Patent Publication Number: US-2002011197-A1

Title: Tabletop with a configuration having safety features and maximum seating capacity for a given size

Description:
FIELD OF THE INVENTION  
       [0001] This invention generally relates to surfaces of objects and the shapes thereof, which are two-dimensional, closed, curvilinear and three-dimensional, closed, curviplanar in nature, and which have outer boundaries that are defined according to a mathematical expression, such that the shapes have no discontinuities, irregularities, inflection or transition points about their periphery that result in potentially unsafe or dangerous angular, sharp corners or edges in the contours of the shapes, and are otherwise also generally ergonomically beneficial and aesthetically pleasing. More particularly, the invention relates to such shapes as used for items of furniture, windows, doors, and floor coverings. Still more particularly, the invention relates to such shapes that have closed, curvilinear outer boundaries and that are substantially two-dimensional, with no angular corners or sharp edges in their contour, for use as tabletops, wherein the number of individuals, who may be seated around the perimeter of a table incorporating such a tabletop, is maximized for a table of given dimensions, as measured by a total continuous linear dimension, with each individual to be seated thereat being allocated a predetermined amount of linear space around the periphery of the table.  
       BACKGROUND OF THE INVENTION  
       [0002] Plane surfaces such as windows, doors, carpets, tables, and the surface of furniture and other items have previously been designed so that those surfaces have boundaries which are rectangular, square, circular, elliptical, or a combination of these geometric shapes.  
       [0003] In general, the closed outer boundary of a geometric shape is described by the equation (1):  
                   x   u       A   u       +       y   v       B   v         =   1           (   1   )                       
 
       [0004] where x and y are coordinates of points on the outer boundary of the geometric shape, with reference to x and y axes defining an ordinary Cartesian coordinate (x, y) system; and A and B are the coordinates of the points at which the curve described by the above equation intersects the x and y axes. Specifically, the curve intersects the positive x axis at (A,  0 ), the negative x axis at (−A,  0 ), the positive y axis at ( 0 , B), and the negative y axis at ( 0 , −B). The exponents u and v are the degrees or orders of the closed curve, and may be any rational numbers, not just integers. If u=v=2 and A=B, the curve is a circle. If u=v=2 and A is not equal to B, the curve is an ellipse. If u=v=∞ and A=B, the closed curve is a square. Finally, if u=v=∞ and A is not equal to B, the closed curve is a rectangle.  
       [0005] Although the circle and ellipse have commonly been used in the surfaces previously described, they lack surface area present with respect to comparably dimensioned squares and rectangles, respectively. On the other hand, the square and rectangular shapes, although providing maximum surface area for a given A and B, are not aesthetically pleasing and have the ergonomic disadvantage of sharp corners which may cause injury or other discomfort to users.  
       [0006] Thus, there exists a need for surfaces shaped such that more surface area is available than the traditional circular or elliptical shapes, while eliminating the unattractiveness of and the hazard of sharp corners of square or rectangular shapes.  
       SUMMARY OF THE INVENTION  
       [0007] The invention comprises surfaces, such as table tops, with boundaries defined by the general analytical curve ( 1 ), shown above, but with the further restriction that 2&lt;u&lt;10 and 2&lt;v&lt;10 ( 2 ). The lower limit of 2 defines a circle (when A=B) or an ellipse (when A B). The upper limit of 10 has been found empirically to be the limit after which the boundary shape approximates a rectangle or square, albeit with slightly rounded corners. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0008]FIG. 1 is a graphical representation of two comparative closed curves for the case of A=4 and B=3 where u=v=2 for one curve and u=v =3 for the other curve.  
     [0009]FIG. 2 is a graphical representation of two comparative closed curves for the case of A=4 and B=3 where u=v=2 for one curve and u=v=4 for the other curve.  
     [0010]FIG. 3 is a graphical representation of two comparative closed curves for the case of A=4 and B=3 where u=v=2 for one curve and u=v=6 for the other curve.  
     [0011]FIG. 4 shows a plan view of a table top of an elliptical shape (u=v=2) where A=46 inches and B=23 inches, and shows outlines of 23 inch width spaces, each space representing the width needed for the seating of one person, arranged around the table top.  
     [0012]FIG. 5 shows a plan view of a table top of a shape where u=v=4 and where A=46 inches and B=23 inches, and shows outlines of 23 inch width spaces, each space representing the width needed for the seating of one person, arranged around the table top.  
     [0013]FIG. 6 is a three-dimensional perspective view of the table of FIG. 6. 
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS  
     [0014] Equation (1) can be converted into the following equation by multiplying both sides by the factor A u B v :  
       B   v   x   u   +A   u   y   v   =A   u   B   v   (3) 
     [0015] Setting x=0, one obtains:  
     A y y v =A u B v   (4) 
     y=B  (5) 
     [0016] Letting y=0, one further obtains:  
     B v x u =A u B v   (6) 
     x=A  (7) 
     [0017] Differentiating equation (3) with respect to x results in:  
                   B   v          ux     u   -   1         +       A   u          vy     v   -   1                 y          x           =   0           (   8   )                       
 
