Abstract: A exercise device for exercising muscles of the upper body. The exercise device includes a generally U-shaped housing with a base portion adapted for resting on a floor, a main portion upwardly extending from a rear of the base portion, and an upper portion outwardly extending from an upper end of the housing. A pulley system is housed in the housing. A continuous rope extends through the pulley system and through holes in the upper and base portions of the housing. Movement of the rope is resisted by a friction member adjustably engaging portions of the rope wrapped about a cylindrical pulley.

Abstract: A wrist and forearm exerciser for exercising the weak areas of the hands, arms, wrists, and forearms. The wrist and forearm exerciser includes an elongate handle rod and a line with opposite upper and lower ends. The upper end of the line is coupled to the handle rod. A weight support means is coupled to the lower end of the line. A plurality of weights are supported by the weight support means.

Abstract: A one piece face and neck exerciser (10), specifically configured to tone the muscles of the forehead, face, chin and neck, comprising a head band (12) to which two elongated beginners head resistance bands (88) are attached into head grooves (20) extending outwardly and down to be permanently attached to the lateral parts of the chin cup (38). Also comprising a neck band (57) to which two elongated beginners neck resistance bands (104) are attached into neck grooves (64) extending outwardly and up to be permanently attached to the lateral sides of the chin cup (42). The frontal head band (14), and lateral head band (16), the frontal neck band (58) and lateral neck band (60) are made of a strong, durable, lightweight material, the posterior head band (32), and posterior neck band (76) are made of velcro fasteners (34), (36), (78), and (80).

Abstract: A portable in-bed exercising machine is provided for strengthening and/or strength maintenance of the musculature and ligaments associated with an extremity. The exercise apparatus employs the principles of closed kinetic chain exercise of the extremity in both concentric and eccentric modes and for isometric, isotonic and isokinetic exercise. Each extremity is engaged separately allowing for bilateral, unilateral and reciprocal motion. As exemplified for lower extremities, force is actively applied to the feet with each lower extremity having the force level continuously adjustable from zero force to a force equal to or greater than body-weight. The exercise motions and force applications use the same major muscle groups as functional activities such as standing from a chair, climbing stairs, walking, jumping and jogging. Vertical support is provided for the lower extremity to stabilize and control the lower extremity motion within safe limits.

Abstract: A rope pull exercising device in which the rope follows a tortuous path around a figure eight braking bar of the type used conventionally in rock climbing. The rope threads through one of the rings of the figure eight bar at least once and around the connecting region that joins through the rings. Mounting structure secures the second ring to a stationary object such as, for example, a door frame.

Abstract: A weighted vest capable of comfortably conforming to the body of the wearer. A top loaded vest is provided which extends over the wearer's thorax but terminates entirely above the wearer's abdomen so that full body movement is permitted and breathing is not restricted. Stretchable shoulder harnesses, or straps, are utilized so that the vest is supported comfortable on the shoulders of the wearer and so that the vest can conform to the body during various full range body movements. A pair of securement straps gird the wearer's chest to ensure that the weights and vest do not bounce or shift during exercise.

Abstract: A variable friction resistance exercise machine 10 including a pair of friction wheels 31 rotatably suspended by axles 30, 30' in a framework member 20. The axles 30, 30' are operatively connected to coiled lengths 41 of cable attached to one end to clutch rewind assemblies 40. The friction wheels 31 are provided with adjustable friction means 33, 35, 36, 37, and 38 to vary the frictional resistance required to rotate each friction wheel 30, 30' relative to the framework member 20.

Abstract: The present invention discloses an enclosure 10 having a base with rubber edges 16 to prevent slippage on the floor. A boot 12 for a human foot is provided also. Straps 18 connect boots 12 holding the human feet which straps connect to each other through cables 20 traveling over a pulley means 22 connecting to a multiple spring assembly 40 which cause the straps to be tensioned and to communicate with each other. The multiple springs 40 allow the tension to be adjusted on the straps 18 so that the tension on the boots 12 can be varied, thereby varying the exercise tension on the foot 52. The boots 12 are provided with cleats 50 and hook material 48 for holding the straps 18 thereon.

Abstract: An energy expenditure garment is made of a variable density fabric having a plurality of zones of differing resistance characteristics. This is accomplished by utilizing threads in the fabric of differing resistance characteristics so that the zones are seamless and blend into each other.

