Force feedback mechanism for analog simulation equipment

Abstract

The invention relates to the technical field of simulation operation training equipment, in particular to a force feedback mechanism for analog simulation equipment, which comprises a front-back state feedback assembly, a left-right state feedback assembly and a containing sleeve mounted on a support through a pitch shaft, and the pitch shaft is connected with a pitch state feedback assembly; the operating mirror is inserted into the containing sleeve and moves forwards or backwards, and the moving distance and the rotating angle of the operating mirror are detected through the laser sensor; the operation mirror and the containing sleeve integrally rotate in a pitching mode, the containing sleeve and the pitching state feedback assembly integrally rotate left and right, and the pitching rotation angle and the left-right rotation angle are detected through a potentiometer. The force feedback mechanism provided by the invention has tactile feedback of four degrees of freedom and three degrees of freedom response, obtains the in-out position, the rotation angle and the corresponding speed of the operation mirror, cooperates with software 3D modeling scenes, simulates the position and the posture of an instrument in a human body 3D model, and controls a motor to output different torques through an algorithm, so that the simulated operation is closer to the real hand feeling.

Description

Technical Field

[0001] The invention relates to the technical field of simulated surgery training equipment, and in particular to a force feedback mechanism for simulating a simulation equipment. Background Art

[0002] Medical training, and particularly surgical training, involves a great deal of hands-on learning, allowing students, as well as experienced physicians learning new skills, to practice and develop the skills they need in a hands-on and immersive training environment to become proficient in medical and surgical procedures.

[0003] In recent years, hysteroscopic surgery technology and related equipment have made great progress. At the same time, the demand for hysteroscopic surgery medical workers has increased year by year. This requires training more qualified hysteroscopic surgery medical workers in a short period of time. In response to this situation, the hysteroscopic surgery simulation system came into being.

[0004] Generally, hysteroscopic surgery simulation systems require force feedback systems to create a virtual surgical environment as realistically as possible, which requires the force feedback system to be as simple and compact as possible and have sufficient flexibility and a large range of motion. Existing hysteroscopic surgery virtual simulation training systems mostly use a simple series structure, which is limited by the mechanical structure and has a small actual range of motion. If a torque motor or other driving mechanism is added to each joint, the volume and weight of the mechanism will increase. The increase in mass will further reduce the operator's force perception, thereby affecting the training effect.

[0005] Based on the technical problems existing in the prior art, the present invention provides a force feedback mechanism for simulating a simulation device. Summary of the invention

[0006] The purpose of the present invention is to provide a force feedback mechanism for simulating simulation equipment to solve the technical problems in the prior art that the simulation force feedback system is heavy and bulky, making it difficult to obtain a good clinical training experience similar to that of real equipment, thus affecting the training effect.

[0007] The technical solution of the present invention is: a force feedback mechanism for simulating a simulation device, comprising a bearing sleeve, the bearing sleeve is mounted on a bracket through a pitch axis, and a pitch state feedback component is connected to the pitch axis;

[0008] The operating mirror is inserted into the carrying sleeve along the length direction. Under the action of external force, the operating mirror moves forward or backward in the carrying sleeve, and the front and rear position or posture of the operating mirror is detected by the front and rear state feedback component assembled at the rear of the carrying sleeve, and force feedback is performed under control; under the action of external force, the operating mirror and the carrying sleeve rotate as a whole relative to the pitch axis, and the pitch position or posture of the operating mirror is detected by the pitch state feedback component, and force feedback is performed under control; the bracket drives the carrying sleeve and the pitch state feedback component to rotate as a whole, and the left and right position or posture of the operating mirror is detected by the left and right state feedback component, and force feedback is performed under control.

[0009] Preferably, the pitch axis clamp is on the carrying sleeve, and the axis of the pitch axis is orthogonal to the axis of the carrying sleeve; the carrying sleeve comprises an inner core tube and an outer core tube, an oblique transmission assembly is provided on the side wall of the inner core tube, and the inner core tube is hollow and straight to form an operating channel; the friction force between the inner wall of the operating channel and the operating mirror is transmitted through the oblique transmission assembly to drive the inner core tube to rotate, and a potentiometer is connected to the pitch axis, and the potentiometer is fixed on the side panel of the bracket, and is used to detect the rotation angles of the operating mirror left and right and pitch rotation.

