1461119035-245ddba5-bc5d-4bd6-ba62-ed386574e550

1. An internal blow out preventer comprising:
a ball valve comprising:
a tubular assembly comprising a central passageway;
an upper seat mounted within the tubular assembly;
a lower seat mounted within the tubular assembly;
a ball rotatably received between the upper and lower seats and comprising an external surface and a central opening, wherein the central opening comprises an undercut; and
a protective coating applied to both the external surface and the undercut of the ball.
2. The internal blow out preventer of claim 1, wherein the protective coating comprises a hard protective coating capable of reducing wear between the ball and the seats.
3. The internal blow out preventer of claim 1, wherein the protective coating comprises tungsten carbide.
4. The internal blow out preventer of claim 1, wherein the upper seat comprises a central opening axially aligned with both the central passageway of the tubular assembly and the central opening of the ball; and wherein the central opening of the ball is larger than the central opening of the upper seat.
5. The internal blow out preventer of claim 4, wherein the lower seat comprises a central opening axially aligned with both the central passageway of the tubular assembly and the central opening of the ball; and wherein the central opening of the lower seat is larger than the central opening of the ball.
6. The internal blow out preventer of claim 1, wherein the undercut is in the upper end of the ball.
7. The internal blow out preventer of claim 6, further comprising an undercut in the lower end of the ball, and wherein the protective coating is applied to each of the external surface, the upper undercut and the lower undercut of the ball.
8. An internal blow out preventer comprising:
a ball valve comprising:
a tubular assembly comprising a central passageway;
an upper seat mounted in the central passageway of the tubular assembly;
a lower seat mounted in the central passageway of the tubular assembly;
a ball rotatably received between the upper and lower seats, and comprising a central opening and an upper contacting surface that contacts a contacting surface of the upper seat, and wherein the central opening comprises an upper undercut at an upper end of the ball; and
a protective coating applied to both the upper contacting surface and the upper undercut of the ball.
9. The internal blow out preventer of claim .8, wherein the protective coating comprises a hard protective coating capable of reducing wear between the ball and the seats.
10. The internal blow out preventer of claim 8, wherein the protective coating comprises tungsten carbide.
11. The internal blow out preventer of claim 8, wherein the upper seat comprises a central opening axially aligned with both the central passageway of the tubular assembly and the central opening of the ball; and wherein the central opening of the ball is larger than the central opening of the upper seat.
12. The internal blow out preventer of claim 11, wherein the lower seat comprises a central opening axially aligned with both the central passageway of the tubular assembly and the central opening of the ball; and wherein the central opening of the lower seat is larger than the central opening of the ball.
13. The internal blow out preventer of claim 8, wherein the ball comprises a lower contacting surface that contacts a contacting surface of the lower seat; and wherein the central opening comprises a lower undercut at a lower end of the ball; and wherein the protective coating is applied to each of the upper and lower contacting surfaces and the upper and lower undercuts of the ball.
14. An internal blow out preventer comprising:
a first ball valve comprising:
a first tubular assembly comprising a central passageway;
a first upper seat mounted in the central passageway of the first tubular assembly;
a first lower seat mounted in the central passageway of the first tubular assembly;
a first ball rotatably received between the first upper and lower seats, and comprising a central opening and an upper contacting surface that contacts a contacting surface of the first upper seat, and wherein the central opening comprises an upper undercut at an upper end of the first ball; and
a first protective coating applied to both the upper contacting surface and the upper undercut of the first ball; and

an second ball valve comprising:
a second tubular assembly threadably connected to the first tubular assembly and comprising a central passageway;
a second upper seat mounted in the central passageway of the second tubular assembly;
a second lower seat mounted in the central passageway of the second tubular assembly;
a second ball rotatably received between the second upper and lower seats, and comprising a central opening and an upper contacting surface that contacts a contacting surface of the second upper seat, and wherein the central opening comprises an upper undercut at an upper end of the second ball; and
a second protective coating applied to both the upper contacting surface and the upper undercut of the second ball.
15. The internal blow out preventer of claim 14, wherein the first and second protective coatings each comprise a hard protective coating capable of reducing wear between the ball and the seats.
16. The internal blow out preventer of claim 14, wherein the first and second protective coatings each comprise tungsten carbide.

The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.

