US2840924A

US2840924A - Hyperbolic paraboloid generator

Info

Publication number: US2840924A
Application number: US655279A
Authority: US
Inventors: Donald W Willis
Original assignee: Individual
Current assignee: Individual
Priority date: 1957-04-26
Filing date: 1957-04-26
Publication date: 1958-07-01
Anticipated expiration: 1975-07-01

Description

Juiy 1, 1958 D. w. WILLIS HYPERBOLIC PARABOLOID GENERATOR 3 Sheets-Sheet 1 Filed April 26, 195'! INVENTOR. Donna-W. \flll us ,Wm*.@m ATTORNEYS y 1958 D. w. WILLIS 2,840,924

HYPERBOLIC PARABOLOID GENERATOR Filed April 26, 1957 3 Sheets-Sheet 2 AT T0254 EY$ July 1, 1958 o. w. WILLIS HYPERBOLIC PARABOLOID GENERATOR 3 Sheets-Sheet 3 Filed April 26, 1957 INVENTOR. DQMALD W|Lu= A TOQHEYS t6 win This invention relates to three dimensional mathematical demonstration devices, and more particularly to a device for developing geometrical forms such as hyperbolic paraboloids.

A main object of the invention is to provide a novel and improved device for developing geometrical forms, such as hyperbolic paraboloid surfaces and similar mathematical surfaces, said device being especially useful for instruction in mathematics and solid geometry or for demonstration purposes for illustrating structural designs, for example, thin-shell concrete structures in architecture or in the engineering design of such structures.

A further object of the invention is to provide an improved apparatus for generating mathematical surfaces, such as hyperbolic paraboloids, the device involving relatively simple components, being easy to adjust, and presenting accurate physical realizations of certain theoretimathematical surfaces which have been hitherto very difficult to construct.

A still further object of the. invention is to provide an improved device for developing hyperbolic paraboloids, said device involving inexpensive parts, being durable in construction, and being adjustable so that the developed hyperbolic paraboloid may be formed either on a square or on a parallelogram of a selected shape.

A still further object of the invention is to provide a device for developing certain geometric forms, namely, hyperbolic paraboloids, said device employing straight elastic generating elements which are oriented by suitable adjustment of the device so that they combine to generate the desired mathematical surface.

Further objects and advantages of the invention will become apparent from the following description and claims, and from the accompanying drawings, wherein:

Figure l is a perspective view of a geometrical surfacegenerating device constructed in accordance with the present invention, shown with the collar elements thereof at substantially equal heights and arranged so that the connections of the upstanding rod members thereof define a square on the base of the device.

Figure 2 is an enlarged vertical cross sectional detail view taken on the line 22 of Figure 1.

Figure 3 is a fragmentary, enlarged horizontal cross sectional view with parts broken away, taken on the line 3-6 of Figure 1.

Figure 4 is a vertical cross sectional view taken on the line 4-4 of Figure 3.

Figure 5 is a fragmentary perspective view, similar to Figure 1, but showing the collar elements of the device adjusted so that the straight elastic band elements thereof cooperate to define a hyperbolic paraboloid.

Figure 6 is a digramrnatic view illustrating the configuration defined by the intersection of a vertical plane with the elastic band elements of the device in the position of Figure 5, the vertical plane extending substantially parallel to the vertical plane of the rod members to which the lowermost collar elements are secured.

Figure 7 is a diagrammatic view, similar to Figure 6,

ice

but showing the configuration of the curve defined by the intersection of a vertical plane with the elastic band elements of the device, in the position of Figure 5, with said vertical plane extending substantially parallel to the rod members to which the uppermost collar members are secured.

Referring to the drawings, the device of the present invention is designated generally at 11 and comprises a rigid horizontal base board 12 which may be supported in a slightly elevated position with respect to an underlying surface in any suitable manner, as for example by the provision of a pair of supporting

bars

13, 13 secured to the bottom surface of opposite marginal portions of the base board 12.

Designated at 14 and 15 are a pair of upstanding vertical rod members which are fixedly secured to the base board 12 in any suitable manner at the points designated at D and C in Figures 1 and 5, the points D and C being suitably located adjacent to a longitudinal edge 16 of the base board 12 and on a line parallel to said longitudinal edge, although said points D and C may be located at any other convenient location on the base board, if so desired.