[0010] Preferably, a accommodating groove is provided on the side wall of the inner core barrel, and the oblique transmission assembly includes a rotating wheel, which is assembled in the accommodating groove through a pin shaft, and the outer rim of the rotating wheel protrudes from the inner wall surface and the outer wall surface of the inner core barrel; wherein, the angles and postures of the assembly of several rotating wheels on the same busbar direction of the inner core barrel are the same.

[0011] Preferably, the pitch axis includes a snap-on axis and a connecting axis, the snap-on axis is arranged in a C shape, and the two arms of the C-shaped opening are connected to the connecting axis on both sides, and the straight lines where the connecting axis on both sides are located are orthogonal to the axis of the load-bearing sleeve; the connecting axis on both sides are connected to a pair of side plates of the bracket through bearings.

[0012] Preferably, a photoelectric sensor and a laser sensor are provided at the entrance end of the containing sleeve for inserting the operating mirror, the photoelectric sensor is used to detect whether the operating mirror is inserted, and the laser sensor is used to obtain the rotation angle and movement distance of the operating mirror.

[0013] Preferably, the front and rear state feedback assembly adopts a combination of a first motor, a first synchronous wheel and a first synchronous belt;

[0014] The second synchronous wheel is coaxially connected to the inner core tube, and the rotation of the inner core tube drives the first synchronous wheel and the first synchronous belt to run; wherein, when the first motor applies voltage, it simulates output resistance to the operating mirror.

[0015] Preferably, the pitch state feedback assembly adopts a set of third motors, bevel gears and motor gears meshing with the bevel gears, and the third motor is fixed to a side plate of one side of the bracket;

[0016] The bevel gear is mounted on the pitch axis and can follow the bearing sleeve and the pitch axis to rotate synchronously in pitch as a whole. Furthermore, when the third motor applies voltage, it simulates output resistance to the operating mirror.

[0017] Preferably, the left and right state feedback components are arranged at the bottom of the bottom plate of the bracket, and the left and right state feedback components adopt a combination of a second motor, a second synchronous wheel and a second synchronous belt;

[0018] The second synchronous wheel is connected to the base plate via a fixed shaft, the bracket and the bearing sleeve on the bracket rotate synchronously as a whole, and when the second motor applies voltage, it simulates output resistance to the operating mirror.

[0019] Preferably, after the operating mirror is inserted into the containing sleeve, it rotates left, right, up, and down, and the rotation angles γ and β are obtained respectively by the potentiometer, and the moving distance L of the operating mirror is obtained by the laser sensor. The rotation angle and force feedback are calculated as follows:

[0020] The position of the spatial coordinate point at the end of the operating mirror is (L cosβcosγ, L cosβsinγ, L sinβ);

[0021] Force feedback determines the direction and magnitude of force applied to a point based on its position in space, i.e., its position in the 3D scene. The in-and-out resistance F is the component force of the rotating wheel rolling through the pin:

[0022]

[0023] Wherein, T is the torque of the first motor, Z1 is the diameter of the upper wheel body of the first synchronous wheel, Z2 is the diameter of the lower wheel body of the first synchronous wheel, r is the radius of the rotating wheel, and φ is the installation angle of the rotating wheel deviating from the generatrix.

[0024] Compared with the prior art, the advantages of the present invention are:

[0025] The present invention sets a carrying sleeve, sets a front and rear state feedback component, a left and right state feedback component and a pitch feedback component on the side of the carrying sleeve, inserts the operating mirror into the carrying sleeve, moves forward or backward, and detects the position and movement distance of the operating mirror through a laser sensor; the operating mirror and the carrying sleeve rotate in pitch as a whole, the bracket drives the carrying sleeve and the pitch state feedback component to rotate left and right as a whole, and detects the pitch rotation angle and left and right rotation angle through a potentiometer. The force feedback mechanism provided by the present invention has tactile feedback with 4 degrees of freedom and 3 degrees of freedom response, obtains the in and out position, rotation angle and corresponding speed of the operating mirror, cooperates with the software 3D modeling scene, simulates the position and posture of the instrument in the 3D model of the human body, controls the motor to distribute different force values, and makes the simulated surgery closer to the real feel. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0027] Figure 1 is a three-dimensional schematic diagram of the force feedback mechanism of the present invention;

