slacks may have bottoms that are modeled with a mathematical definition , and be illustrated using descriptive geometry with perspective representations of that mathematical definition . the mathematical definition may be adjustable for various sized persons , various shoe styles , and more particularly for variations in clearance to the shoe heel or throat , so that the slacks do not drag along the ground . the definition may also be adjusted for clearance with the shoe throat to prevent excessive sag therein . fig1 and 2 show front and side views of an article of manufacture , being slacks 10 , and having legs ( 11 l and 11 r ) with bottoms 12 that were surface - modeled , cut , and assembled according to a method of the current invention . as seen in the enlarged side view ( fig3 ), front view ( fig4 ), and rear view ( fig5 ), the slacks may have custom contouring , which may be defined using mathematical definitions , with those mathematical definitions being communicable to a computer numerically controlled ( cnc ) fabric cutting machine . examples of such cnc fabric cutting machines are currently produced by autometrix precision cutting systems , inc ., and by the eastman machine company . the slack bottom 12 may be surface modeled in a simple form as a cylindrical surface , but may be more accurately represented as a cylindrical surface that transitions into either an elliptical conical surface or a circular conical surface , where the front and rear portions of the slack bottoms may be specially trimmed to prevent the rear slack bottom from dragging on the ground or sagging at the heel , and prevent excessive slack - to - shoe - throat sag . the shoe throat ( see fig9 ) is that part of the shoe between the toe and the top of the tongue . manufacturing according to the disclosed method permits creation of a library of slack bottoms that may be tailored to coordinate to persons of different sizes , persons having different shoe sizes , and for varying styles of shoes , such as flat heeled shoes versus boot heeled or high heeled shoes . the process works for both men &# 39 ; s and women &# 39 ; s slack bottoms , and for denim jeans , or other fabrics . surface modeling of a leg of the slack is shown in a simplified form in fig8 . the pant leg is surface modeled as a cylindrical surface , whose definition may be given by the equation : where “ x ” is a variable describing a coordinate position on the cylinder surface relative to the x - axis , “ y ” is a variable describing the coordinate position on the cylinder surface relative to the y - axis , and “ r ” is the radius of the cylinder . the z variable , which does not appear in the cylinder surface equation , will , in later equations , describe the coordinate position on a surface relative to the z - axis . the slack may remain accurately represented by the cylindrical surface all the way until reaching the shoe , or , as is common with many garments , the slack may terminate in a flared bottom . it may also terminate in a reduced diameter cylindrical bottom . the case of the flared bottom may be more accurately represented by the cylindrical surface transitioning into an elliptical conical surface , or the circular conical surface shown at the bottom of fig8 . where x , y , and z are variables as previously described . to eliminate confusion herein , a , b , and c in the equation may be replaced by the greek alphabetical equivalents of α , β , and γ , so the equation would appear as follows : where α is the maximum x - coordinate value of the elliptical conical surface at the cone bottom , β is the maximum y - coordinate value of said elliptical conical surface , and γ is the maximum z - coordinate value of the elliptical conical surface . the origin of the cone ( coordinate being x = 0 , y = 0 , and z = 0 ) may be conveniently chosen in relation to the maximum α , β , and γ values to produce a flare of desired proportions , and would commonly be at some point above the transition . since the slack is normally pressed to have creases and naturally thereafter tends towards an elliptical conical shape , the flared bottom of the surface model for the slack may be more simply represented by the special case of the elliptical cone , where β is equal to a to form a circular conical surface , which is described by the equation : ( x / α ) 2 +( y / α ) 2 −( z / γ ) 2 = 0 trimming of the cylindrical surface or the circular conic surface of the slack bottom to achieve the aforementioned clearances may be accomplished using different cutting shapes . the simplest means of trimming the cylindrical surface or the circular conical surface may be to cut along a planar surface to create a forward and rear edge contour on the slack bottom . planar intersections of a cone may form three different kinds of curves , which are illustrated in fig7 . the planar intersection where the plane is normal to the axis of the cone ( the z - axis in fig8 ) produces a circular entity . the planar intersection where the plane is angled with respect to the cone axis produces an ellipse , and where the angle is located to pass through the end of the cone at the maximum cone diameter , a parabola results . the representation of the slack bottom in fig8 may have the lower rear and forward portions of the circular conical surface trimmed with the planar cuts as shown , for the aforementioned clearance . the lower front portion of the slack may preferably be trimmed by a first carefully selected plane , while the rear portion of the slack may preferably be trimmed by another , different , plane . the front trim plane may be described by the equation : since the planar cut may likely be symmetric with respect to the x - z plane ( see the profile view in fig1 ), the plane equation reduces to : where the equation &# 39 ; s coefficients are denoted with the upper - case greek characters of a , b , and γ , rather than a , b , and c , and δ is the perpendicular distance of the plane to the origin . the planar cut for the rear slack bottom portion may be similarly described by the equation : trimming the circular conical surface according to those planar definitions may produce the forward and rear parabolic edge contours seen in fig8 , and which appear planar in the profile view of fig1 . it should be pointed out that either of , or both of , the front and rear portions of the circular conical surface model of the slack bottom may be trimmed as just described . the bottom may also be trimmed according to other mathematical definitions . as seen in the profile view in fig1 , either of , or both of , the front