Abstract: An energy expenditure garment incorporates various techniques for providing resistance to the bending movements of the wearer. Such techniques include having a plurality of different types of resistance means. One of the resistance means is weights.

Abstract: An energy expenditure garment includes a body portion and limb portions made of a base fabric and incorporating at least one elastic resistance band which is free of any limb extremity anchor structure. The garment includes structure for retarding slippage of the garment during an exercise while still permitting some degree of slippage.

Abstract: Energy conservation or expenditure garments are designed for use by cyclists. The garment includes in one embodiment an air cooling system through use of various air passageways in the energy conservation garment. In another embodiment the energy expenditure garment includes pockets to offer drag or resistance to the user. An indicator may be provided at the back of the hand portion of the garment to indicate some parameter of exercise.

Abstract: Weighted suit to be used in athletic training, physical therapy, muscle toning and weight reduction. The suit has a bottom and top that can be adjusted to accommodate the physical characteristics of the wearer, as well as the activity to be performed by the wearer while the suit is worn. The suit also has weighted gloves which may be attached to the sleeve portion of the top of the suit. The suit is constructed from spandex material that accommodates stretch in one direction and resists it in another direction approximately perpendicular to the first. Weight compartments are located about the suit into which weight units may be installed. Variable weight packets may be used to selectively apply different resistance experienced during the user's activity. The weights are located away from the user's joints, and protective components may be integrally included into the suit to cover vulnerable knee and elbow joints.

Abstract: An exercise machine mounted above one end of a bed, occupied by a bedridden patient or other supine user, has one or more double-ended ropes, each rope having a handle or a stirrup on each of its ends and a central loop tractively encircling a torque drum. Each torque drum is mounted for rotation against adjustably variable braking torque, which may be canceled in one direction of angular rotation by a one-way clutch. The handles and stirrups have adjustable rope engaging features, achieving infinitely variable rope lengths. Pulleys guide the ropes for substantially horizontal pulling reciprocation by the supine user, whose position may be braced or stabilized by torso-positioning rigid arms, such as adjustable telescoping spacer tubes, extending from the exercise machine to the user.

Abstract: A motion restricting device restraining rectilinear movement of the ends of a rope and comprised of a post with retainment rings passing the rope at one or both opposite ends of the post, and having at least one and preferably a plurality of radially projecting adjustment bits intermediate the ends of the post for hooked engagement of a section of the rope drawn circumferentially around the post, the post and rings and bits being integrally formed and smooth surfaced for controlled frictional engagement with said rope.

Abstract: An exercise apparatus which isometrically works the upper body muscles while the person is engaged in an aerobic activity such as walking or while the person remains in a standing or sitting stationary position. The exercise apparatus includes a belt member having a first end, a second end, and a body section disposed between the first and second ends. The first and second ends are securable to each other for fastening the belt member around the torso of a person. A pulley housing is mounted to the body section of the belt. A pulley wheel is rotatably mounted within the pulley housing. An adjustable tension mechanism is attached to the pulley housing and the pulley wheel for selectively adjusting an amount of force needed to rotate the pulley wheel within the pulley housing.

Abstract: A passive exercise apparatus includes an interface member for coupling to a user's limb. An arm member is coupled to the interface member by a wrist joint. The wrist joint has rotational motion about three axes. A first link is coupled to the arm member by a sliding joint. The sliding joint allows translational motion of the arm member relative to the first link. A second link is rotatably coupled to the first link by a first rotary joint. A third link is rotatably coupled to the second link by a second rotary joint. A first brake is coupled to the first rotary joint for resisting movement of the first rotary joint. A second brake is coupled to the second rotary joint for resisting movement of the second rotary joint. A third brake is coupled to the arm member for resisting movement of the arm member. The user is capable of interfacing with the apparatus to exercise a six degree of freedom motion.

Abstract: A fitness equipment including a base, an oblique support equipped with bottom wheels and pivoted to the base to hold a bench with a seat cushion and a headrest for supporting the user, two handlebars coupled to the oblique support, two actuating rods respectively pivoted to the handlebars, a bracing collapsibly connected between the base and the oblique support, two collapsible pedals pivoted to the base and the oblique support, upper links and intermediate links and bottom links respectively pivotably connected between the oblique support and the actuating rods, a damping mechanism connected between the oblique support, upper and intermediate links, wherein when the user sits on the seat cushion with the back supported on the bench and the hands holding on the handlebars and the legs pedaling the pedals, the bottom links and the intermediate links and the upper links are alternatively driven to move the actuating rods and the connecting frames, enabling the user to simulate the action of swimming.