[0028] Figure 2 An exploded schematic diagram of the force feedback mechanism of the present invention;

[0029] Figure 3 It is a schematic diagram of the partial explosion connection of the containment sleeve of the present invention;

[0030] Figure 4 It is a schematic diagram of the connection between the carrying sleeve and the pitch state feedback assembly on the bracket according to the present invention;

[0031] Figure 5 It is a structural schematic diagram of the inner core tube of the present invention;

[0032] Figure 6 It is a schematic diagram of assembling the rotating wheel on the inner core cylinder of the present invention;

[0033] Among them: 1. Load-bearing sleeve; 2. Pitch axis; 3. Bracket; 4. Pitch state feedback component; 5. Front and rear state feedback component; 6. Left and right state feedback component; 7. Bearing;

[0034] 11. Inner core barrel; 12. Outer sleeve; 13. Oblique transmission assembly; 14. Photoelectric sensor; 15. Laser sensor; 16. Port seat; 21. Buckle shaft; 22. Connecting shaft; 31. Side plate; 32. Bottom plate; 33. Fixed shaft; 41. Bevel gear; 42. Motor gear; 43. Third motor;

[0035] 100, operation channel; 101, receiving groove; 131, rotating wheel; 132, pin shaft;

[0036] 201, front end cover; 202, rear end cover; 203, potentiometer; 501, first synchronous wheel; 502, first synchronous belt; 601, second synchronous wheel, 602, second synchronous belt. DETAILED DESCRIPTION

[0037] The present invention is further described in detail below in conjunction with specific embodiments:

[0038] like Figure 1 As shown, an embodiment of the present invention provides a force feedback mechanism for simulating a simulation device, comprising a containing sleeve 1, which is mounted on a bracket 3 via a pitch axis 2. The operating mirror is pluggable, and a front end cover 201 is provided at the front entrance side of the containing sleeve 1, and a rear end cover 202 is provided at the rear end to protect the containing sleeve 1.

[0039] See attached Figure 2 The bracket 3 of this embodiment includes a pair of side plates 31 and a bottom plate 32. The side plates 31 are placed vertically and parallel to each other. The carrying sleeve 1 is placed between the two side plates 31. The two ends of the pitch axis 2 are vertically assembled on the side plates 31 through bearings 7. The pitch state feedback component 4 is connected to the pitch axis 2.

[0040] The containing sleeve 1 comprises an inner core sleeve 11 and an outer sleeve 12. The inner core sleeve 11 is hollow and straight to form an operating channel 100 for inserting and moving the operating mirror.

[0041] A photoelectric sensor 14 and a laser sensor 15 are installed at the inlet end of the containing sleeve 1 through the port seat 16. Figure 3 As shown, the photoelectric sensor 14 is arranged at the top of the entrance end, and is used to detect whether the operating mirror is inserted and reset the value of the laser sensor 15. The laser sensor 15 obtains the rotation angle of the operating mirror, the in and out value (moving distance) during the movement, and the rotation angle value, so that the forward distance L and the rotation amount α of the operating mirror can be obtained.

[0042] An oblique transmission assembly 13 is provided on the side wall of the inner core tube 11. During operation, under the action of external force, the operating mirror moves forward or backward in the carrying sleeve 1, and the friction force between the inner wall of the operating channel 100 and the operating mirror is transmitted through the oblique transmission assembly 13 to drive the inner core tube 11 to rotate, and the front and rear state feedback assembly 5 assembled at the rear of the carrying sleeve 1 detects the force feedback state of the operating mirror.

[0043] The side wall of the inner core barrel 11 is provided with a receiving groove 101, and the oblique transmission assembly 13 includes a rotating wheel 131, and the rotating wheel 131 is assembled in the receiving groove 101 through a pin shaft 132. Figure 6The outer rim of the rotating wheel 131 protrudes from the inner and outer wall surfaces of the inner core tube 11; the operating mirror enters the operating channel 100 and contacts the rotating wheel 131, generating mutual friction. Under the action of the friction force, the inner core tube 11 rotates, driving the front and rear state feedback component 5 of the tail to operate for detection.

