced in (2) above, i.e., a plate-shaped body whose width changes along the conductor direction in proportion to the radius of curvature, or a plate-shaped body whose thickness changes along the conductor direction in proportion to the radius of curvature, is curved along the conductor direction and fixed in a curved state at least two points opposite each other across the center in the conductor direction, thereby producing a parabolic cylinder with the desired focal length (step S3 in Figure 3).

[0036] One method for curving a plate is to connect and fix both ends of the plate along the conductor (i.e., the length). The plate is fixed so that both ends of the conductor lie on the desired parabola when the center of the plate in the length direction is positioned at point 0 on the x-axis. For a plate having the values ​​shown in Table 1, both ends of the plate (each 225 mm long from the center) are fixed at points on the parabola corresponding to x = 166.07 and -166.07. By fixing both ends of the plate so that they lie on the parabola in this way, the remaining portions of the plate automatically conform to the desired parabolic shape, resulting in a parabolic cylinder with this parabola as its conductor. The fixing points can also be any two points on the parabola closer to the origin than the ends. In this case, the portion between the two fixed points is a parabolic cylinder with a shape conforming to the desired parabola, and the portion outside these fixed points is not curved.

[0037] The two ends of the plate-like body can be fixed in the required position, i.e., on the desired parabola, for example, by connecting the two ends with one or more screw shafts and nuts, or by passing a tape, string, etc. Alternatively, for example, the two ends can be fixed in the required position by preparing a flat plate corresponding to the size of the plate-like body, fixing the central part of the plate-like body in the longitudinal direction to the flat plate, and supporting the two ends of the plate-like body with support members extending from the surface of the flat plate.

[0038] As described above, according to the present invention, it is possible to obtain a parabolic cylinder having a cross-sectional area determined along the direction of the conductor so as to be proportional to the radius of curvature calculated along the direction of the conductor. By varying the width of a plate of uniform thickness along the direction of the conductor, it is possible to obtain a parabolic cylinder whose width is determined so as to be proportional to the length of the generatrix perpendicular to the conductor, i.e., the radius of curvature. Furthermore, by varying the thickness of a plate of constant width along the direction of the conductor, it is possible to obtain a parabolic cylinder whose generatrix perpendicular to the conductor is constant and whose thickness is determined so as to be proportional to the radius of curvature.

[0039] (4) Comparison with a Rectangular Cylinder with No Ends Removed: A cylinder whose width is determined to be proportional to the radius of curvature according to the method of the present invention is compared with a cylinder whose widthwise ends are not removed, i.e., a cylinder obtained by simply curving a rectangular plate. Figure 7 is a graph comparing the curved states of the two. In Figure 7, line B is a line tracing the edges of a cylinder obtained by curving a rectangular plate of the same size as the above example, i.e., 450 mm long x 150 mm wide x 1 mm thick, while line A is a line tracing the unremoved edges of a cylinder of the same size as the cylinder of line B, with one widthwise end removed according to Table 1 (i.e., the cylinder of Figure 2(b)). The dots in the graph of Figure 7 represent the calculated values ​​of a parabola using equations (1) and (4) with f = 50. Both ends of both lines lie on the parabola.

[0040] The line A of the cylinder fabricated by the method of the present invention closely matches the calculated value of the parabola (●), which is the desired curved shape, while the line B bulges outward from the parabola between the fixed part and the origin. This is because the cross-sectional area of ​​the cylinder of line A varies with the longitudinal position, i.e., the cross-sectional area is determined so that it conforms to the parabola when curved, whereas the cross-sectional area of ​​the cylinder of line B does not change with the longitudinal position. Therefore, the cross-sectional area of ​​the cylinder of line B is larger than that of the cylinder of line A at the same longitudinal position, and when fixed at both longitudinal ends and curved, it experiences a stronger tendency to expand outward than the cylinder of line A. This result also shows that, according to the present invention, it is possible to fabricate a cylinder with function f as a conductor by previously varying the cross-sectional area A along the curve of a differentiable function f so that it is proportional to the radius of curvature r obtained from the function f.

[0041] (Fabrication of Other Cylindrical Bodies) In the above, a parabola has been described as an example of an arbitrary differentiable curve, but such curves are not limited to parabolas, and a cylindrical body can be fabricated similarly for various curves. For example, an elliptical cylindrical surface can be fabricated in the same way as the parabolic cylindrical surface described above by calculating the radius of curvature from the following ellipse formula (9), fabricating a plate-like body whose cross-sectional area is determined along the direction of the conductor so that it is proportional to the radius of curvature, and then curving the plate-like body.

