3 Moved image away from left margin.

I hesitate to suggest this because you already mentioned Zoll surfaces. But for what it's worth, in Besse's book, Manifolds All of Whose Geodesics Are Closed, (Ergebnisse der Mathematik und ihrer Grenzgebiete, 93. Berlin: Springer-Verlag, 1978), Section D of Chapter 4, he gives an explicit embedding into $\mathbb{R}^3$ of a Zoll surface of revolution via parametric equations $\lbrace x,y,z\rbrace (r,\theta)$, and computes the cut locus from a particular point (it takes the shape of a 'Y').

Edit. Taking Bill Thurston's point re a "graphical representation of the distance function, and/or diagrams" to heart, I found this elegant image of the Zoll cut locus in the paper "Thaw: A Tool for Approximating Cut Loci on a Triangulation of a Surface" by Jin-ichi Itoh and Robert Sinclair, Experiment. Math. Volume 13, Issue 3 (2004), 309-325:

2 Added image of the cut locus from a paper by Itoh & Sinclair.

I hesitate to suggest this because you already mentioned Zoll surfaces. But for what it's worth, in Besse's book, Manifolds All of Whose Geodesics Are Closed, (Ergebnisse der Mathematik und ihrer Grenzgebiete, 93. Berlin: Springer-Verlag, 1978), Section D of Chapter 4, he gives an explicit embedding into $\mathbb{R}^3$ of a Zoll surface of revolution via parametric equations $\lbrace x,y,z\rbrace (r,\theta)$, and computes the cut locus from a particular point (it takes the shape of a 'Y').

Edit. Taking Bill Thurston's point re a "graphical representation of the distance function, and/or diagrams" to heart, I found this elegant image of the Zoll cut locus in the paper "Thaw: A Tool for Approximating Cut Loci on a Triangulation of a Surface" by Jin-ichi Itoh and Robert Sinclair, Experiment. Math. Volume 13, Issue 3 (2004), 309-325:

1

I hesitate to suggest this because you already mentioned Zoll surfaces. But for what it's worth, in Besse's book, Manifolds All of Whose Geodesics Are Closed, (Ergebnisse der Mathematik und ihrer Grenzgebiete, 93. Berlin: Springer-Verlag, 1978), Section D of Chapter 4, he gives an explicit embedding into $\mathbb{R}^3$ of a Zoll surface of revolution via parametric equations $\lbrace x,y,z\rbrace (r,\theta)$, and computes the cut locus from a particular point (it takes the shape of a 'Y').