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Inkscape is a free, open-source vector editing program that is available for Windows, Mac and Linux. You can download it from inkscape.org. Inkscape includes a bitmap tracing tool that will attempt to automatically convert your raster image into a vector. Click "File" → "Open" and browse for the bitmap image that you want to convert into a vector. Simple images and logos are best suited for this process. You'll have a difficult time getting adequate results when attempting to auto-trace a photograph. After loading the bitmap, you'll need to click it to select it on the canvas. Once you have the bitmap selected, you can open the auto-tracing tool. Click "Path" → "Trace Bitmap", or press ⇧ Shift+Alt+B. There are three different modes available for creating a single path. The "path" is the line tracing of your image. Selecting each one will update the Live Preview, allowing you to get a feel for what they do.  These three options will help you to create a basic outline which you can then manually edit yourself. Use the "Threshold" options to adjust how the scan is processed. For the "Brightness cutoff", a threshold of 0.0 will be black, while a threshold of 1.0 will be white. For the "Edge detection", the threshold is the same and determines whether the pixel will be counted as part of the border or not. Increase the number of scans to increase the number of colors that appear. This can be an inaccurate tool, and may not be able to provide good results for complex images. Once you're happy with the results, click OK to apply them to the image. Click "Path" → "Simplify" or press Ctrl+L to reduce the number of nodes. This will result in a lower-quality image, but the end result will be much easier to edit. This is located in the toolbox on the left side of the screen. You can also press F2. This will allow you to select and move nodes to edit the image. For detailed instructions on editing nodes in Inkscape, see this article. You can use the tools in the left toolbar to create shapes and lines. Use these to touch-up or replace parts of the trace that didn't turn out right. Once you've created a shape or line, you can use the nodes to adjust the curve and shape of the object. Click the "Convert selected object to path" button (⇧ Shift+Ctrl+C) to edit the nodes shapes that you've created. Once you are satisfied with your new vector art, you can save it as a vector image. Click "File" → "Save As" and choose one of the vector file formats from the drop-down menu. SVG is one of the most common vector formats.

Summary:
Download and install Inkscape. Open the bitmap image in Inkscape. Select the image. Open the tracer. Select the path mode. Select the "Colors" option if you are trying to create an exact copy of a simple image. Optimize the paths once you're finished. Click the "Edit paths by nodes" button. Add your own lines. Save the file.