Description

Field of the art

The present invention refers to a machine that can be used both by an artisan (plumber, etc.) and at industrial level, for bending metal pipes. In particular, different from the machines of the prior art, the machine subject of the present invention allows obtaining - on only one machine - bends of any type in the three-dimensional space, without having to pass from one station to the other or from one machine to the other according to the type of bending to be imparted to the metal pipe.

Prior art and the disadvantages thereof

Known machines comprise an external casing/body which houses and protects a series of reduction gears. The drive motor, installed within the external casing or body of the machine initially has an angular speed of the output shaft thereof which is generally in the order of 1400 revolutions per minute. This speed of rotation is progressively reduced by the reduction gears up to obtaining, on the operating shaft, (the one that provides the torque required for bending the pipe) a mechanical torque of about 1400-1600 Nm (purely indicative value) and a number of revolutions per minute of 1.4 (a thousand times lower than the initial value on the output shaft of the drive motor). Obviously, according to the known formula which provides the power, the product M · ω would be constant were the losses to be ignored. The number of revolutions (also for the machine of the present invention) is purely indicative and may vary widely, for example even in the order of 10 or more. In any case this depends on the dimensions of the machine and the pipes/sections intended to be bent. Similar arguments can also be raised regarding the value of the torque provided by the output shaft, which also depends on the particular application (size of the pipes or sections and the dimensions of the machine) or other parameters, for example the resistance of the pipes/sections.

The operating shaft which provides the power required for bending the metal pipe is made integral to the matrix (substantially a wheel with peripheral groove which receives part of the pipe). The operating shaft projects from the external body of the machine, vertically upwards, perpendicularly to the upper metal plate of the external body, and it for example bears an end with square-shaped head (or hexagonal-shaped head, etc.) to be introduced into a central hole of the matrix, with shape complementary and concentric to the geometric axes of the matrix itself (as well as the operating shaft after mounting). In this manner, the matrix is driven in rotation by the operating shaft. The pipe to be bent is locked on a bracket integral with the matrix (which cooperates with a counter-matrix), and the bracket drives the pipe therealong bending it. This system is generally known.

The counter-matrix is driven by an opposition screw so that the pipe moves away from the groove.

Other systems for bending pipes are known in the art but they do not regard the present invention. For example, in the so-called "thrust" system the pipe tends to flatten on the external side during the bending given that the drawing of the material cannot be controlled, given the absence of an opposition element like the counter-matrix which "copies" the second half of the pipe (the first is "copied by the matrix").

Thus, in the art regarding the present invention, the function of the counter-matrix is actually that of reducing the lateral deformations of the pipe. The pipe obtained through this process is usually much more rigid in the bending area of the pipe with respect to the straight area of the pipe, given that it has been denied of a part of the extension thereof. Thus, the fact that the pipe bending method, by utilizing the matrix and the counter-matrix, is preferred to other alternative methods, such as the thrust technique in which there is substantially just one bending/flexure of the material without drawing the same, is also due to this reason.

In another method, there is used a core within the pipe which prevents the pipe from collapsing inwards, maintaining the internal diameter of the pipe unvaried in the critical area immediately after the point of tangency in which the curve starts to form. The internal cores have been made in various forms such as for example adjustable and they are suitably held so as not to be driven therealong by the pipe during the bending, specifically by the great forces involved in the bending process. However, the method used by the present invention differs also from the latter prior art given that it does not use any internal core.

In other words, the present invention exclusively regards a machine in which the bending is obtained simply through a matrix and a counter-matrix, without the use of a core within the metal pipe; in order to avoid confusion with other known methods, such method shall hereinafter be referred to as "inherent prior art".

A first problem of the machines of this "inherent" prior art lies in the fact that they are little versatile given that they do not allow to easily vary the type of matrix using the same machine. It would be desirable to be able to utilize the same machine, and thus only one power take-off of the operating shaft, making the machine adaptable to various types of matrix and counter-matrix, also for example of the previously known type and manufactured by various specialized companies. In this manner, the machine would be directly reconfigurable according to the various types of the prior art machines.

