Method for approximation of caps of smooth line segments

A method for plotting an ending cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array, said line segment having a mathematical end point and a thickness, the method comprising:

• determining a mathematical end point corrected for said thickness by computing a square having said mathematical end point at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical end point corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical end point corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical end point corrected for said thickness does not completely cover based on the area of said mathematical end point corrected for said thickness lying within each shaded pixel.

A method for plotting a starting cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array with an x-axis, wherein said x-axis in a positive x-direction has a direction of 0 degrees, said line segment having a mathematical origin, a direction, and a thickness, the method comprising:

• determining a mathematical origin corrected for said thickness by computing a square having said mathematical origin at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical origin corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical origin corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical origin corrected for said thickness does not completely cover based on the area of said mathematical origin corrected for said thickness lying within each shaded pixel, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the same direction as said direction of said line segment if said direction is between −45 degrees and 135 degrees, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the opposite direction as said direction of said line segment if said direction of said line segment is between 135 degrees and 180 degrees or between −45 degrees and −180 degrees.

A method for plotting an ending cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array with an x-axis, wherein said x-axis in a positive x-direction has a direction of 0 degrees, said line segment having a mathematical end point, a direction, and a thickness, the method comprising:

• determining a mathematical end point corrected for said thickness by computing a square having said mathematical end point at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical end point corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical end point corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical end point corrected for said thickness does not completely cover based on the area of said mathematical end point corrected for said thickness lying within each shaded pixel, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the opposite direction as said direction of said line segment if said direction is between −45 degrees and 135 degrees, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the same direction as said direction of said line segment if said direction of said line segment is between 135 degrees and 180 degrees or between −45 degrees and 180 degrees.

A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to plot a starting cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array, said line segment having a mathematical origin and a thickness, the method comprising:

• determining a mathematical origin corrected for said thickness by computing a square having said mathematical origin at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical origin corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical origin corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical origin corrected for said thickness does not completely cover based on the area of said mathematical origin corrected for said thickness lying within each shaded pixel.

A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to plot an ending cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array, said line segment having a mathematical end point and a thickness, the method comprising:

• determining a mathematical end point corrected for said thickness by computing a square having said mathematical end point at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical end point corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical end point corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical end point corrected for said thickness does not completely cover based on the area of said mathematical end point corrected for said thickness lying within each shaded pixel.

A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to plot a starting cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array with an x-axis, wherein said x-axis in a positive x-direction has a direction of 0 degrees, said line segment having a mathematical origin, a direction, and a thickness, the method comprising:

• determining a mathematical origin corrected for said thickness by computing a square having said mathematical origin at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical origin corrected for said thickness falls, said group of shaded pixels having four corners;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical origin corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical origin corrected for said thickness does not completely cover based on the area of said mathematical origin corrected for said thickness lying within each shaded pixel, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the same direction as said direction of said line segment if said direction is between −45 degrees and 135 degrees, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the opposite direction as said direction of said line segment if said direction of said line segment is between 135 degrees and 180 degrees or between −45 degrees and −180 degrees.

A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to plot an ending cap of a line segment on a computer display, said computer display comprising a plurality of pixels in a rectilinear array with an x-axis, wherein said x-axis in a positive x-direction has a direction of 0 degrees, said line segment having a mathematical end point, a direction, and a thickness, the method comprising:

• determining a mathematical end point corrected for said thickness by computing a square having said mathematical end point at one corner of said square, said square being aligned horizontally and vertically with said rectilinear array of pixels and each side of said square having a length equal to said thickness;
• determining a group of shaded pixels, said shaded pixels being pixels in which said mathematical end point corrected for said thickness falls, said group of shaded pixels having four corner;
• choosing a temporary origin at a corner of said shaded pixels;
• assigning a full intensity value to any shaded pixel that said mathematical end point corrected for said thickness completely covers;
• and
• assigning intensity values to said shaded pixels that said mathematical end point corrected for said thickness does not completely cover based on the area of said mathematical end point corrected for said thickness lying within each shaded pixel, said intensity value being zero for any shaded pixels on a side of said group of shaded pixels in the opposite direction as said direction of said line segment if said direction is between −45 degrees and 135 degrees, said intensity values being zero for any shaded pixels on a side of said group of shaded pixels in the same direction as said direction of said line segment if said direction of said line segment is between 135 degrees and 180 degrees or between −45 degrees and 180 degrees.