1. A wireless base station system, operative to split a plurality of radio transceiver chains between a backhaul link and a radio access network, comprising:
a baseband subsystem comprising N digital ports, operative to synthesize N digital baseband signals; and
N radio transceiver chains, each connects to one of the N digital ports of the baseband subsystem via an analog-digital interface;
wherein the system is configured to:
split the N radio transceiver chains into a first set of K radio transceiver chains and a second sets of N minus K radio transceiver chains;
synthesize, by the baseband subsystem, the N digital baseband signals according to the split, such that K digital baseband signals are operative to support a backhaul link, and N minus K digital baseband signals are operative to support a radio access network; and
input the N digital baseband signals to the N radio transceiver chains via the corresponding N digital ports and the corresponding analog-digital interfaces, thereby transmitting both the backhaul link and the radio access network.
2. The system of claim 1, wherein N is equal to at least 3, and therefore at least one of (i) the backhaul link and (ii) the radio access network are transmitted using at least two radio transceiver chains.
3. The system of claim 1, wherein at least two of the N digital baseband signals driving at least two of the radio transceiver chains comprise at least two multiple input multiple output signals, thereby transmitting at least one of (i) the backhaul link and (ii) the radio access network using a multiple input multiple output scheme.
4. The system of claim 1, wherein at least two of the N digital baseband signals driving at least two of the radio transceiver chains comprise at least two phased-array signals, thereby transmitting at least one of (i) the backhaul link and (ii) the radio access network using a phased-array scheme comprising the at least two of the radio transceiver chains.
5. The system of claim 1, wherein at least two of the N digital baseband signals received from at least two of the radio transceiver chains comprise at least two signal types selected from a group consisting of maximal ratio combining, minimum mean square error and maximum likelihood.
6. The system of claim 1, wherein the baseband subsystem comprises a baseband processor, and the baseband processor is operative to simultaneously generate both the backhaul link and the radio access network, according to the setting of K.
7. The system of claim 6, wherein the baseband processor comprises at least one device selected from a group consisting of application specific integrated circuit, field programmable gate array and digital signal processor, and simultaneous generation of the backhaul link and the radio access network is performed, at least in part, by the at least one device according to the setting of K.
8. The system of claim 7, wherein the at least one device is a digital signal processor, and simultaneous generation of the backhaul link and the radio access network is done, at least in part, in software running on the digital signal processor, according to the setting of K.
9. The system of claim 1, wherein the baseband subsystem comprises at least two baseband processors, and the at least two baseband processors are operative to simultaneously generate the backhaul link and the radio access network, via the corresponding K baseband signals and N minus K baseband signals, according to the setting of K.
10. The system of claim 9, wherein one of the at least two baseband processors is operative to generate the backhaul link, and another one of the at least two baseband processors is operative to generate the radio access network.
11. The system of claim 9, wherein a configurable digital interconnect subsystem is used to interconnect each of the at least two baseband processors with at least some of the N digital ports, according to the setting of K and according to the assignment of the N radio transceiver chains between the backhaul link and the radio access network, such that the K radio transceiver chains are connected to one of the baseband processors, and the N minus K radio transceiver chains are connected to another of the baseband processors.
12. The system of claim 1, wherein the N radio transceiver chains are connected to N omni-directional antennas respectively, thereby allowing both the backhaul link and the radio access network to span a 360 degrees coverage area around the wireless base station, regardless of assignment of radio transceiver chains between the backhaul link and the radio access network.
13. A method for sharing a plurality of radio transceiver chains between a backhaul link and a radio access network, comprising:
operating, by a wireless base station, N radio transceiver chains;
splitting, by the wireless base station, according to a first criterion, the N radio transceiver chains into two sets of radio transceiver chains, wherein the first set of radio transceiver chains is allocated to a backhaul link and the second set of radio transceiver chains is allocated to a radio access network; and
communicating (i) a first set of data between the wireless base station and a core network data source via the backhaul link employing the first set of radio transceiver chains, and (ii) a second set of data between the wireless base station and at least one subscriber station via the radio access network employing the second set of radio transceiver chains.
14. The method of claim 13, further comprising:
determining a minimum number of radio transceiver chains required by the wireless base station to communicate the first set of data, wherein the minimum number of radio transceiver chains is equal to at least one, and the minimum number of radio transceiver chains is equal to at most N minus one; and
setting the number of radio transceiver chains in the first set of radio transceiver chains to the minimum number determined.