A second problem of the inherent prior art lies in the fact that the respective metal pipes bending machines are little versatile, given that they can bend the pipe only to the right or only to the left. As of date there are machines which perform the bending of the pipe both to the right and to the left. Thus, currently, during an operation of multiple bending of a pipe (on only one plane or even in three dimensions) it should be transferred manually from a right bending machine to a left bending machine, or vice versa, even several times. Objectively even the costs deriving from the purchase of two different machines (or with several operating stations) represent an additional cost, especially, but not only, for a simple artisan (for example a plumber). The latter shall be forced, for example after taking on site the measurements of the various bends and straight portions the future pipe to be installed in the system in question is meant to have, to return to the workshop to concretely obtain concretely the metal pipe with all the bends and straight portions thereof. It would thus be desirable to have only one machine capable of bending a pipe both to the right and to the left, on one plane and in 3D ("crooked bends"). In this case, besides reducing the costs for a company that may eventually purchase the machine in question, this would also bring benefits also to the artisan or installer of domestic appliances. Actually, a smaller machine, could be installed in a van and thus actually become a mobile workshop capable of bending the pipe even "in situ".

Actually, a machine capable of bending both to the right and to the left becomes fundamental when the bends are made in three dimensions, given that at some point, after some bends, the operator could be faced with the impossibility to obtain a further bend given that the machine would physically obstruct such operation (the pipe would impact against the machine).

Thus, generally the aim of the present invention is to provide a machine more versatile than that of the inherent prior art, that is adjustable/modulable as regards the type of matrices and counter-matrices that can be used, and which can also bend a pipe even in a multiple manner and in 3D ("crooked bends") without having to change the machine or the operating station.

Description of the invention

The objects of the invention are attained by providing a metal pipes bending machine having different characteristics. A first characteristic is that of obtaining an operating shaft of the "removable" and "interchangeable" type. Substantially, this operating shaft may be literally "dropped" vertically through the horizontal plate (operating plane) of the external body, in a vertical hollow shaft located inside - the external body - of the machine. The operating shaft has - at the lower part - a first male part - bearing a key for the rotatable driving - which is coupled with the vertical hollow shaft, and - at the upper part - a second male part - bearing a profile adaptable to the matrix (for example a square or hexagonal head), which is coupled with the matrix, for example with a complementary through hole obtained in the matrix. This would allow providing a machine adaptable to various types of processes, to various matrices and counter-matrices. Between the first male part and the second male part of the interchangeable operating shaft according to the present invention, preferably there is an abutment flange which, interacting with a counter-abutment, defines the vertical height of the top part of the operating shaft.

Still according to the present invention, the motorization comprised in the external body of the machine, i.e. the reduction gear with cascade of gears, shall be invertible. Furthermore, it shall preferably have - on the power take-off - the same intensity of the mechanical torque, both in one direction and the other direction of rotation of the operating shaft. Thus, the power take-off, constituted by the hollow shaft or by the operating shaft, shall develop the same mechanical torque both in one direction and in the other, so as to be able to equally bend a pipe both to the right and to the left. Changing the direction of the bending operation performed on the metal pipe, from right to left, and vice versa, requires tilting the matrix and connecting it again to the power take-off (operating shaft). This is not difficult given that the type of connection is perfectly symmetric both on one side and on the other side of the matrix (preferably the matrix has, as previously mentioned, a through hole, complementary to the shape of the upper surface of the second male part of the operating shaft).

Should the various bends to be made on the metal pipe are not all on the same plane, then it is required from time to time, between "one bend and the other", to invert the direction of rotation of the power take-off i.e. of the matrix (by actuating a switch present on one side of the machine) after removing the pipe from the machine, tilting the matrix, and repositioning the pipe in the demountable pipe driving bracket (U-bolt). Actually, contrary to the inherent prior art, wherein the driving bracket forms a single piece with the matrix, according to the present invention the driving bracket is also demountable still for allowing demounting and re-mounting the metal pipe in the driving bracket or on the matrix. Like in the inherent prior art, the counter-matrix is rotatably mounted around a vertical rotation axis integral to a movable support (linearly) towards the matrix. The counter-matrix is moved adjacent to the matrix with a positioning and quick adjustment system, operating on the movable i.e. mobile support of the counter-matrix (for example a toggle system). The counter-matrix may rotate and thus follow (within given limits, given that there is a limit angle of rotation of the counter-matrix which can be adjusted through an abutment pin against the mobile support) the movement/rotation of a part of pipe opposite with respect to the part of pipe that is bent. According to the present invention, even the counter-matrix is obviously repositioned on the mobile support upon passing from the right bending operation towards the left bending operation, and vice versa.

