The task can be performed with 3D software which allows vector object mappings to surfaces. I tried it in an entry level 3D modeller named MoI3D. It somehow resembles a premium package named Rhinoceros. MoI3D has radically different user interface, poor coloring options and no photorealistic rendering. But it can make the wanted curve projection in the same way as Rhinoceros:
The curves to be mapped are placed onto a square surface. The square presents which area should be placed onto the sphere. The mapping function is named Deform > Flow. Only the curve on the square is selected as the object to be flown and the square is selected to be the base surface. I clicked the matching points so that the midpoint is on the visible side.
The projected 2D scene can be saved as Adobe Illustrator file or PDF. I opened the PDF in Inkscape and recolored the strokes:
There's a couple of minor disadvantages:
- the outline of the sphere is an incredibly dense polyline
- everything is grouped, the groupings can be very complex with complex objects
The sphere outline is easy to redraw and one can get rid of unwanted too complex groupings with Extension > Arrange > Deep Ungroup
There's no numeric control for the mapping in MoI3D, the only controlling method is to change the sizes and forms of the objects.
In the next image the sphere is the same, but the curve is smaller. The square is even smaller, so the result is bigger on the sphere:
There's perspective, but it can be turned OFF in MoI3D. Inkscape saves as SVG. The original plane curves can be imported to MoI3D as PDF if one doesn't want to draw them in MoI3D.
As well as to Inkscape you can take the image to Illustrator which offers more coloring possibilities. For example there's no gradient along a path in Inkscape. But if the final result must be SVG, you must stick with things supported by the SVG format.
The presented method needs some carefully drawn guide lines if one wants to apply exactly the same transformation to different shapes. There's another answer which is based on advanced math scripting. I recommend to use it if perfect repeatability with different patterns is a must. My method can be useful if one wants to adjust shapes and the view purely visually