The Earth's orbital speed around the Sun is about 30 km/s. So we can calculate the intersection of two spheres offset by 30 km. Suppose we put the first sphere at the origin [0,0,0], and the second sphere at [30,0,0]. That is, the x-axis is aligned with the orbital velocity vector.
Then, from Wolfram, the intersection of the spheres is a curve lying parallel to the z-y plane at a single x-coordinate ($x=d/2$), described by $$y^2+z^2=r^2-d^2/4$$ where $d$ is the distance between the spheres, and $r$ is the radius of the spheres. So, you can find specific points by varying $\theta$ between $0$ and $2\pi$ with:
In words instead of equations: only a subset of points on the Earth's surface will be in the same place in a Sun-centered coordinate system one second later. These points will be on a plane perpendicular to the orbital velocity vector, and offset from the center of the Earth in the direction of the orbital velocity vector by 15 km. One second later, a new set of points will occupy the same space. The new set of points will also be in a plane perpendicular to the orbital velocity vector, but now offset opposite the direction of the orbital velocity vector by 15 km. Here is a conceptual picture Not To Scale.
Do you want to account for the revolution of the Earth during that time (a point on the equator moves about .46 km in a second)? Then apply a rotation prior to calculating the latter set of points using a rotation matrix from axis and angle where your angle is $2\pi/86,400$ radians, and your axis is a unit vector specifying the orientation of the Earth's rotational axis from a Sun-centered coordinate system. At summer solstice, this is about $[0,-\sin(23.5),\cos(23.5)]$.
- The Earth's orbital path doesn't curve over the order of a second.
- The Earth is a perfect sphere.
- You could get a higher fidelity model using an orbital simulator like Universe Sandbox.
- You could get a higher fidelity model by using DTED data instead of assuming the Earth is a sphere. Then you might have to use a numerical engine to find point intersections.