[0044] Specifically, in this embodiment, the front and rear state feedback assembly 5 adopts a combination of a first motor, a first synchronous wheel and a first synchronous belt, and the second synchronous wheel is coaxially connected to the inner core barrel 11.

[0045] When the operator uses the operating mirror, the operating mirror rotates in the operating channel 100 of the inner core tube 11, and the rotation of the inner core tube 11 drives the first synchronous wheel 501 and the first synchronous belt 502 to run. At this time, if the first motor applies voltage, a driving force or resistance is generated to simulate the actual resistance generated during movement or surgery, and force feedback is performed. The moving distance of the operating mirror is obtained by calculation. According to the 3D modeling scene in the hysteroscopic surgery, the first motor can be controlled to distribute different force values ​​according to different situations or different positions, so that the simulated surgery is closer to the real feel.

[0046] Among them, refer to the attached Figure 5 As shown, the plurality of rotating wheels 131 on the same busbar direction of the inner core tube 11 are assembled at the same angle and posture, and the rotating wheels 131 arranged on adjacent busbars have the same deflection angle and posture.

[0047] The inclination angle of the rotating wheel 131 relative to the busbar is within the range of 45 degrees. The angle is related to the resistance. The larger the angle, the greater the resistance. If the angle is too large, the rotation is not smooth and is prone to jamming. Moreover, within the set angle range of the rotating wheel 131, when moving the same distance, the greater the number of motor rotations, the greater the resistance.

[0048] The pitch axis 2 includes a buckle shaft 21 and a connecting shaft 22, and the connecting shafts 22 on both sides are connected to a pair of side plates 31 of the bracket 3 through bearings 7. The buckle shaft 21 is set in a C shape, and the two arms of the C-shaped opening are connected to the connecting shafts 22 on both sides, the C-shaped opening faces downward, and the clamp is connected to the load-bearing sleeve 1. The straight line where the connecting shafts 22 on both sides are located is orthogonal to the axis of the load-bearing sleeve 1, ensuring that the center hole of the pitch axis 2 and the center axis of the load-bearing sleeve 1 are in the same horizontal plane to overcome the torque generated by the deadweight.

[0049] In this embodiment, the pitch state feedback assembly 4 uses a set of third motors 43, bevel gears 41 and motor gears 42 meshing with the bevel gears 41. The third motor 43 is fixed on a side plate 31 of the bracket 3, and the bevel gears 41 are installed on the pitch axis 2. Figure 4 as shown in .

[0050] When the operator uses the operating mirror, the operating mirror rotates in the operating channel 100 of the inner core tube 11, and the operating mirror and the carrying sleeve 1 rotate relative to the pitch axis 2 in the vertical plane, driving the pitch axis 2 to rotate synchronously. The potentiometer 203 installed on the pitch axis 2 is fixed in position. When the pitch rotates, the resistance changes and the rotation angle is detected. Compared with the encoder, no reset action is required, which has the effect of reducing cost and increasing gain.

[0051] At this time, if the third motor applies voltage, it generates a driving force or resistance, driving the bevel gear 41 and the motor gear 42 to rotate, simulating the actual resistance generated during movement or surgery, and realizing force feedback. According to the 3D modeling scene in hysteroscopic surgery, the third motor can be controlled to distribute different force values ​​according to different situations or different positions, making the simulated surgery closer to the real feel.

[0052] Under the action of external force, the operating mirror is in the containing sleeve 1, and the bracket 3 drives the containing sleeve 1 and the pitch state feedback component 4 to rotate as a whole, and detects the left and right rotation angles of the operating mirror through the left and right state feedback component 6.

[0053] The left and right state feedback component 6 is arranged at the bottom of the base plate 32 of the bracket 3. The left and right state feedback component 6 adopts a combination of a second motor, a second synchronous wheel and a second synchronous belt; the second synchronous wheel is connected to the base plate 32 through a fixed shaft 33, and the bracket 3 and the load-bearing sleeve 1 on the bracket 3 rotate synchronously as a whole, and the potentiometer 203 detects the rotation angle.

[0054] At this time, if the second motor applies voltage, it will generate driving force or resistance; it will simulate the actual resistance generated during movement or surgery, and perform force feedback. According to the 3D modeling scene in hysteroscopic surgery, it can control the second motor to allocate different force values ​​according to different situations or different positions, making the simulated surgery closer to the real feel.