The base board 12 is formed opposite the respective rod connection points D and C with a pair of arcuate slots 1'7 and 18 concentric with the respective connection points D and C and at equal radial distances from said connection points, said radial distances being the same as the distance between the connection points D and C. The

slots

17 and 18 are spaced apart by the same distance as the connection points D and C, whereby a pair of corresponding points in the

solts

17 and 18 define an equal-sided parallelogram, namely, a rhombus, with respect to the points of connection D and C of the

rod members

14 and 15 to the base board 12. The upper terminal ends of

slots

17 and 18, as viewed in Figure 1, define a square with respect to the connection points D and C of the

vertical rod members

14 and 15. The board 12 is inscribed with angle calibration lines radiating from the points D and C toward the slots 17 and 1%, said lines being shown at 19, and being calibrated in angular values to indicate corersponding points along the

slots

17 and 18.

Designated at 20 and 21 are a pair of additional upstanding vertical rod members which are adjustably secured to the base board 12 at the

slots

17 and 18. Thus, as shown in Figure 4, each of the rods 20 and 21 may be provided with a collar member 22 rigidly secured thereto and a threaded depending portion extending through the associated

arcuate slot

17 or 18, shown at 23. A wing nut 24 is threadedly engaged on the threaded depending lower portion of the rod, bearing against a clamping washer 25 engaged on the rod and disposed against the bottom surface of the base board 12, whereby the rod will be rigidly secured to the base board when be associated wing nut 24 is tightened.

The top ends of the

rods

14, 15, 20 and 21 have reduced threaded portions 26, and engaged on the shoulders thus defined on the

rods

14 and 15 are the ends of a connecting bar 27. A similar connecting bar 28 is engaged on the shoulder thus defined on the top end of the rod 21 and on the portion of rod 27 engaged on the rod 15.

The

bars

27 and 28 are suitably apertured at their ends to receive the reduced threaded portions 26 of the

bars

14, 15 and 21.

A pair of extensible bars 29 and 30 connect the top portions of the bars 20 and 14, and of the bars 20 and 21. Thus, as shown in Figure 4, each of the connecting bars 29 and 30 comprises a pair of overlapping sections, the upper section being formed with a longitudinal slot 32 and the lower section being provided with an up 3 standing threaded stud 33 slidably engaged through the slot and provided with a wing nut 34, which, when tightened, clamps the two sections together in adjusted relationship. Similar wing nuts 35 and provided on the threaded reduced top portions 26 of the respective upstanding

vertical rod members

14, 15, 20 and 21.

Slidably engaged on each of the

rod members

14, 15, 20 and 21 is a vertically adjustable collar assembly designated generally at 36. Each collar assembly comprises an annular top member 37 having an internally threaded annular flange portion 38 in which is threadedly engaged the upstanding shank portion 39 of an annular bottom clamping sleeve 40. As shown, the clamping sleeve 40 is slidably engaged on its associated upstanding vertical rod member is threadedly secured to the associated top ring element 37, a pair of

intermediate ring members

41 and 42 being engaged on the shank portion 39 of the lower member 40. The top member 37 is provided with a wing screw 43 which, when tightened, clampingly engages the associated upstanding vertical rod member, for example, the rod member 20 in Figure 2, to secure the collar assembly 36 in the selected position on the vertical rod member. The vertical rod member is provided with suitable calibrations 44, whereby the associated collar assembly 36 may be accurately secured at a desired elevation on the rod member.

The upper ring members 42 on the rod members 20 and 21 are connected by an extensible link member, designated generally at 45, and the upper ring members 42 on the rod members 14 and are similarly connected by an extensible link member designated generally at 46. The lower ring members 41 on the rods and 14 are connected by an extensible link member 47, and the lower ring members 41 on the upstanding

vertical rods

21 and 15 are connected by a similar extensible link member 48. Each of the

extensible link members

45, 46, 47 and 48 comprises a sleeve portion 49 and a rod portion 50 telescopically received in the associated sleeve portion 49. Each of the elements 49 and 50is formed at its connection end with a vertical pivot lug 51 which is pivotally connected to the associated

ring elements

42 or 41 by being received between a pair of pivot lugs 52 formed on the ring element and pivotally connected therebetween by a transverse pivot pin 53 extending through the vertical lug 51 and secured in the pivot lugs 52, as shown in Figure 2.

As is further shown in Figure 2, the top collar element 38 engages against a limiting shoulder 55 formed on the shank portion 39 of the associated bottom collar element 40, whereby to provide sufficient clearance between the flange 38 and the flanged portion of the lower member 40 so that the

rings

41 and 42 are free to rotate on the shank 39 to allow the respective

extensible link members

45, 46, 47 and 48 to adjust themselves angularly relative to each other when the collar assemblies are vertically adjusted on their associated vertical rods. Similarly, the rods 50 are freely telescopic in their associated sleeves 49 so that the respective extensible link assemblies 45, 46, 47 and 48 are free to adjust their lengths responsive to the vertical adjustments of the collar assemblies 36 on their associated supporting

rods

14, 15, 21 and 20.