                    y          x       =         -     B   v            ux     u   -   1             A   u          vy     v   -   1                   (   9   )                       
 
     [0018] Evaluating  
            y          x                   
 
     [0019] at x=0, one obtains:  
                    y          x                               =   0                 x   =   0                   (   10   )                       
 
     [0020] Finally, evaluating  
            y          x                   
 
     [0021] represented by the equation (9) at y=0, one obtains:  
                    y          x                               =   ∞                 y   =   0                   (   11   )                       
 
     [0022] It has been empirically determined that u and v should not be allowed to reach the value of 10.  
     [0023] In general, the surface area available increases as the degree of the curve increases, with the most dramatic increase occurring from the elliptical shape to the third degree shape.  
     [0024] FIGS.  1 - 3 , respectively, graphically show the closed curves for u=v=3, u=v=4, and u=v=6, where A=4 and B=3 for all of FIGS.  1 - 3 . The closed curves are drawn with respect to a Cartesian coordinate system, wherein the x axis is a first axis of symmetry,  2 , for the curves, and the y axis is a second axis of symmetry,  4 , for the curves. Being the axes of a Cartesian coordinate system, the x and y axes are perpendicular to each other. In all of FIGS.  1 - 3 , the closed curve for u=v=2 and A=4 and B=3, which is an ellipse, is also drawn with respect to the x and y axes for the purpose of comparison to the curves for u=v=3, u=v=4, and u=v=6.  
     [0025] The use of these boundaries of degree three or higher up to degree nine can be applied in a myriad of plane surfaces commonly used. Examples of such applications are tables, doors, windows, carpets, any plane surface of furniture, or indeed any plane surface of any object imaginable. Furthermore, although the surface enclosed by such a shaped boundary may not, in fact, be planar, but rather curved in three dimensional space (curviplanar surface), the shape of the boundary described herein can still be used with like advantages for increased surface area yet pleasing aesthetic appearance and beneficial ergonomic characteristics.  
     [0026] A specific example of a tabletop with a shape defined by the boundary of a closed curve, with A and B being set at actual physical dimensions, such that A=46 inches and B=23 inches, and u=v=4 is shown in FIG. 5. Standard individual seating spaces of 23-inch width are arranged about the table. As can be seen in FIG. 5, a total of ten such seating spaces can be arranged around the table. In contrast, a table top with a boundary, described by a closed curve with the same values of A and B as FIG. 5, but with u=v=2 (an ellipse), is shown in FIG. 4. FIG. 4 shows that only eight 23-inch seating spaces can be arranged comfortably about that table. The 14-inch seating spaces remaining at each relatively curved end of the table do not afford comfortable seating spaces.  
     [0027]FIG. 6 is a three-dimensional (3-D) perspective view of the table according to FIG. 5, showing the way in which ten persons may comfortably be seated.