15. The method of claim 13, further comprising:
determining a minimum number of radio transceiver chains required by the wireless base station to communicate the second set of data, wherein the minimum number of radio transceiver chains is equal to at least one, and the minimum number of radio transceiver chains is equal to at most N minus one; and
setting the number of radio transceiver chains in the second set of radio transceiver chains to the minimum number determined
16. The method of claim 13, further comprising:
using, by at least most of the N radio transceiver chains, substantially omni-directional antennas, such that (i) the first set of radio transceiver chains supports the backhaul link in substantially any direction, (ii) the second set of radio transceiver chains supports the radio access network in substantially any direction, and (iii) substantially any split of the N radio transceiver chains is supported regardless of the directions of the radio access network and backhaul links.
17. The method of claim 13, further comprising:
determining that the number of radio transceiver chains in the first set is not sufficient to maintain the backhaul link; and
increasing the number of radio transceiver chains in the first set, at the expense of the number of radio transceiver chains in the second set, in order to improve the backhaul link.
18. The method of claim 13, further comprising:
determining that the number of radio transceiver chains in the second set is not sufficient to maintain the radio access network; and
increasing the number of radio transceiver chains in the second set, at the expense of the number of radio transceiver chains in the first set, in order to improve the radio access network.
19. The method of claim 13, further comprising:
having a capability, in the N radio transceiver chains, to operate in at least two frequency bands;
setting the radio transceiver chains in the first set to operate in a first frequency band operative to support the backhaul link; and
setting the radio transceiver chains in the second set to operate in a second frequency band operative to support the radio access network.
20. A method for boosting performance of a backhaul link associated with a wireless base station, comprising:
operating, by a wireless base station, K radio transceiver chains associated with a backhaul link, and M radio transceiver chains associated with a radio access network;
detecting that the K radio transceiver chains are not sufficient to maintain a predetermined level of performance associated with the backhaul link; and
increasing the number of radio transceiver chains associated with the backhaul link from K to at least K plus one, at the expense of the M radio transceiver chains.
21. The method of claim 20, further comprising:
using the K radio transceiver chains in a multiple-input-multiple-output configuration;
detecting that the K radio transceiver chains are not sufficient to maintain a predetermined wireless data capacity associated with the backhaul link; and
using the at least K plus one radio transceiver chains to increase the level of the multiple-input-multiple-output configuration, thereby boosting the wireless data capacity associated with the backhaul link.
22. The method of claim 20, further comprising:
using the K radio transceiver chains to realize a wireless reception scheme selected from a group consisting of phase-array reception, maximal ratio combining reception, minimum mean square error reception, and maximum likelihood reception;
detecting that the K radio transceiver chains are not sufficient to maintain a predetermined wireless sensitivity associated with the backhaul link; and
using the at least K plus one radio transceiver chains to increase the level of the wireless reception scheme, thereby boosting the wireless sensitivity associated with the backhaul link.
23. The method of claim 20, wherein the K and M radio transceiver chains operate in a first band, thereby implementing in-band-backhaul communication scheme.
24. The method of claim 20, wherein:
the K and M radio transceiver chains operate in two separate bands respectively, thereby operating the backhaul link in a different band than the radio access network; and
at least one of the M radio transceiver chains is capable of operating in both the first and the second bands; and
the at least one of the M radio transceiver chains is reset from the first band to the second band before being assigned to the backhaul link, thereby increasing the number of radio transceiver chains associated with the backhaul link from K to the at least K plus one.

1461119025-ce5ebd8c-07a1-40a6-bf83-b97624189d2d

1. A method for managing one or more secure gateway virtual private network, VPN, devices in a secure VPN for cryptographically separated and tunnelled VPN communication, the method comprising:
receiving (401) VPN configuration data provided by a management system; and
encapsulating the received VPN configuration data and a domain type forming one or more data packets, said domain type identifying an administrative network domain for cryptographically separated and tunnelled management communication with a hardware separated administrative controller of said one or more secure gateway VPN devices, exclusively for management of said one or more secure gateway VPN devices by means of said VPN configuration data.
