If you have a parametric curve c(t) = (x(t), y(t)), it’s easy to see that for each parameter value t, the tangent of the curve is going to point towards dc/dt = (dx/dt, dy/dt). You can take a perpendicular vector to this tangent by rotating it 90 degrees, and the two vectors will form a so-called tangent frame at t.

If you want to bend your bitmap around this curve, the key idea is to transform the domain of the bitmap. That is, instead of putting your bitmap pixel at (x, y), you would put it at something like c(x) + (dc/dt)(y) (some scaling factors omitted for brevity).

(In practice, you would want to invert this mapping, and map the pixels around the curve back to the domain of the bitmap, to avoid holes, but that’s an implementation detail. The key idea is thinking in terms of warping the bitmap’s domain.)