Abstract: A method for an authentication server to authenticate Voice over Internet Protocol (VoIP) devices includes receiving a request from a unverified VoIP device to verify its global telephone number over a computer network, searching for verified VoIP devices that have authenticated identities and caller ID service, and providing the telephone numbers of the verified VoIP devices to the unverified VoIP device, wherein the unverified VoIP device calls the verified VoIP devices and the verified VoIP devices detect telephone numbers of the unverified VoIP device. The method further comprises receiving the detected telephone numbers from the verified VoIP devices and determining if the detected telephone numbers match a telephone number provided by the unverified VoIP device, wherein the unverified VoIP device becomes verified if the telephone numbers match the telephone number provided by the VoIP device.

Abstract: Each device (A, B) keeps an IP cache list storing IP address of devices it has communicated with in the past over an IP network (106). When the IP address of a device changes, it provides its new IP address to the devices in its IP cache list. As a result, any two devices that have communicated in the past are able to locate each other as long as their IP addresses do not change about the same time. A device can also provide its IP cache list to another device so the other device can query the devices in the IP cache list for the updated IP address of the device.

Abstract: A method for an authentication server to authenticate Voice over Internet Protocol (VoIP) devices includes receiving a request from a unverified VoIP device to verify its global telephone number over a computer network, searching for verified VoIP devices that have authenticated identities and caller ID service, and providing the telephone numbers of the verified VoIP devices to the unverified VoIP device, wherein the unverified VoIP device calls the verified VoIP devices and the verified VoIP devices detect telephone numbers of the unverified VoIP device. The method further comprises receiving the detected telephone numbers from the verified VoIP devices and determining if the detected telephone numbers match a telephone number provided by the unverified VoIP device, wherein the unverified VoIP device becomes verified if the telephone numbers match the telephone number provided by the VoIP device.

Abstract: A method is provided for approximating a quadratic Bezier curve represented by two anchor points and a control point that are fixed-point binary numbers. If a first flatness of a line formed between the two anchor points of the Bezier curve is less than a threshold, the method includes replacing the Bezier curve with an edge between the two anchor points. If the first flatness is not less than the threshold, the method includes subdividing the Bezier curve into halves and determining a second flatness of one of the halves. The first flatness is approximated by a triangular area formed by the two anchor points and the control point, and the triangular area is approximated by multiplication and subtraction operations. The second flatness of the half is approximated as the triangular area divided by 8, where the division by 8 is determined by shifting 3 bits to the right.

Abstract: A method for a first device to communicate with a second device over a public computer network includes detecting a user dialing a telephone number of the second device and searching a pairing list for a pairing including the telephone number and an IP address of the second device. If the pairing is found in the pairing list, the method then includes using the IP address in the pairing to establish a computer network connection with the second device over the public computer network and exchanging data packets with the second device. If the pairing is not found in the pairing list, the method then includes using a peer-to-peer lookup protocol to find the pairing. If the pairing is not found using the peer-to-peer lookup protocol, the method then includes establishing a telephone connection with the second device over a telephone network to receive the IP address.

Abstract: A method is provided for approximating a quadratic Bezier curve represented by two anchor points and a control point that are fixed-point binary numbers. If a first flatness of a line formed between the two anchor points of the Bezier curve is less than a threshold, the method includes replacing the Bezier curve with an edge between the two anchor points. If the first flatness is not less than the threshold, the method includes subdividing the Bezier curve into halves and determining a second flatness of one of the halves. The first flatness is approximated by a triangular area formed by the two anchor points and the control point, and the triangular area is approximated by multiplication and subtraction operations. The second flatness of the half is approximated as the triangular area divided by 8, where the division by 8 is determined by shifting 3 bits to the right.

Abstract: A method for determining areas of sub-pixel regions formed by edges in a pixel includes receiving a first fill style to a first side of an edge, a second fill style to a second side of the edge, and a projected area of the edge to the second side of the edge. The method further includes determining if there is a sub-pixel region in the cell having the first fill style. If so, the method further includes (1) incrementing an area of the sub-pixel region by the difference between the unit area of the pixel and the projected area, and (2) modularizing the area by the unit area of the pixel. If not, the method further includes saving in the cell a sub-pixel region having (1) the first fill style and (2) an area equal to a difference between a unit area of the pixel and the projected area.

Abstract: A method for determining areas of sub-pixel regions formed by edges in a pixel includes receiving a first fill style to a first side of an edge, a second fill style to a second side of the edge, and a projected area of the edge to the second side of the edge. The method further includes determining if there is a sub-pixel region in the cell having the first fill style. If so, the method further includes (1) incrementing an area of the sub-pixel region by the difference between the unit area of the pixel and the projected area, and (2) modularizing the area by the unit area of the pixel. If not, the method further includes saving in the cell a sub-pixel region having (1) the first fill style and (2) an area equal to a difference between a unit area of the pixel and the projected area.

Abstract: A method for approximating a quadratic Bezier curve represented by a first anchor point, a control point, and a second anchor point includes determining a flatness of a line formed between the first anchor point and the second anchor point. The flatness is a first quotient of (1) a first triangular area formed by the first anchor point, the control point, and the second anchor point divided by (2) a first distance between the first anchor point and the second anchor point. If the flatness is less than a threshold, the method further includes replacing the quadratic Bezier curve with an edge between the first anchor point and the second anchor point.