2. The method of claim 1, further comprising:
establishing the cryptographically separated and tunnelled management communication with the hardware separated administrative controller of said one or more secure gateway VPN devices; and
communicating (404) the data packets including the encapsulated VPN configuration data and domain type to the hardware separated administrative controller of said one or more of the secure gateway VPN devices by means of the cryptographically separated and tunnelled management communication over a virtual administrative network defining the administrative network domain for management of the one or more of the secure gateway VPN devices according to the VPN configuration data.
3. The method of claim 1, wherein encapsulating the received VPN configuration data and the domain type comprises: preparing data packets including a data filed with the VPN configuration data, a domain type identifying the administrative network domain, a header filed with a header and a trailer filed with a trailer.
4. The method of claim 1, wherein encapsulating the received VPN configuration data and the domain type further comprises: protecting the integrity of the VPN configuration data and the domain type.
5. The method of claim 4, wherein protecting the integrity of the VPN configuration data and the domain type further comprises: generating a digital signature of the data packets using certificates.
6. The method of claim 4, wherein protecting the integrity of the VPN configuration data and the domain type further comprises:
calculating a hash value of the VPN configuration data and the domain type; and
encrypting the domain type, the VPN configuration data, and the hash value filed by using a secret key.
7. The method of claim 6, further comprising: preparing data packets including a Key ID field indicating to the receiving secure gateway VPN device which key to be used for decryption.
8. The method of claim 4, wherein protecting the integrity of the VPN configuration data and the domain type further comprises:
calculating a keyed hash value of the VPN configuration data and the domain type; and
encrypting the VPN configuration data and the domain type by using a secret key.
9. A method for managing secure gateway virtual private network, VPN, devices in a secure VPN for cryptographically separated and tunnelled VPN communication, the method comprising:
receiving data packets comprising encapsulated VPN configuration data and a domain type, said domain type identifying an administrative network domain for cryptographically separated and tunnelled management communication with a hardware separated administrative controller of said one or more secure gateway VPN devices, exclusively for management of one or more secure gateway VPN devices;
extracting the VPN configuration data and the domain type from the data packets;
identifying the domain type;
if the domain type is identified to be the administrative network domain, configuring the one or more secure gateway VPN devices according to the received VPN configuration data.
10. An administration gateway device for managing one or more secure gateway virtual private network, VPN, devices in a secure VPN for cryptographically separated and tunnelled VPN communication, the administration gateway comprising:
a first communication interface adapted to receive VPN configuration data provided by a management system; and
an administrative gateway controller operatively connected to the first communication interface and a second communication interface, wherein the administrative gateway controller is adapted to:
encapsulate the received VPN configuration data and a domain type, said domain type identifying an administrative network domain for cryptographically separated and tunnelled management communication with a hardware separated administrative controller of said one or more secure gateway VPN devices, exclusively for management of said one or more secure gateway VPN devices by means of the VPN configuration data.
11. The administration gateway device of claim 10, wherein the administrative gateway controller is operatively connected to a second communication interface and is further adapted to establish the cryptographically separated and tunnelled management communication with the hardware separated administrative controller of said one or more secure gateway VPN devices; and
the second communication interface is adapted to communicate the encapsulated VPN configuration data and domain type to the hardware separated administrative controller of said one or more of the secure gateway VPN devices by means of the cryptographically separated and tunnelled management communication over a virtual administrative network (106) defining the administrative network domain exclusively for management of the one or more of the secure gateway VPN devices according to the VPN configuration data.
12. A secure gateway virtual private network, VPN, device for cryptographically separated and tunnelled VPN communication comprising:
a first communication interface adapted to receive data packets comprising encapsulated VPN configuration data and a domain type, said domain type identifying an administrative network domain for cryptographically separated and tunnelled management communication, via the cryptographically separated and tunnelled management communication separate from said cryptographically separated and tunnelled VPN communication, exclusively for management of the secure gateway VPN device by means of the VPN configuration data;
a hardware separated administrative controller operatively connected to the first communication interface and adapted to:
identify only the administrative network domain and to only extract configuration data from data packets of the administrative network domain; and
configure the secure gateway VPN device according to the received VPN configuration data, if the domain type is an administrative network domain.