[0042] Specifically, the radius of curvature r (of the ellipse) calculated using the ellipse formulas (9) and (2) is expressed by the following formula (10).

[0043] Based on this formula (10), the width is calculated along the conductor direction so that it is proportional to the radius of curvature, and an elliptical cylinder is created using the ellipse as the conductor. The values ​​used to create an elliptical cylinder by physically removing one end from the original rectangular plate are shown in Table 2. These values ​​were calculated for an ellipse with a major axis of 2a and a minor axis of 2b, with a = 60 and b = 30.

[0044] In this table, p is the x-coordinate value on the ellipse (p = 0 corresponds to (0, b) on the ellipse). L is the arc length (mm) of the ellipse from the coordinate (0, b) or (0, -b) to the position corresponding to the value of x = p. The arc length of the ellipse corresponds to the length along one side from the longitudinal end of the plate-like body when the elliptical cylinder is expanded. r is the radius of curvature (mm) of the ellipse at the position corresponding to the value of p. In the table, r(p) is the radius of curvature of the ellipse at the position x = p, and r(pe) is the radius of curvature of the ellipse at the position p = 0. r(p) / r(pe) indicates the ratio of the radius of curvature at the position p to the radius of curvature at the position p = 0, and therefore the width of the plate-like body can be determined according to this ratio.

[0045] When the elliptical cylinder is expanded into a plate-like body, the length of the plate-like body from the position of x = 0 on the ellipse (coordinates (0, b) or (0, -b)) to the position corresponding to x = p on the ellipse (i.e., the arc length of the ellipse) can be calculated, for example, by equation (11), which is well known to those skilled in the art for calculating the arc length of an ellipse. Here, θ is the angle between the line connecting the point on the ellipse where x = p and the origin and the minor axis of the ellipse (the central angle of the sector). In order to simplify the table, not all values ​​of p and their corresponding values ​​are listed in Table 2, but only representative values ​​are listed. The granularity of p can be determined depending on the size of the elliptical cylinder, the focal length, and other factors.

[0046] By bending the plate thus fabricated, i.e., a plate whose width varies in proportion to the radius of curvature along the wire direction, and fixing both ends together, an elliptical cylinder with an elliptical wire can be fabricated. Figure 8 shows a photograph of a prototype elliptical cylinder fabricated in this way. The wire of the prototype elliptical cylinder nearly coincides with the elliptical equation (9). An elliptical cylindrical mirror using an elliptical cylinder fabricated in this way can be used in its entirety or, as needed, a portion along the wire direction, and applied as a reflector in a floating zone furnace used in the floating zone method, a single crystal growth method.

[0047] Similarly, a bipolar cylinder can be created based on the following formula:

[0048] In general, according to the present invention, a cylinder can be fabricated using any differentiable curve as a conductor. A curve having an inflection point can be expressed by the following formula: When a cylindrical body having a curved line with an inflection point as a conductor is to be curved, it is preferable that the plate-like body be fixed not only at both ends but also at the inflection point.

[0049] As an example, a parabolic cylindrical reflector with a focal length f = 500 mm was fabricated, and an optical system was constructed using this reflector. An actual landscape was photographed using the constructed optical system. To fabricate the parabolic cylindrical reflector, an acrylic mirror with a length of 270 mm, width of 150 mm, and thickness of 2 mm was prepared. The parabola with a focal length f = 500 mm can be expressed by the following equation (14) from equations (1) and (4): y = (5 x 10 －４ ) x ２ (14) The radius of curvature of this parabola is calculated from equation (3) as follows: Based on this formula, the following values ​​were calculated, as in Table 1. In Table 3, not all p's and their corresponding values ​​are listed in order to simplify the table.

[0050]

[0051] The width of the plate was calculated based on the ratio calculated in this way, and a line indicating the position of the edge to be removed was marked on the surface (back surface) of the prepared acrylic mirror opposite the reflective surface. Figure 9 is a diagram tracing the back surface of an actual acrylic mirror with the line indicating the removal position. Next, the edge of the acrylic mirror was removed along this line. Both longitudinal ends of the acrylic mirror from which the edge was removed were fixed with two screw shafts and nuts. By using both screw shafts and nuts, fine adjustments could be made so that the longitudinal end was positioned on the desired parabola. The parabolic cylindrical reflector thus fabricated was used to construct the optical system shown in Figure 10. The optical system of Figure 10 is a realization of the optical system shown in Figure 1(a). It uses the fabricated parabolic cylindrical reflector as a primary mirror that reflects light from a subject, and includes a cylindrical convex lens (focal length f = 200 mm) that is positioned on the optical axis of the light reflected from the primary mirror and serves as a secondary mirror that transmits and refracts the light reflected by the primary mirror M, and an oblique mirror as a light receiving unit that receives the light transmitted through the cylindrical convex lens.