Thus, according to the present invention, repositioning the parts interested by the bending operation, i.e. the driving bracket, the matrix (tilting of the matrix) and the counter-matrix (repositioning on the mobile support), it is possible to pass from the right bending operation towards the left bending operation and vice versa.

According to the present invention, the machine also allows an "under head rotation"; this means that a part (26) of the piece (metal pipe) may pass under the head (25) of the machine (head where the bending group with matrix and counter-matrix is mounted), given that such head (25) for supporting the bending group, according to a preferred embodiment of the invention is mounted cantilevered, i.e. "it is empty beneath" (27). In case of the previously described right-left bending system, there is no need to have the cantilever on the head which supports the bending group, given that there can be obtained also with a very small portion between one bend and the subsequent counter-bend; however the cantilever speeds the pipe bending operations, in that at times the presence thereof requires the tilting of the matrix and repositioning all the parts of the bending group so as to obtain a counter-bend (28).

Demounting the pipe form the machine also obviously requires returning the mobile support of the counter-matrix towards the initial idle position thereof, so as to create the free space required for lifting (extraction) of the pipe, between the matrix and the counter-matrix. Thus, the mobile support shall be required to have a travel at least equivalent to the diameter of the pipe, so as to extract it from the matrix and counter-matrix after demounting the U-bolt (driving bracket).

In the inherent prior art, the driving bracket is made in a single cast piece together with the matrix, and it is not symmetric; the pipe may be removed from the matrix at the end of the bending operation given that the bracket is open at the upper part, but should the matrix be tilted the driving bracket would be closed at the upper part and it would be possible to extract the pipe after the bending thereof. Therefore, the inherent prior art neither contemplates the possibility of tilting the matrix nor the possibility of inverting the direction of rotation of the matrix for bending the pipe in the opposite direction in any manner whatsoever.

As known to those skilled in the art, the groove of the matrix and the groove of the counter- matrix do not define, together, a perfect circle, when they are at approached operating position, but a sort of oval section which allows eliminating the return effect of the material, after the machining.

The speed of rotation of the matrix may be adjustable, and in this case the machine has - externally - a knob for adjusting the number of revolutions of the drive motor, preferably located in proximity of the switch for selecting the direction of rotation of the drive motor. Actually, as a function of the material and the diameter of the pipe, and other process parameters, it is often useful to be able to adjust the speed of rotation of the motor.

It should be observed that the inversion of the motion of the drive motor requires that the stages of the gearing down be obtained suitably, given that inverting the motion of the motor does not necessarily imply that the mechanical torque on the power take-off remains the same. I.e., the reduction gear should be made suitably, but there is no need to get deeper into these details given that they are within the reach of an average man skilled in the art and/or they represent systems known by gear motor manufacturers.

Brief description of the drawings

The following detailed description shall better outline how the present invention may be obtained concretely, according to the non-limiting or non-binding illustrative embodiments thereof, shown in the attached drawings, wherein:

FIGURE 1 is a perspective view of the metal pipes bending machine, according to the present invention, during the beginning of the left bending operation; FIGURE 2A is the plan view (form above) of the left bending operation, performed using the machine of Fig 1;

FIGURE 2B is the plan view (from above) of the right bending operation;

FIGURE 3 is a lateral view of the machine of Fig 1 and it shows the utility of the cantilever (of the head for supporting the bending group) when performing several bends on the pipe.

Detailed description of the present invention In the following detailed description a same detail is indicated in all the figures still using the same reference number, so as to avoid confusion.

Furthermore, though being more detailed than the previous general description of the inventive concept of the present invention, the following description is however limited to the details strictly useful for the understanding of the invention and the applications thereof, without getting into details that are already known or obvious to a man skilled in the art.