13. The secure gateway virtual private network, VPN, device according to claim 12, comprising a plain text side controller operatively connected to the first communication interface and adapted to only identify plain text data packets of a red domain received by the first communication interface and forward the data packets to a red network via a red communication interface of the secure gateway VPN device, wherein data packets of any other domain type than the red domain are blocked.
14. The secure gateway virtual private network, VPN, device according to claim 12, wherein the encapsulated VPN configuration data and the domain type forming data packets including a data filed with the VPN configuration data, a domain type identifying the administrative network domain, a header filed with a header and a trailer filed with a trailer.
15. The secure gateway virtual private network, VPN, device according to claim 12, wherein the received VPN configuration data and the domain type forming data packets protecting the integrity of the VPN configuration data and the domain type.
16. The secure gateway virtual private network, VPN, device according to claim 12, wherein the integrity of the VPN configuration data and the domain type is protected by a digital signature of the data packets using certificates.
17. The secure gateway virtual private network, VPN, device according to claim 12, wherein the integrity of the VPN configuration data and the domain type is protected by a hash value calculated of the VPN configuration data and the domain type; and wherein the domain type, the VPN configuration data, and the hash value filed are encrypted by means of a secret key and forming part of the data packets.
18. The secure gateway virtual private network, VPN, device according to claim 17, wherein the data packets comprises a Key ID field indicating to the receiving secure gateway VPN device which key to be used for decryption.
19. The secure gateway virtual private network, VPN, device according to claim 12, wherein the integrity of the VPN configuration data and the domain type is protected by a keyed hash value of the VPN configuration data and the domain type; and the PN configuration data and the domain type are encrypted by means of a secret key.
20. A computer program comprising program instructions for causing a computer to perform the method of claim 1, when said program is run on a computer.
21. A computer program on a carrier and comprising computer executable instructions for causing a computer to perform the method according to claim 1, when said program is run on a computer.
22. The computer program according to claim 21, wherein said carrier is a record medium, computer memory, read-only memory, computer-readable medium or an electrical carrier signal.
23. A computer program product comprising a computer-readable medium, having thereon: computer program code means, when said program is loaded, to make the computer execute the method of claim 1.

The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.

1. A computer-implemented method, comprising:
assigning a respective set of barycentric coordinates to each of three vertices of a triangle, wherein the triangle comprises a control triangle for a quadratic B\xe9zier curve, and wherein each set of barycentric coordinates comprises three values; and
rendering the quadratic B\xe9zier curve using a graphics processing unit (GPU), wherein the rendering comprises evaluating a function of the barycentric coordinates.
2. The method as recited in claim 1, wherein the quadratic B\xe9zier curve is a rational quadratic B\xe9zier curve.
3. The method as recited in claim 1, further comprising:
sending to the GPU only two of the three values in the set of barycentric coordinates for each corresponding vertex of the triangle.
4. The method as recited in claim 1, further comprising:
determining a set of barycentric coordinates for each of a plurality of points on edges of the triangle and within the triangle, wherein the determining comprises interpolating between the sets of barycentric coordinates assigned to the three vertices of the triangle.
5. The method as recited in claim 1, wherein rendering the quadratic B\xe9zier curve using the GPU comprises evaluating the function of the barycentric coordinates to determine whether each point in the triangle lies above or below the quadratic B\xe9zier curve.
6. The method as recited in claim 1, wherein the function of the barycentric coordinates is evaluated in a pixel shader.
7. The method as recited in claim 1, wherein assigning the set of barycentric coordinates to each of the three vertices of the triangle comprises:
assigning a first set of barycentric coordinates (1,0,0) to a first vertex of the three vertices;
assigning a second set of barycentric coordinates (0,1,0) to a second vertex of the three vertices; and
assigning a third set of barycentric coordinates (0,0,1) to a third vertex of the three vertices.
8. The method as recited in claim 1, further comprising:
determining an integer representing each set of barycentric coordinates as a type of the corresponding vertex, wherein the integer comprises an index into a global array which stores a plurality of vertex parameters;
sending the integer representing each set of barycentric coordinates to the GPU as a texture parameter associated with the corresponding vertex of the triangle.
9. A computer-readable storage medium, comprising program instructions, wherein the program instructions are computer-executable to implement:
assigning a respective set of barycentric coordinates to each of three vertices of a triangle, wherein the triangle comprises a control triangle for a quadratic B\xe9zier curve, and wherein each set of barycentric coordinates comprises three values; and
rendering the quadratic B\xe9zier curve using a graphics processing unit (GPU), wherein the rendering comprises evaluating a function of the barycentric coordinates.
