Both the equator and the ecliptic are great circles on the celestial sphere.
The appearance of each on a map depends on the map projection.
In an equirectangular projection centered on the equator, the equator (brown) is a straight line, and the ecliptic (blue) is a sinusoid.
If the same projection is centered on the ecliptic instead, the ecliptic is a straight line and the equator is a sinusoid.
Lucey provides both views too, but his ecliptic-centered map doesn't show the equator.
The axes are labeled in ecliptic rather than equatorial coordinates.
He also provides a stereographic projection centered on the midday zenith at 55°N.
Here both equator (red) and ecliptic (green) are mapped as circular arcs.