Thus, the object of the present detailed description is exclusively that of allowing an average man skilled in the art to understand the attained technical progress, and be able to apply, if willing, the present invention. The specific processes of manufacturing the machine, the materials used, the variability intervals of the various process parameters (mechanical torque required for bending, the speed of rotation of the motor and of the matrix), as well as the specific shape of the matrix and of the counter-matrix, of the drawing bracket, etc, the external shape of the machine and/or the sizes thereof, the details of the means for adjusting the position of the counter-matrix, etc represent details that the man skilled in the art may modify at will without departing from the scope of protection of the present invention. That said, Fig 1 shows a general view of the machine of the invention according to any non- limiting or non-binding preferred embodiment of the present invention. The machine, indicated in its entirety with reference n° 1, comprises a lower part la which houses the drive motor (not visible given that it is arranged inside the external body 2 of the machine) and an upper part lb which substantially houses the reduction gear unit (or reduction gear), also not visible Fig 1 given that it is concealed inside the external body 2 of the machine 1.

The machine also comprises a base 3 and an upper cover plate 4, removable but rigidly fixed to the external body 2, at the upper part of the latter. The base 3 can for example be fixed to the floor of a van used by a plumber, or alternatively, the base 3 can be fixed (or be made integral) to a metal structure or metal box which increases the vertical dimensions of the machine, for use in an industrial workshop, so as to save the operator the effort of bending when machining the piece (metal pipe 5), or for other purposes. It is thus understood that the machine 1 is of the type that can be adapted to various situations. For example, should the pipe 5 be very long, it may be advantageous to manufacture a machine 1 with extremely small vertical dimensions, so as to be possibly able to place the distal end (with respect to the machine) of the pipe 5 on a small temporary support in the workshop, in proximity to the floor, or even thereon. It is thus clear that the machine can be configured variously.

The upper plate 4 is traversed by the operating shaft (not shown) which is simply dropped into the hollow shaft (also not shown) constituting the last stage of the of the gear motor. The operating shaft is vertical, the lower end thereof is engaged, through a lateral key, with the hollow shaft (also directed vertically); thus, the operating shaft rotatably drives a matrix 6 which receives, through a central hole thereof, the upper surface 7 (triangular-shaped in this case) of the operating shaft. The operating shaft is thus preferably interchangeable, according to the present invention. However, in another alternative embodiment it is possible to obtain a machine 1 - for bending pipes both to the right and to the left - with an operating shaft made in a single piece with the vertical output shaft of the gear motor. It is thus observable that the machine 1 may be configured variously. Thus, as regards the present patent application, it is essential that the machine 1 of the invention be provided with means for both right and left bending of a metal pipe 5. In the prior art which utilizes an internal core for bending the metal pipes, there are already provided machines which bend the pipes both to the right and to the left. The present invention represents a valid alternative to this prior art. The internal cores may be quite complex and they also require to be positioned accurately and held suitably firmly so as to prevent them from moving during the pipe bending operation. Thus, the method for bending metal pipes utilizing an internal opposition core, which maintains the internal diameter of the pipe constant, is not always advantageous. Thus, present invention offers a valid alternative solution.

Furthermore, it should be observed that as regards the present invention, the "inherent prior art", does not include, by definition, the pipe machining operations including the use of an internal core.

A possible embodiment of such right and left bending means according to the invention, shall now be described in detail. Reference shall also be made to the plan views of figures 2 A and 2B with the aim of outlining the left and right bending technique better. The arrow F always indicates the direction of advancement of the pipe 5 between the matrix 6 and the counter-matrix 8. The idea consists in the use of tools that can be inverted in a specular manner. This concept shall now be clarified further.

A movable support i.e. mobile 10 (Fig 1) of the counter-matrix 8 allows approaching the latter to the matrix 6, thus also performing the fine adjustment of the position of this counter- matrix 8. The adjustment of the position of the mobile support 10 occurs through a toggle lever system 1 1 (rough adjustment) cooperating with a screw adjustment system 14 (fine adjustment). The latter system 14, upon performing the fine adjustment to bend a pipe 5 of a given diameter, is no longer positioned, while the actuation of the toggle system 11 allows - each time - extracting and repositioning the metal pipe 5 so as to perform the desired bending at the desired points. Thus, the travel of the mobile support 10 and hence that of the counter-matrix 8, determined by the actuation of the toggle 11, is adapted to the diameter of the pipe 5 to be bent, as mentioned previously. In the present preferred but non-binding embodiment, the mobile support 10 is U-shaped wherein the base of the U-shape is vertical (facing the observer in Fig 1) and integral with a stem/pushing element 13 of the toggle 11. The mobile support 10 bears an extractable vertical pin 12 (Figs 2 A and 2B) which is inserted into a through hole of the counter-matrix 8 and into holes aligned to such hole, provided on the horizontal sides of the U-shaped element of the mobile support 10. Thus, the counter-matrix 8 may freely rotate around the vertical geometric axis of the extractable vertical pin 12, but only by a predefined maximum angle, which is set through a screw (not shown) which can be fastened and unfastened in the hole 15 (see Figs 2A and 2B) obtained on a projection 16 of the counter-matrix 8 (see Figs 2 A and 2B). The abutment of the adjustment screw - of the rotation of the counter-matrix 8 - against the U-shaped mobile support 10, defines the end stop of the rotation of the counter- matrix 8, which can be "set" from time to time by the operator, depending on the type of pipe 5 and machining operation.