Connecting the sleeve portions 49 of the

opposing link assemblies

45 and 46 are uniformly spaced elastic bands 56, said bands comprising strips of any suitable elastic material, such as rubber or the like, secured at their ends to the respective sleeve elements 49 of the

extensible link assemblies

45 and 46 at uniformly spaced locations along said sleeve elements so as to define a series of parallel strip elements connecting the

extensible link members

45 and 46 when the respective collar assemblies on the upstanding

vertical rods

14, 15, 20 and 21 are of equal heights. Connecting the sleeve elements 49 of the opposing

extensible link assemblies

47 and 48 are similar uniformly spaced elastic bands 57, said bands extend- '4' ing beneath and being transverse to the bands 56, as is clearly shown in Figures 3 and 4. The bands 57 are spaced apart by the same distances as the bands 56 and thus define a rectangular grid when viewed from above, as in Figure 3.

As will be readily apparent, each of the collar assemblies 36 is independently adjustable on its associated vertical supporting rod, so that it is possible, by setting the collar assemblies to respective predetermined selected positions on their associated supporting rods to define any one of an infinite number of different geometrical surfaces, said surfaces being developed by the

elastic strips

56 and 57 connecting the respective pairs of opposed

extensible link assemblies

45, 46 and 47, 48. Further adjustment of the geometrical surfaces developed is provided by the adjustability of the upstanding vertical rod members 20, 21 along their

arcuate connection slots

17 and 18.

Figure 5 illustrates one example of a hyperbolic paraboloid which may be developed by the apparatus. In the arrangement of Figure 5, the collar assemblies 36 on the diagonally opposed rod members 20 and 15 are adjusted to equal heights on the rod members 20 and 15 at positions near the upper ends of said rod members, whereas the collar assemblies 36 on the remaining diagonally opposed vertical rod members 14 and- 21 are adjusted to similar positions adjacent the base board 12. Therefore, the extensible link assemblies 47 and 46 are inclined downwardly and forwardly toward the connection point D, whereas the remaining extensible link assemblies and 48 are inclined downwardly and rearwardly toward the connection point B in Figure 5. In the arrangement illustrated, the connection points, shown at A, B, C and D, define a square on the base board 12. The transversely overlapping straight

elastic strips

56 and 57 thereby define a geometrical surface which can be proven to be a hyperbolic paraboloid. A characteristic of this surface is that any vertical plane passing through the surface will define a parabola at its intersection therewith. Thus, a vertical plane passing through the surface generated in Figure 5 and arranged, for example, substantially parallel to the upstanding

vertical rod members

14 and 21 will intersect the hyperbolic paraboloid at a curve 60, shown in Figure 6, which can be proven to be a parabola. Since the terminal points of the parabola are adjacent the base board 12, the parabola is concave downwardly. If the cutting plane is substantially parallel to the other vertical rod members 15 and 20, the curve at the intersection of the vertical plane with the hyperbolic paraboloid is a parabola, shown at 61 in Figure 7, which is concaved upwardly, since its terminal generating elements are the collar assemblies 36 on the rods 15 and 20. As above stated, the vertical planes need not be parallel to the planes of the diagonally opposed

vertical rod members

15, 20 or 14, 21, to thus define parabolas at their intersections with the hyperbolic paraboloid.

Horizontal planes will intersect the hyperbolic paraboloid at curved lines which are hyperbolas.

As above explained, the vertical rods 20 and 21 may be positioned at corresponding points along their connecting

slots

17 and 18, whereby the geometrical surface generated may be formed with a rhombus or parallelogram as a base, instead of on a square. A considerable further degree of variation is possible, as above explained, since the collar elements 36 may be adjusted independently to desired settings on their associated supporting rods. For example, the collar assemblies 36 on the vertical rods 15 and 20 may be adjusted to positions adjacent the base board 12 and the remaining collar members 36 on the

rods

14 and 21 may be adjusted to positions near the top ends of said

rods

14 and 21, to define a hypo'bolic paraboloid similar to that shown in Figure 5, but reverse in its configuration.

The device above described is extremely useful for instructions in mathematics and solid geometry and is also useful in visualizing certain structural designs, such as the proportions of thin shelled concrete structures wherein it provides the ability to visually observe proportions of slopes, heights, or other geometrical features of a proposed design. The device also may be employed as an aid in making scale model structures for architectural use.