[0055] After the operating mirror is inserted into the containing sleeve 1, it can rotate left and right, as well as up and down. The rotation angle is passed through a set of gear sets, so that the potentiometer 203 obtains a larger rotation angle, so as to improve the detection accuracy through the amplification effect. The rotation angles γ and β can be obtained respectively through the potentiometer 203, and the moving distance L of the operating mirror is obtained through the laser sensor 15. The rotation angle and force feedback calculation method are as follows:

[0056] The position of the spatial coordinate point at the end of the operating mirror is (L cosβcosγ, L cosβsinγ, L sinβ);

[0057] Force feedback determines the direction and magnitude of force required to be applied to a point based on its posture in space, that is, its position in the 3D scene.

[0058] The resistance force F of the in and out movement is the component force of the rolling friction of the rotating wheel 131 through the pin 132:

[0059]

[0060] Wherein, T is the torque of the first motor, Z1 is the diameter of the upper wheel body of the first synchronous wheel, Z2 is the diameter of the lower wheel body of the first synchronous wheel, r is the radius of the rotating wheel 131, and φ is the installation angle of the rotating wheel 131 deviating from the generatrix.

[0061] The force feedback mechanism provided by the present invention is the core component of the hysteroscopic surgery simulation device. When in combined motion, it can be regarded as the synthetic motion of the pitch state feedback component 4, the front-back state feedback component 5, and the left-right state feedback component 6. The spatial posture and position of the end of the operating mirror are obtained through the potentiometer 203 and the laser sensor 15.

[0062] This force feedback mechanism has tactile feedback with 4 degrees of freedom and 3 degrees of freedom response. It can detect the entry and exit position, rotation angle and corresponding speed, and the spatial position of the end of the instrument when the operator uses the resectoscope or operating mirror for simulated training. It cooperates with the 3D modeling scene of the hysteroscopic surgery process constructed by software to simulate the position and posture of the operating mirror in the human body during hysteroscopic surgery and distribute different force values ​​by controlling the motor to simulate the effect of the operating mirror touching the inner wall of the tissue, making the simulated surgery closer to the real feel.

[0063] The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with this technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the scope of protection of the present invention. For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and the present invention can be implemented in other specific forms without departing from the spirit or basic features of the present invention. Therefore, no matter from which point of view, the embodiments should be regarded as exemplary and non-restrictive. The scope of the present invention is limited by the attached claims rather than the above description, and it is intended to include all changes within the meaning and scope of the equivalent elements of the claims in the present invention.

Claims

1. A force feedback mechanism for simulating a simulation device, characterized in that: It comprises a containing sleeve (1), the containing sleeve (1) is mounted on a bracket (3) via a pitch axis (2), and a pitch state feedback component (4) is connected to the pitch axis (2); The operating mirror is inserted into the containing sleeve (1) along the length direction. Under the action of external force, the operating mirror moves forward or backward in the containing sleeve (1), and the front and rear position or posture of the operating mirror is detected by a front and rear state feedback component (5) assembled at the rear of the containing sleeve (1), and force feedback is performed in a controlled manner; under the action of external force, the operating mirror and the containing sleeve (1) rotate as a whole relative to the pitch axis (2), and the pitch position or posture of the operating mirror is detected by the pitch state feedback component (4), and force feedback is performed in a controlled manner; the bracket (3) drives the containing sleeve (1) and the pitch state feedback component (4) to rotate as a whole, and the left and right position or posture of the operating mirror is detected by the left and right state feedback component (6), and force feedback is performed in a controlled manner.

2. A force feedback mechanism for a simulation device according to claim 1, characterized in that: The pitch axis (2) is clamped on the containing sleeve (1), and the axis of the pitch axis (2) is orthogonal to the axis of the containing sleeve (1); The carrying sleeve (1) comprises an inner core sleeve (11) and an outer sleeve (12), and an oblique transmission assembly (13) is arranged on the side wall of the inner core sleeve (11). The inner core sleeve (11) is hollow and straight to form an operating channel (100); the friction force between the inner wall of the operating channel (100) and the operating mirror is transmitted through the oblique transmission assembly (13) to drive the inner core sleeve (11) to rotate, and a potentiometer (203) is connected to the pitch axis (2), and the potentiometer (203) is fixed on the side plate (31) of the bracket (3) and is used to detect the rotation angle of the operating mirror in left and right rotation and pitch rotation.