[0052] FIG. 11( a) shows a photograph taken using the optical system of FIG. 10 . This photograph was obtained by capturing an image captured by a camera on the light receiving section of the optical system of FIG. 10 . The roof of the building seen in the center of the scene shown in FIG. 11( b) is clearly visible in this photograph. Note that FIG. 11( b) was taken with a standard digital camera. In the photograph of FIG. 11( a), the roof appears vertically stretched compared to the actual scene. This is a distortion caused by the difference in focal lengths between the primary mirror (parabolic cylindrical reflector) and the secondary mirror (cylindrical convex lens) of this optical system. This distortion can be easily corrected, if necessary, by placing another lens after the secondary mirror (i.e., downstream in the light propagation direction) or by using correction software.

Claims

1. A method for manufacturing a cylindrical body having a desired curved shape from a rectangular plate, comprising the steps of: calculating the radius of curvature along the direction of the conductor for the cylindrical surface having the desired curved shape; producing a plate whose cross-sectional area is determined along the direction of the conductor so as to be proportional to the radius of curvature; and curving the plate along the direction of the conductor.

2. A method for manufacturing a cylindrical body according to claim 1, wherein the step of manufacturing a plate-like body with a predetermined cross-sectional area includes manufacturing a plate-like body with a uniform thickness and a length of a generatrix perpendicular to the conductor that is determined to be proportional to the radius of curvature.

3. A method for manufacturing a cylindrical body according to claim 1, wherein the step of producing a plate-like body with a defined cross-sectional area includes producing a plate-like body in which the length of a generatrix perpendicular to the conductor is constant and the thickness along the direction of the conductor is determined to be proportional to the radius of curvature.

4. A method for manufacturing a cylindrical body according to any one of claims 1 to 3, wherein the conducting wire is an arbitrary differentiable curve.

5. The cylindrical body manufacturing method according to claim 4, wherein the conducting wire is either a parabola, an ellipse, or a hyperbola.

6. The method for manufacturing a cylindrical body according to claim 4, wherein the conducting wire is a curve having an inflection point.

7. A method for manufacturing a cylindrical body according to claim 2, wherein the step of manufacturing a plate-like body having a cross-sectional area determined along the direction of the conductor so as to be proportional to the radius of curvature includes manufacturing a plate-like body having one side that is straight and the other side that is curved along the direction of the conductor.

8. A method for manufacturing a cylindrical body according to claim 2, wherein the step of manufacturing a plate-like body having a cross-sectional area determined along the direction of the conductor so as to be proportional to the radius of curvature includes manufacturing a plate-like body having two curved sides along the direction of the conductor.

9. A method for manufacturing a cylindrical body according to claim 4, wherein the step of curving the plate-like body along the direction of the conductor includes fixing at least two points of the plate-like body that face each other across the center of the plate-like body in the direction of the conductor while maintaining the curvature.

10. A method for manufacturing a cylindrical body according to claim 6, wherein the step of curving the plate-like body along the direction of the conductor includes fixing at least two points on the plate-like body that face each other across the center of the plate-like body in the direction of the conductor, and a point corresponding to the inflection point.

11. A cylinder having a desired curved shape, the cross-sectional area of ​​which is determined along the direction of a lead of the cylinder having the desired curved shape in proportion to a radius of curvature calculated along the direction of the lead.

12. The cylindrical body according to claim 11, wherein the thickness is uniform and the length of a generatrix perpendicular to the conductor is determined to be proportional to the radius of curvature.

13. The cylindrical body according to claim 11, wherein the length of the generatrix perpendicular to the conductor is constant, and the thickness along the direction of the conductor is determined to be proportional to the radius of curvature.

14. The cylindrical body according to any one of claims 11 to 13, wherein the conducting line is an arbitrary differentiable curve.

15. The cylindrical body according to claim 14, wherein the conducting wire is either a parabola, an ellipse, or a hyperbola.

16. The prism body according to claim 14, wherein the conducting wire is a curve having an inflection point.

17. A method for designing a cylindrical body having a desired curved shape from a rectangular plate, the method comprising: a step of calculating the radius of curvature along the direction of a conductor for a cylindrical surface having the desired curved shape; and a step of determining the cross-sectional area of ​​the plate along the direction of the conductor so as to be proportional to the radius of curvature.

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