While a specific embodiment of a device for developing a mathematical surface has been disclosed in the foregoing description, it will be understood that various modifications Within the spirit of the invention may occur to those skilled in the art. Therefore, it is intended that no limitations be placed on the invention except as defined by the scope of the appended claims.

What is claimed is:

1. A device for developing a geometrical surface comprising a horizontal base, four upstanding vertical rod members secured to said base at points defining a foursided polygon on said base, respective vertically adjustable members mounted on said rod members, respective extensible link bars connecting said vertically adjustable members in the vertical planes of the respective sides of said polygon, and spaced elastic elements connecting the opposed pairs of link bars.

2. A device for developing a geometric surface comprising a horizontal base, four upstanding vertical rod members secured to said base at points defining a foursided polygon on said base, respective collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted posi-' tions on the rod members, respective extensible link bars connecting said collar members in the vertical planes of the respective sides of said polygon, each link bar comprising a sleeve element and a rod element telescopically received in the sleeve element, and spaced elastic elements connecting the opposing pairs of link bars.

3. A device for developing a geometrical surface comprising a horizontal base, four upstanding vertical rod members secured to said base at points defining a parallelogram on said base, respective collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted positions on the rod members, respective extensible link bars connecting said collar members in the vertical planes of the respective sides of said parallelogram, each link bar comprising a sleeve element and a rod element telescopically received in the sleeve element, and spaced elastic elements connecting the opposing pairs of link bars.

4. A device for developing a geometric surface comprising a horizontal base, four upstanding vertical rod members secured to said base at points defining an equalsided parallelogram on said base, respective collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted positions on the rod members, respective extensible link bars connecting said collar members, and spaced elastic elements connecting the opposing pairs of link bars.

5. A device for developing a geometrical surface comprising a horizontal base, a first pair of upstanding vertical rod members secured to said base, said base being formed with a pair of arcuate slots concentric with the respective rod members at equal radial distances therefrom, said radial distances being substantially equal to the horizontal spacing between the rod members, said slots being spaced apart by a similar horizontal distance,

whereby a pair of corresponding points in said slots define an equal-sided parallelogram with respect to the points of connection of said rod members to said base, a pair of additional upstanding vertical rod members secured at corresponding points in said slots, respective vertically adjustable collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted positions on the rod members, respective extensible link bars connecting said collar members in the vertical planes of the sides of said parallelogram, and uniformly spaced elastic bands connecting the opposing pairs of link bars.

6. A device for developing a geometrical surface comprising a horizontal base, a first pair of upstanding vertical rod members secured to said base, said base being formed with a pair of arcuate slots concentric with the respective rod members and at equal radial distances therefrom, said radial distances being substantially equal to the horizontal spacing between the rod members, said slots being spaced apart by a similar horizontal distance, whereby a pair of corresponding points in said slots define an equal-sided parallelogram with respect to the points of connection of said rod members to said base, a pair of additional upstanding vertical rod members secured at corresponding points in said slots, respective vertically adjustable collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted positions on the rod members, respective extensible link bars connecting said collar members in the vertical planes of the sides of said parallelogram, each link bar comprising a sleeve member and a rod member telescopically received in the sleeve member, and uniformly spaced elastic bands secured on and connecting the sleeve members of opposing pairs of link bars.

7. A device for developing a geometrical surface comprising a horizontal base, a first pair of upstanding vertical rod members secured to said base, said base being formed with a pair of arcuate slots concentric with the respective rod members and at equal radial distances therefrom, said radial distances being substantailly equal to the horizontal spaces between the rod members, said slots being spaced apart by a similar horizontal distance, whereby a pair of corresponding points insaid slots define an equal-sided parallelogram with respect to the points of connection of said rod members to said base, a pair of additional upstanding vertical rod members, means for clampingly securing said additional vertical rod members at corresponding points in said slots, respective vertically adjustable collar members slidably mounted on said rod members, clamping means on the collar members engageable with the rod members to secure the collar members in vertically adjusted positions on the rod members, respective extensible link bars connecting said collar members in the vertical planes of the sides of said parallelogram, each link bar comprising a sleeve member and a rod member telescopically received in the sleeve member, and uniformly spaced elastic bands secured on and connecting the sleeve members of opposing pairs of link bars.

References Cited in the file of this patent UNITED STATES PATENTS 630,217 Hanstein Aug. 1, 1899 1,810,421 Guyer June 16, 1931 1,981,646 Hamley Nov. 20, 1934 2,168,634 Spencer Aug. 8, 1939 2,312,175 Korotzer Feb. 23, 1943 2,573,946 Armstead Nov. 6, 1951