3. A force feedback mechanism for a simulation device according to claim 2, characterized in that: A receiving groove (101) is provided on the side wall of the inner core barrel (11), and the oblique transmission assembly (13) comprises a rotating wheel (131), the rotating wheel is assembled in the receiving groove (101) via a pin shaft (132), and the outer wheel edge of the rotating wheel (131) protrudes from the inner wall surface and the outer wall surface of the inner core barrel (11); Wherein, the plurality of rotating wheels (131) on the same generatrix direction of the inner core barrel (11) are assembled at the same angle and posture.

4. A force feedback mechanism for a simulation device according to claim 2, characterized in that: The pitch axis (2) comprises a buckle shaft (21) and a connecting shaft (22); the buckle shaft (21) is arranged in a C shape, and two arms of the C-shaped opening are connected to the connecting shafts (22) on both sides; the straight lines where the connecting shafts (22) on both sides are located are orthogonal to the axis of the carrying sleeve (1); the connecting shafts (22) on both sides are connected to a pair of side plates (31) of the bracket (3) through bearings (7).

5. The force feedback mechanism for simulation equipment according to claim 2, characterized in that: A photoelectric sensor (14) and a laser sensor (15) are arranged at the entrance end of the containing sleeve (1) for inserting the operating mirror. The photoelectric sensor (14) is used to detect whether the operating mirror is inserted, and the laser sensor (15) is used to obtain the rotation angle and movement distance of the operating mirror.

6. The force feedback mechanism for simulation equipment according to claim 3, characterized in that: The front and rear state feedback component (5) adopts a combination of a first motor, a first synchronous wheel and a first synchronous belt; The second synchronous wheel is coaxially connected to the inner core barrel (11), and the rotation of the inner core barrel (11) drives the first synchronous wheel and the first synchronous belt to operate; wherein, when the first motor applies voltage, it simulates output resistance to the operating mirror.

7. A force feedback mechanism for a simulation device according to claim 2, characterized in that: The pitch state feedback component (4) adopts a set of third motors (43), bevel gears (41) and motor gears (42) meshing with the bevel gears (41), and the third motors (43) are fixed on a side plate (31) of the bracket (3); The bevel gear (41) is mounted on the pitch shaft (2) and is capable of following the integral synchronous pitch rotation of the carrying sleeve (1) and the pitch shaft (2); and when the third motor applies voltage, it simulates output resistance to the operating mirror.

8. The force feedback mechanism for simulation equipment according to claim 2, characterized in that: The left-right state feedback component (6) is arranged at the bottom of the bottom plate (32) of the bracket (3), and the left-right state feedback component (6) adopts a combination of a second motor, a second synchronous wheel and a second synchronous belt; The second synchronous wheel is connected to the base plate (32) via a fixed shaft (33); the bracket (3) and the carrying sleeve (1) on the bracket (3) rotate synchronously as a whole; and when a voltage is applied to the second motor, a simulated output resistance is applied to the operating mirror.

9. The force feedback mechanism for simulation equipment according to claim 5, characterized in that: After the operating mirror is inserted into the containing sleeve (1), it rotates left and right as well as up and down. The rotation angles γ and β are obtained respectively through the potentiometer (203). The moving distance L of the operating mirror is obtained through the laser sensor (15). The rotation angle and force feedback are calculated as follows: The position of the spatial coordinate point at the end of the operating mirror is (L cosβcosγ, L cosβsinγ, L sinβ); The force feedback determines the direction and magnitude of the force to be applied to the point according to the point's posture in space, that is, the point's position in the 3D scene. The in-and-out resistance F is the component force of the rolling friction of the rotating wheel (131) through the pin (132): Wherein, T is the torque of the first motor, Z1 is the diameter of the upper wheel body of the first synchronous wheel, Z2 is the diameter of the lower wheel body of the first synchronous wheel, r is the radius of the rotating wheel (131), and φ is the installation angle of the rotating wheel (131) deviating from the generatrix.

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