I think about this sometimes, so I like the idea, but how do you define “straight” on an oblate spheroid? Great circle, constant direction (e.g. “due east”), or something else?

The mathematical field of Differential Geometry can answer this question precisely: https://en.wikipedia.org/wiki/Geodesic#Affine_geodesics

An oblate spheroid is an example of a Riemannian manifold: a smooth object that looks like a plane (or, in general, any ℝ^n) locally, and has a way to measure angles between vectors in that local plane.

All Riemannian manifolds have an object called the Levi-Cevita connection, which defines how vectors in the local plane (tangent space) most naturally map to vectors in other tangent spaces in the immediate neighborhood.

Standing at a point on the Earth and looking in a certain direction gives us 1) a point on the manifold, and 2) a direction in that point's tangent space.

We then take an infinitesimally small step forward, and apply the Levi-Cevita connection to get from the old tangent space to the (infinitesimally nearby) new tangent space, and repeat. This defines an ordinary differential equation. Integrating the differential equation gives us a curve through the manifold.

Within some neighborhood of the initial point, this curve is a geodesic, i.e. the shortest path between the initial point and all subsequent points on the curve. This matches our typical intuition of "straight".

(Disclaimer: I am currently learning about this topic, but am not an expert.)

edit: https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid goes into some interesting specifics about the results of this process on ellipsoids.

I went with great circles since that feels like the most “natural” straight line on a sphere — the path you’d walk if you just kept going forward without steering. You could define "straight" as a constant compass direction (I think it's called a "rhumb") -- that would look straight on a Mercator map but would actually require regular steering adjustments to maintain the bearing.

Probably not scientifically accurate or anything, but if you point somewhere, then "straight" is in that direction. I guess it'll loose accuracy as you get further and further in the distance of the direction, but probably for most people would be good enough for "straight in that direction" :)