For the Clebsch Diagonal Cubic (related to a Quanta mag article on Hilbert's 13th Problem), I want to generate a point at will on any of these lines along the surface. Wolfram calls these "Solomon's Seal Lines" but I've had much better luck searching "27 lines on a cubic surface", e.g. this analysis.
Despite having EngCalc~4, I am a bit overwhelmed by the (graduate-level) technical terms in the topic (and so...apologies if my terminology is also broken here). Are there explicit polynomials for these so I can iterate a point along a line across the surface (or even just hit an Eckardt point)? Page 186 in The Geometry of Cubic Hypersurfaces (PDF author Daniel Huybrechts) is the closest I've gotten to a definition of the lines.
If an application helps, my ultimate goal is to do a passive transformation / translation of axes to have the translated origin follow a line's path, so I can analyze coefficients under the new origin. Here's a (wip) Desmos 3D graph where I'm doing an origin translation (all subscript-2 items are passive, where $(h_{2}, i_{2}, j_{2})$ is the new passive origin (in green) and $coe_{2(x,y,z)}$ are the new coefficients for the same surface, but with respect to the new origin).
EDIT: In case comments expire: Added short Desmos 2D explanation of iterating a point (the alias origin) along a path, this method upscales to 4D and is what I want to do along the surface. Then I found how to plot [lines] on Clebsch surface (mathematica.se), with comments by "bmf" leading to a possible parameterization.
EDIT 2 Via the AnalyticPhysics model, I am able to move the aliased origin along one of the lines by setting (in the Desmos 3D linked above) $j_2 = \frac{1}{3}$ ; $i_2 = (\frac{2}{3} - h_2)$ and then sliding $h_2$ at will. Calculated $d_{istance} = 0$ always (so it stays on the surface) and it appears to be an instance of what I am seeking, but I am not able to write my own answer at this time.