Figures 2A and 2B show the perfect specularity of the arrangement of the components 6, 8, 9 (reference 9 indicates the driving bracket) which allows the bending of the pipe 5, once to the left (Fig 2 A) and once to the right (Fig 2B).

According to the present invention, the matrix 6 is shaped symmetrically with respect to an ideal horizontal plane, passing through the bottom of the peripheral groove 17 thereof, hence it can operate both in the condition shown in Fig 1 and 2A (to perform the left bending of the pipe 5) and in the condition shown in Fig 2B (to perform the right bending of the pipe 5); it will be sufficient to tilt the matrix 6 and insert it "capsized" on the triangular head 7 corresponding to the upper end 7 of the operating shaft, to pass from one bending direction to the other. Specifically, as illustrated by Figs 2A and 2B, the matrix 6 supports a removable U-bolt 9 (U-shaped bracket), which can be demounted by removing a pin 18. The arms or lateral sides of the U-bolt 9 rest in respective flat seats (flat recesses) of the upper and respectively lower lateral surface of the matrix 6; thus, the arms of the U-bolt 9 are arranged flushed with respect to the upper and respectively lower surface of the matrix (Fig 1). It should be observed that Fig 1 shows the initial position regarding the components of the group for bending the pipe 5, before performing the left bending (the U-bolt 9 for driving the pipe 5 arranged adjacent i.e. approached to the counter-matrix 8), while Fig 2A shows a position in which the U-bolt is rotated by about 90° around the axis (7) of the operating shaft, i.e. at the end of the left bending of the pipe 5. Obviously, the figures are solely for exemplifying surfaces given that the bend obtained in the pipe could subtend an angle up to 180°, as known to a man skilled in the art.

Bending the metal pipe 5 in the opposite direction to perform a "counter-bend", requires first extracting the pin 18 of the U-bolt 9 so as to remove the U-bolt 9 from the matrix 6, then the counter-matrix 8 is moved away from the matrix 6 through a toggle system 1 1 (or any other similar movement system), and lastly the pipe 5 is removed. Subsequently, the pin is removed 12 from the mobile support 10 and the counter-matrix 8 is positioned the other way (i.e. in a specular fashion, symmetrically with respect to the axis X-X defined by the toggle system 11), re-inserting the stop pin 12. Then, the matrix 6 is tilted and it is rotated until it reaches the initial position (seat of the U-bolt 9 near the counter-matrix 8), and then the U- bolt 9 is inserted in the seats thereof by re-introducing the pin 18, after introducing the pipe 5 into the groove 17 (in the desired direction F so as to perform the counter-bend). Lastly, the counter-matrix 8 is approached to the matrix 6. Then the switch 20 is activated to select the direction of rotation of the drive motor. The button 19, clearly visible and with bright colors (for example red), is used for the emergency stop. A speed adjuster 21 (small knob) is set for selecting the speed of rotation of the matrix 6 most suitable for the type of machining process being carried out (such speed may be as a function of the diameter and the thickness of the pipe, the material the pipe is made of, the desired bending angle, the type of matrix, the pipe-matrix friction, etc.). The ideal speed of rotation is selected according to the return of the material, bearing in mind that the slower the rotation of the matrix the greater the nullifying effect of the return of the material. It should be observed that the counter-matrix 8, as mentioned previously (and as deducible by a man skilled in the art), rotates slightly around the pin 12 to accompany the movement of the metal pipe 5 in the straight part thereof, which tends to radially move away from the matrix 6, as known in the industry. This is not shown in Figs 2A and 2B, which are used for the general illustration of the inventive concept. The slight rotation of the counter-matrix 8 allows the metal pipe 5 to "select" the optimal point of tangency on the peripheral groove 17 of the matrix 6 during the bending operation, such point of tangency depending (as known) on the radius of the matrix.