Abstract: An exercise apparatus includes a plurality of inelastic, retractable cords, each terminating at a handgrip. A retracting mechanism is provided for retracting the cords. Separate resistance mechanisms are provided to resist each of the cords as the cords are being withdrawn from the retracting mechanism. The resistance mechanisms operate independently of one another. At least one removable disk unit is connectible to the resistance mechanism of the exercise apparatus for incrementally increasing resistance force with respect to unwinding movement of the cords when operably connected to the resistance mechanism. The removable disk units can be preloaded with a predetermined biasing resistance force when assembled during manufacturing, and the biasing resistance force can be maintained while not connected to the main unit of the exercise apparatus.

Abstract: An improved friction type exercise device of the type in which a housing has an outer wall forming a hollow cavity with a central axis. A central shaft is positioned within the cavity and positioned to allow axial and rotative movement relative to the housing when moved in one axial direction and preventing axial rotative movement relative to the housing when moved in the other axial direction. A hub is positioned in the cavity and fixed to the shaft to move axially therewith while allowing rotation of the shaft relative to the hub, the hub being nonrotatable with respect to the housing and includes a first and a second opening. A rope-like member extends through the first opening in the hub, around the shaft and out the second opening in the hub whereby when the rope-like member is pulled frictional resistance is exerted on the rope-like member in proportion to the number of turns around the shaft.

Abstract: It is critical to prevent muscular damage to an athlete in a machine that produces a restraining force that must be overcome in the initial part of an exercise stroke. This is achieved by a precise and accurate type of control system by which the athlete safely and surely control the initial restraining force magnitude. With a reliable vernier type control device the athlete then may "inch-by-inch" increase the restraining force as the muscles become able to overcome greater forces in the manner the proverbial strong man lifts a calf each day so that he can lift a full grown cow eventually. In a preferred embodiment of this vernier type control the restraining device comprises an electromagnet with a retained magnetic member coupled to the athlete by an intermediate muscular operated linkage.

Abstract: A multi-purpose exercising apparatus includes a mounting base securely provided with a horizontal seat cushion and a vertical backrest. A hollow top side of a backrest tubular body at a rear side of said backrest is connected to two adjusting mechanisms which are respectively connected to a support arm. A tensile arm is disposed below the seat cushion at a front edge thereof. A damping mechanism is disposed within the mounting base. The damping mechanism consists of three resilient ropes wound around revolving shafts, in which two of the ropes respectively pass around pulley sets to be pulled out from their corresponding support arms. The two ropes are then fixedly connected to two rings or a lever. The third rope is fixedly connected to the tensile arm. By means of pressing a button of the adjusting mechanism to adjust the angles of the support arms, the user may exercise the triceps or muscles of the arms, chest, legs or abdomen as desired.

Abstract: A multifunction exercise and aerobic bench includes an elongated support member having a head end and a foot end. Legs extend below each of the head end and foot ends to elevate the support member. Handles are secured to the support member at the head end and extend upwardly above the support member. The legs at the foot end are movably mounted so that the elevation of the foot end can be lower than the head end to selectively incline the support member. A bar is pivotally mounted at the foot end for moving in an arc.

Abstract: A portable exerciser having an adjustable constant resistance to employment thereof. The exerciser includes a shaft and a housing slidably disposed on the shaft. A gear in the housing engages the shaft and a member frictionally engages the gear so that movement of the housing relative to the shaft rotates the gear against the friction force applied by the member.

Abstract: An apparatus for exercising the muscles surrounding the cervical spine of a person, comprising: a belt adapted to be secured about the waist of a person, and including frontal and posterior portions; a pair of suspenders connecting at their frontal ends of opposing frontal portions of the belt, extending over opposing shoulders of the person, and connecting at their posterior ends to opposing posterior portions of the belt; a chin cup adapted to fit around the person's chin; a head band adapted to fit around the person's head and including frontal and posterior portions; a first elastic strap adapted to provide a tensile load between the chin cup and the frontal portion of the belt; and a plurality of elastic straps adapted to provide tensile loads between the frontal portion of the head band, across the person's head, and the posterior portion of the belt or suspenders.