and rear portions of the circular conical surface model of the slack bottom may be trimmed by a cylindrically defined cut , with respective equations of : ( x − x r ) 2 / r r 2 +( z − z r ) 2 / r r 2 = 1 ( x − x f ) 2 / r f 2 +( z − z f ) 2 / r f 2 = 1 , where x r and z r comprise the coordinate location of the cylinder axis of a cylinder having a radius or r r . a surface , which may be a cylindrical surface or another surface having a discrete geometric definition , usable for cutting the circular conical surface ( or cylindrical surface if so used to surface model pant legs ) may be oriented so as to be positioned so as to be orientated non - orthogonal relative to the x / y / z axes , and therefore have an arbitrary axis . an example of where this may be desirably exploited through the current invention , is the case where surface modeling is used for a front cutting surface for the front slack bottom , where the front cutting surface represents the front portion of particular styles of shoes or boots , particularly for the likely shoe contact area - the throat . the mathematics involved becomes more substantial than those for profiling along an axis by a cylinder ( fig1 - 12 and 13 - 15 ), but is discussed briefly in the following paragraph to enable such a trim contour , particularly as it may also be advantageously communicable to the cnc fabric cutting machine . the front portion of a shoe , namely the tongue and throat area , may be approximated by a cylindrical surface . in fig1 a , a cutout is created by using a cylindrical surface having an axis parallel to the x - axis , where the x - axis is into the page . the cylinder may pass through the point ( y shoe , z shoe ), having a radius r shoe , and be defined by the equation : the cylindrical cut may better approximate the shape of the shoe &# 39 ; s tongue and throat , by having an axis which is angled with respect to the x - axis . the mathematics involved for the angled cylinder may be described using a unit vector ( u 1 , u 2 , u 3 ), or alternatively using parametric equations , the mathematics of which may be found in relevant math or geometry texts , papers or websites ( e . g ., http :// mathforum . org / library / drmath / view / 51734 . html ), which is incorporated herein by reference , as it may be beyond the scope of this presentation . it should also be noted that an irregular trim surface may also be defined mathematically and utilized for slack bottom contouring . also , for simplicity , the front and back of the slack bottom may be trimmed so as to be identical . the mathematical definition of the surface for the slack may be unwrapped into a flat pattern , permitting cutting of fabric according to that flat pattern on a cnc fabric cutting machine . the described slack surface definition may be generated using a processor of computer , which may then be unwrapped using the same computer , for transmission of that definition to the cnc machine . use of the computer in relation to the trimming of the slack bottom requires description of an exemplary computer for enablement of the claims using such a processor . an exemplary computing unit 201 may include a data bus 224 or other communication mechanism for communicating information across and among various parts of computing unit 201 , and a central processing unit (“ processor ” or cpu ) 222 coupled with a bus 224 for processing information and performing other computational and control tasks . computing unit 201 may also include a volatile storage 225 , such as a random access memory ( ram ) or other dynamic storage device , coupled to bus 224 for storing various information as well as instructions to be executed by processor 222 . the ram may be dynamic random access memory ( dram ), or static ram ( sram ), or any other similar type of ram known in the art . the volatile storage 225 also may be used for storing temporary variables or other intermediate information during execution of instructions by processor 222 . computing unit 201 may further include a read only memory ( rom ) or an erasable programmable memory ( eprom ) 227 or other static storage device coupled to bus 224 for storing static information and instructions for processor 222 , such as basic input - output system ( bios ), as well as various system configuration parameters . a persistent storage device or non - volatile memory 226 , such as a magnetic disk , optical disk , or solid - state flash memory device is provided and coupled to bus 224 for storing information and instructions . computing unit 201 may be coupled via bus 224 to a touch screen display 221 , such as a plasma display , or a liquid crystal display ( lcd ), for displaying information to a user of the computing unit 201 . if desired , the computing unit 201 may also be coupled via bus 224 to an external display screen 245 , which may further comprise a cathode ray tube ( crt ). an external input device 244 , including alphanumeric and other keys , may also be coupled to bus 224 for communicating information and command selections to processor 222 . another type of user input device is cursor control device 243 , such as a mouse , a trackball , or cursor direction keys for communicating direction information and command selections to processor 222 and for controlling cursor movement on display 245 , if desired . also , a cursor control device 243 may also be utilized for the pc 261 of the network resources 203 . an external storage device 242 may be connected to the computing unit 201 via bus 224 to provide an extra or removable storage capacity for the computing unit 201 . in an embodiment of the computer system 200 , the external removable storage device 242 may be used to facilitate exchange of data with other computer systems . according to one embodiment of the invention , the techniques described herein are performed by computing unit 201 in response to processor 222 executing one or more sequences of one or more instructions contained in the volatile memory 225 . such instructions may be read into volatile memory 225 from another computer - readable medium , such as persistent storage device or non - volatile memory device 226 . execution of the sequences of instructions contained in the volatile memory 225 causes processor 222 to perform the process steps described herein . in alternative embodiments , hard - wired circuitry may be used in place of or in combination with software instructions to implement the invention . thus , embodiments of the invention are not limited to any specific combination of hardware circuitry and software . the term “ computer - readable medium ” as used herein refers to any medium that participates in providing instructions to processor 222 for execution . the computer - readable medium is just one example of a machine - readable medium , which may carry instructions for implementing any of the methods and / or techniques described herein . such a medium may take many forms , including but not limited to , non - volatile media , volatile media , and transmission media . non - volatile media includes , for example , optical or magnetic disks , such as storage device 226 . volatile media includes dynamic memory , such as volatile storage 225 . transmission media includes coaxial cables , copper wire and fiber optics , including the wires that comprise data bus 224 . transmission media can also take the form of acoustic or light waves , such as those generated during radio - wave and infra - red data communications . common forms of computer - readable media include , for example , a floppy disk , a flexible disk , hard disk , magnetic tape , or any other magnetic medium , a cd - rom , any other optical medium , punchcards , papertape , any other physical medium with patterns of holes , a ram , a prom , an eprom , a flash - eprom , a flash drive , a memory card , any other memory chip or cartridge , a carrier wave as described hereinafter , or any other medium from which a computer can read . various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 222 for execution . for example , the instructions may initially be carried on a magnetic disk from a remote computer . alternatively , a remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem . a modem local to computer system 200 can receive the data on the telephone line . the bus 222 may carry the data to the volatile storage 225 , from which processor 222 retrieves and executes the instructions . the instructions received by the volatile memory 225 may optionally be stored on persistent storage device 226 either before or after execution by processor 222 . the instructions may also be downloaded into the computing unit 201 via internet using a variety of network data communication protocols well known in the art . the computing unit 201 may also include a communication interface , such as network interface card 223 coupled to the data bus 222 . communication interface 223 provides a two - way data communication coupling to a network link that may be connected to a local network 262 . for example , communication interface 223 may be an integrated services digital network ( isdn ) card or a modem to provide a data communication connection to a corresponding type of telephone line . as another example , communication interface 223 may be a local area network interface card ( lan nic ) to provide a data communication connection to a compatible lan . wireless links , such as well - known 802 . 11a , 802 . 11b , 802 . 11g and bluetooth may also used for network implementation . in any such implementation , communication interface 223 sends and receives electrical , electromagnetic or optical signals that carry digital data streams representing various types of information . network link 223 typically provides data communication to other network resources . for example , the network link may provide a connection through local network 262 to a host computer 261 , or the computing unit 201 may connect directly to the host computer 261 . alternatively , the network link 223 may connect through gateway / firewall 263 to the wide - area or global network 264 , such as an internet . thus , the computing unit 201 can access network resources located anywhere on the internet 264 . on the other hand , the computing unit 201 may also be accessed by others , with permission , who are located anywhere on the local area network 262 and / or the internet 264 . the other users may themselves be operating a platform similar to computer system 200 . local network 262 and the internet both use electrical , electromagnetic or optical signals that carry digital data streams . the signals through the various networks and the signals on network link and through communication interface 262 , which carry the digital data to and from computing unit 201 , are exemplary forms of carrier waves transporting the information . computing unit 201 can send messages and receive data , including program code , through the variety of network ( s ) including the internet 264 and lan 262 , network link and communication interface 233 . in the internet example , when the computing unit 201 acts as a network server , it might transmit a requested code or data for an application program running on pc 261 through the internet 264 , gateway / firewall 263 ; local area network 262 and communication interface 223 . similarly , it may receive code from other network resources . the received code may be executed by processor 222 as it is received , and / or stored in persistent or volatile storage devices 226 and 225 , respectively , or other non - volatile storage for later execution . in this manner , computer system 200 may obtain application code in the form of a carrier wave . the examples and descriptions provided merely illustrate a preferred embodiment of the present invention . those skilled in the art and having the benefit of the present disclosure will appreciate that further embodiments may be implemented with various changes within the scope of the present invention . other modifications , substitutions , omissions and changes may be made in the design , size , materials used or proportions , operating conditions , assembly sequence , or arrangement or positioning of elements and members of the preferred embodiment without departing from the spirit of this invention .
a method for creating a series of size - proportioned trouser bottoms comprises each of the slack bottoms being approximated as a cylindrical surface transitioning into a circular conical surface . the back lower portion of the circular conical surface is trimmed along a cutting plane to produce a parabolic - shaped rear edge , to provide rear clearance above a shoe heel . the front lower portion of the circular conical surface is trimmed along second cutting plane producing a unique parabolic - shape front edge for clearance above a shoe throat . trimming may be by a cnc fabric cutting machine , using the descriptive geometric definition of the circular conical surface , 2 + 2 2 = 0 , and using planar cuts for the parabolic edge trim defined by the equation , ax + γz + δ = 0 . the defined surface may be unwrapped and supplied to the cnc machine as a flat pattern .