10. The computer-readable storage medium as recited in claim 9, wherein the quadratic B\xe9zier curve is a rational quadratic B\xe9zier curve.
11. The computer-readable storage medium as recited in claim 9, wherein the program instructions are computer-executable to implement:
sending to the GPU only two of the three values in the set of barycentric coordinates for each corresponding vertex of the triangle.
12. The computer-readable storage medium as recited in claim 9, wherein the program instructions are computer-executable to implement:
determining a set of barycentric coordinates for each of a plurality of points on edges of the triangle and within the triangle, wherein the determining comprises interpolating between the sets of barycentric coordinates assigned to the three vertices of the triangle.
13. The computer-readable storage medium as recited in claim 9, wherein rendering the quadratic B\xe9zier curve using the GPU comprises evaluating the function of the barycentric coordinates to determine whether each point in the triangle lies above or below the quadratic B\xe9zier curve.
14. The computer-readable storage medium as recited in claim 9, wherein the function of the barycentric coordinates is evaluated in a pixel shader.
15. The computer-readable storage medium as recited in claim 9, wherein assigning the set of barycentric coordinates to each of the three vertices of the triangle comprises:
assigning a first set of barycentric coordinates (1,0,0) to a first vertex of the three vertices;
assigning a second set of barycentric coordinates (0,1,0) to a second vertex of the three vertices; and
assigning a third set of barycentric coordinates (0,0,1) to a third vertex of the three vertices.
16. The computer-readable storage medium as recited in claim 9, wherein the program instructions are computer-executable to implement:
determining an integer representing each set of barycentric coordinates as a type of the corresponding vertex, wherein the integer comprises an index into a global array which stores a plurality of vertex parameters;
sending the integer representing each set of barycentric coordinates to the GPU as a texture parameter associated with the corresponding vertex of the triangle.
17. A system, comprising:
at least one processor;
a memory coupled to the at least one processor, wherein the memory is configured to store program instructions executable by the at least one processor to assign a respective set of barycentric coordinates to each of three vertices of a triangle, wherein the triangle comprises a control triangle for a quadratic B\xe9zier curve, and wherein each set of barycentric coordinates comprises three values; and
a graphics processing unit (GPU) coupled to the at least one processor, wherein the GPU is executable to render the quadratic B\xe9zier curve, wherein the rendering comprises evaluating a function of the barycentric coordinates.
18. The system as recited in claim 17, wherein the quadratic B\xe9zier curve is a rational quadratic B\xe9zier curve.
19. The system as recited in claim 17, wherein the program instructions are further executable by the at least one processor to:
send to the GPU only two of the three values in the set of barycentric coordinates for each corresponding vertex of the triangle.
20. The system as recited in claim 17, wherein the GPU is further executable to:
determine a set of barycentric coordinates for each of a plurality of points on edges of the triangle and within the triangle, wherein the determining comprises interpolating between the sets of barycentric coordinates assigned to the three vertices of the triangle.
21. The system as recited in claim 17, wherein, in rendering the quadratic B\xe9zier curve, the GPU is executable to evaluate the function of the barycentric coordinates to determine whether each point in the triangle lies above or below the quadratic B\xe9zier curve.
22. The system as recited in claim 17, wherein the function of the barycentric coordinates is evaluated in a pixel shader.
23. The system as recited in claim 17, wherein, in assigning the set of barycentric coordinates to each of the three vertices of the triangle, the program instructions are executable by the at least one processor to:
assign a first set of barycentric coordinates (1,0,0) to a first vertex of the three vertices;
assign a second set of barycentric coordinates (0,1,0) to a second vertex of the three vertices; and
assign a third set of barycentric coordinates (0,0,1) to a third vertex of the three vertices.
24. The system as recited in claim 17, wherein the program instructions are further executable by the at least one processor to:
determine an integer representing each set of barycentric coordinates as a type of the corresponding vertex, wherein the integer comprises an index into a global array which stores a plurality of vertex parameters;
send the integer representing each set of barycentric coordinates to the GPU as a texture parameter associated with the corresponding vertex of the triangle.