Abstract: An apparatus for exercising the muscles surrounding the cervical spine of a person in an isokinetic manner, comprising: a headband adapted to fit around the person's head; a breast plate; a back plate; at least one strap adapted to hold the breast plate and back plate in position on the person's body; a belt adapted to fit around the person's lower torso and lower back, wherein the belt is coupled to the breast plate or back plate; a lumbar belt adapted to fit around the person's waist; a plurality of adjustable non-elastic straps with isokinetic tensioners adapted to provide tensile loads between the headband and the breast and back plates; a plurality of adjustable elastic straps adapted to provide tensile loads between the headband and the breast and back plates; and a plurality of adjustable elastic straps adapted to provide tensile loads between the breast and back plates and the lumbar belt.

Abstract: An exercise device is described comprising frame means, a wheel mounted to said frame means, a second wheel mounted to said frame means spaced from the first wheel, and an endless rope, or cord, passing over the wheels, which are spaced to keep the rope, or cord, taut. Means are provided for applying friction to one of the wheels mounted to said frame means which will apply a restraint to the rope, or cord, which must be overcome by an exercising person pulling on the rope, or cord. The frame means is rotatably mounted to a sub-frame by a central pivot support for rotation relative to said sub-frame and locking means is provided to permit a desired angle of rotation to be held for particular arm related exercises.

Abstract: A one way clutch is rotatable with a member (e.g. helical spring) and a sleeve while the spring is being constrained by the manual winding of a cable on the sleeve. The sleeve is constructed to wind the cable uniformly on the sleeve. During the constraint imposition, the clutch drives a shaft and a disc. One disc surface abuts bearing liners affixed to a casing. The other disc surface is separated from the spring. Calibrated numbers are printed on the casing periphery. A dial externally supported by, the casing is rotatable relative to the casing to any of the calibrated numbers to adjust the friction force between the liners and the disc when the disc is rotated. The casing is vented at its opposite ends. When the spring is being constrained or the constraint is being released, air flows through the vents and through the space between the successive turns in the spring and past the disc to cool the disc.

Abstract: A pivoting exercise apparatus comprising a structural frame, a thigh engagement member, and optional resistance device and motor, whereby the user may engage the thigh engagement member with the thigh portion of their legs and perform lower body exercise routines while in a relatively reclined position. The thigh engagement member is pivotally coupled to the structural frame to pivot in the forward, backward, and side directions. The user may manually produce motion in the thigh engagement member. A resistance may be included to provide a resistance force against this pivoting motion. A motor may also be used to provide automatic pivoting motion of the thigh engagement member.

Abstract: A chest expander includes a first handle having two distal ends, a substantially U-shaped tubular housing including a first line and a second line extended from two limb ends thereof and connected to two distal ends of the first handle, a second handle connected between the two limbs of the U-shaped tubular housing, and a load adjusting mechanism being received in one corner of the U-shaped tubular housing allowing a user to adjust the load of the chest expander. The two handles of the chest expander are manually extended and retracted repeatedly via the two lines.

Abstract: A portable exercise device includes a base having a flat central section on which the user may either stand, sit or lie and a strap housing on each side of the central section. The base comprises two sections joined by mating extrusions such that the base may be disassembled to provide greater portability of the exercise device. One of the strap housings contains a supply/takeup reel onto which left and right straps are wound and from which they are unwound during a workout by the user. The right strap is routed underneath the base and around an idler pulley contained within the other housing. Each of the straps exits its respective housing and is connected to one end of an exercise bar. A rewind spring, a one-way clutch, and a band brake are coupled to the supply/takeup reel to adjust the force required of the user to pull the straps during a workout and to rewind the straps when the user releases the bar.

Abstract: An apparatus is disclosed for stretching tendons, muscles and associated soft tissues of human joints. The apparatus includes a generality planar frame having two sides, two ends, a lower surface adapted to be supported by a ground support surface and an upper surface. An elongated upright support has a first end which is secured to the frame so that the second end of the upright support extends generally vertically upwardly from the frame. At least one pulley is secured adjacent the upper end of the upright support. Thereafter, an elongated cord has one end adapted to be secured to the limb of a human user and this cord extends through the pulley. The human user, by pulling on the cord, stretches the limb of the human user in the desired fashion. By attaching the cord to different limbs and assuming different positions, the human user can stretch his or her tendons, muscles and associated soft tissues of numerous joints and a numerous stretching positions.

