Parametric plot doesn't work as expected

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I'm working on some visualizations. In fact, I'd like to vizualize lines and curves on the projective plane. I started with something very easy: draw a line on sphere. I figured out the following.

Compute intersection of a plane with the unit sphere:

sage: x, y, z = var('x y z')
sage: sol = solve([x + y + z == 0.0, x^2 + y^2 + z^2 == 1.0], y, z)
sage: sol
[[y == -1/2*x - 1/2*sqrt(-3*x^2 + 2), z == -1/2*x + 1/2*sqrt(-3*x^2 + 2)],
 [y == -1/2*x + 1/2*sqrt(-3*x^2 + 2), z == -1/2*x - 1/2*sqrt(-3*x^2 + 2)]]

Draw the plane, the sphere, and their intersection:

sage: p = implicit_plot3d(x + y + z, (x, -5, 5), (y, -5, 5), (z, -5, 5), color="blue", opacity=0.2)
sage: p += implicit_plot3d(x^2 + y^2 + z^2 - 1, (x, -5, 5), (y, -5, 5), (z, -5, 5), color="red", opacity=0.4)
sage: for s in sol: 
....:     p += parametric_plot3d([x, y.subs(s[0]), z.subs(s[1])], (x, -5.0, 5.0), color="green")
sage: show(p)

I believe the computation is correct.

However I'm not satisfied with the resulting plot. Instead of a smooth line (like a tilted equator) I have strange "triangles" at opposite points of the sphere.

What can I do about it?