A block 23 can be inserted in a drawer-like manner into a respective compartment of the lateral wall of the machine. The block 23 represents the electrical and electronic part (control unit) of the machine 1 for bending pipes 5. Reference n° 22 indicates a flexible element for the protection of the electrical cables for supplying the control unit 23.

Lastly, reference n° 24 is used to indicate a handle (gripping element) which facilitates the transport of the machine 1, if transportable (actually, should the machine be larger, it may also not be transportable manually).

Now, with reference to Fig 3, it shows another important characteristic of the machine 1 of the present invention. When the bends and counter-bends to be obtained on the metal pipe 5 are not too close, the fact that the head 25 of the machine (which supports the previously described bending group) is mounted cantilevered, it may be advantageous to allow the operation of bending the pipe 5; actually, a part of the pipe (indicated with reference n° 26) may be arranged under the cantilever of the head 25, in an area 27 below said cantilever, thus actually allowing the operation of bending the pipe 5. Figure 3 and the position acquired by the U-bolt 9 in this figure, at the end of the operation of bending the pipe 5, immediately show that without the cantilever of the head 25 the pipe portion 26 would have ended up against the lateral wall (facing the observer) of the machine 1.

It should be observed that the machine 1 for bending pipes, described up to now, is also generally adapted to be used conventional matrices (for example having a member for driving the metal pipe obtained a single cast piece together with the matrix). In other words, the machine 1 described up to now may serve as a "right bending machine" and simultaneously as a "left bending machine", just by changing matrix, i.e. utilizing two specular matrices with respect to the axis X-X of the machine (see Figs 2 A and 2B). More precisely, the counter-matrix 8 and all other previously described systems/components of the machine 1 may remain unvaried, but conventional matrices couplable with the operating shaft (a pair of matrices, one for right bending and the other specular one for left bending) shall be used instead of the matrix 6 with removable U-bolt 9 described beforehand. In the particular case of claims 10 and 11 there shall be the maximum flexibility of the innovative machine for bending metal pipes, given that simply by selecting a suitable interchangeable operating shaft, the machine may also be used with any matrix of the prior art (with central hole of any shape), for bending pipes both to the right and the to the left.

However, using a matrix (possibly symmetric with respect to the plane passing through the peripheral groove like the described matrix 6) having a removable drive member 9 of the metal pipe, as described, allows bending pipes both to the right and the to the left using a single machine and specifically with a single matrix. Obviously, the matrix and the counter- matrix shall be changed in case of bending pipes with different diameters.

The machine 1 is obviously provided with a clutch or an analogous safety system which prevents a rotation of the drive motor in the wrong direction when the matrix 6 is mounted on the operating shaft to rotate in a given way. For example, with reference to Fig 2A (shown in plan view) the safety system prevents the motor from rotating the matrix 6 clockwise (with respect to Fig 2A) starting from the position shown in Fig 1. Such safety system may for example comprise a sensor which detects the position of the matrix 6 on the horizontal plate 4 (and possibly blocks the operation of the drive motor through the control unit 23), or any other safety system that can be implemented by those skilled in the art so as to avoid damaging the motor and the machine should the operator turn the selector 20 in the wrong direction of rotation. List of reference symbols

1 machine for bending pipes

la lower part

lb upper part

2 external body

3 base

4 upper covering plate

5 metal pipe

6 matrix

7 head for the rotatable driving of the matrix

8 counter-matrix

9 U-bolt/driving bracket

10 mobile support (i.e. moveable) of the counter-matrix

11 system with toggle lever

12 pin (of the counter-matrix)

13 thrust stem

14 screw adjustment (fine)

15 hole of 16

16 projection of 8

17 groove of the matrix

18 pin (of the U-bolt)

19 emergency stop button

20 switch for selecting the rotary motion

21 knob for adjusting the speed of rotation

22 flexible element for electrical cables

23 electrical control unit

24 handle, gripping member 25 cantilevered head

26 pipe portion

27 empty space (under the cantilever)

28 counter-bend of the pipe 5

X-X ideal geometric axis between the center of the matrix and the point of tangency