Abstract: A frictional exercising device includes a shell body which can be attached to a substantially stationary object, for instance a door, a doorjamb or a floor. A core cylinder is rigidly mounted in the shell body. The core cylinder has a wall with an arcuate friction surface having bumps formed thereon. A space is defined between the friction surface and the shell body through which a flat strap extends. The flat strap is partially wound around the core cylinder and hugs the friction surface. When one of the ends of the flat strap is pulled and a resistive force is applied at the other end, a frictional force increases between the friction surface and the flat strap.

Abstract: Rope climbing apparatus comprises a support, a length of rope and a rope accumulator supported by the support for storing a variable length of rope. The accumulator includes a dancer trolley and means for force-loading the dancer trolley to a maximum storage position. A segment of rope is presented to a user of the apparatus so that he can draw rope from the accumulator. Variable drag mechanism supported by the support feeds rope pulled by the user back to the accumulator to replenish the accumulator. The drag mechanism inhibits the feeding of rope to the accumulator unless the rope is pulled with a force sufficient to overcome the force-loading on the accumulator dancer trolley and move the dancer from its maximum storage position. The drag mechanism also controls the rate at which rope is fed to the accumulator in accordance with the magnitude of the accumulator force-loading.

Abstract: A rope friction exercise device is described which is in the form of a member 12 (FIG. 2) having multiple bars (51-62) about which a rope (30) can extend in multiple wrappings to provide frictional resistance against pulling of the rope. The member has an elongated longitudinally-extending first side (42) and has a first plurality of bars (51-54) spaced along the first side, with each bar having a first end (80) mounted on the side and a second end (74) which is free, so a rope can be wrapped and unwrapped from a bar by merely installing and removing it from the free end, to thereby change resistance to rope pulling. A plurality of parallel bars enables a rope section to extend over and under the bars in a small angle of wrap to enable a small change in rope pulled resistance, in an arrangement that makes it easy to understand how to change rope pulling resistance. A handle 36 (FIG. 6) allows easy adjustment of rope length.

Abstract: The motions of mountain climbing and running are simulated by an exerciser having a pair of movable treadles which are connected to ropes that extend around plural pulleys and having free ends provided with handles which are gripped by the user and pulled during movement of the treadles to thereby exercise both the arms and legs. The treadles are connected to hydraulic cylinders which permit the treadles to be utilized independently of the ropes.

Abstract: An exercise apparatus which may be utilized to perform a variety of exercise routines and provides a frictional force for resisting the exercise movements of the user. The preferred apparatus has a modular construction which permits assembly of the apparatus in a variety of configurations for performing the various exercise routines. The apparatus may also be disassembled for storage, ease of transportation, and marketing of the apparatus in kit form. A friction rope brake mechanism for the exercise apparatus and other uses.

Abstract: A wrist exercizing device comprises a pair of generally coaxial handles for grasping in the hands and turning, and a reel mounted therebetween. The reel has a reel surface that is eccentric with respect to the axis of rotation of the handles. A cord is wrapped round the reel surface and supports a weight. Rotation of the handles to raise or lower the weight requires a periodically varying effort. Suitably the reel surface has an axis of rotational symmetry that is offset from the axis of rotation of the handles to provide the eccentricity. The degree of offset may be varied so as to vary the ratio of maximum to minimum effort required to raise the load.

Abstract: A combination stationary bicycle and step/stair climber exercise device containing a support surface, at least one exercise biking mechanism mounted to the support surface, at least one stepping mechanism mounted to the support surface, at least one pair of pedals connected to the biking mechanism, at least one pair of steps connected to the stepping mechanism, at least one seat connected to the top of the support surface over the biking mechanism, and at least one pair of handle bars mounted to the support surface for use with at least one of the biking mechanisms and the stepping mechanisms.

Abstract: A device is disclosed for performing a quantitative analysis of a patient during exercise against isometric, isokinetic, or isoinertial resistance, in order to construct a detailed musculoskeletal model of a human subject. To use the device, the subject grasps a handle which is attached to a length of cable, wound upon a cable spool. Adjustable braking means attached to the cable spool supplies isometric, isokinetic, or isoinertial resistance against unwinding of the cable in order to simulate selected lifting motions. Sensing devices detect the longitudinal and lateral angles at which the patient is pulling the cable. A torsion spring is attached to the cable spool for rewinding the cable. In the preferred version, force sensors are attached to an upper surface of the platform to detect the manner in which the subject's weight is distributed between the anterior and posterior regions of the subject's feet, as well as between the subject's right and left feet.

Abstract: An adjustable resistance upper body exerciser includes a generally arcuate belt (2) encircling the waist of a user and having a generally flat, rigid posterior central portion 3 and flexible anterior portions (4, 4a) including buckling means (5). A flexible inelastic cord (7) having a length and left and right ends (8, 8a) passes through an elongated guide means (12) attached to the belt, slidably retaining a central portion of the length the cord and leaving the right and left ends of the cord extending in the anterior direction. A snubber (15) is attached to the rigid posterior portion of the belt, contacting the slidable cord and applying an adjustable friction load to the slidably retained cord. A pair of handles (18, 18a) are attached to left and right ends of the cord, whereby a user wearing the belt encircling the waist may alternately pull said left and right handles with a tension force generally proportional to the friction load applied to the cord by the snubber.

Abstract: A user-manipulated modular exercise machine has individually operable and simultaneously adjustable right and left-hand reel assemblies. Each reel assembly includes a reel, a pull-cord wound on the reel, and a spirally wound spring which applies to the reel a reactive torque of changing magnitude as the reel rotates in response to pulling input force applied to the pull-cord. Each reel assembly has a compensating mechanism for nullifying changes in the magnitude of the reactive torque as the reel rotates and which includes a prewound spiral spring and a belt and cam mechanism for connecting the prewound spring to the reel.

Abstract: A simple type pedaling exerciser which includes two arched, symmetrical side supports connected by a pivot rising behind a transverse channel bar, two foot pedals pivoted on the pivot, an intermediate driving wheel and two opposite side driven wheels arranged inside the transverse channel bar in a line, a steel cable wound around the driving wheel with two opposite ends extended through the driven wheels in reverse directions and inserted through holes on the transverse channel bar and connected to the foot pedals respectively. The mid-portion of the steel cable is fixed to the driving wheel so that pedaling either foot pedal downwards causes the other foot pedal to be rotated upwards.

Abstract: An exercise device for enhanced aerobic walking that coordinates the motion of the arms and legs. The device is worn around the torso by means of a belt mounted yoke which has an extension pole extending downwardly and rearwardly. Both the length of the extension pole and the angle of the extension pole to the yoke can be adjusted. Exercise cords are threaded through guides and brake-lock devices located on the sides of the yoke and a guide loop at the end of the extension pole. At each end of the exercise cord are cuffs, which are attached to the exerciser's extremities. Several threading methods are available for coordinated exercise: left arm to right leg and right arm to left leg; left arm to left leg and right arm to right leg; the left and right arm; and the left and right leg. The brakes can be used to increase the resistance and the locks can be used to fix the exercise cord in a position and use the elasticity of the cord for maximum resistance.

Abstract: An isoinertial lift machine is described which can be used to evaluate and train patients in the static and dynamic lift modes. The machine consists of an arm which can be positioned on a vertical column. At the end of the arm, various size attachments and handles can be attached. The arm also carries a force sensor that measures the forces applied to the handles. In the static mode, the arm is positioned at a desired height and locked in place. The patient lifts on the handle, and the force sensor registers the lifting force. Since the handle does not move during the lift, the mode is called static. In the dynamic mode, the arm/handle moves in the vertical direction during the lift. A resistance mechanism is used to resist the lifting force applied by the user at the handle. The resisting force is always pulling the arm down, both in the lifting and the lowering motion. The resisting force remains constant throughout the range of motion during lifting.

Abstract: A resistance reel for an exerciser includes an extractive band extractible against an adjustable force of resistance and rewound by a return spring. A central pin has mounted thereon an extraction section and a resistance section. The components of such sections that are rotatable relative to the central pin are connected by a clutch means for non-rotatable connection in an extraction direction and are freewheeling for rewinding of the extractive band. One end of the central pin is intended for non-rotatable attachment to a stationary point.