What relationship, if any do you see between the shape of the parallax path...

The shape of the parallax path of a star is closely related to its ecliptic latitude. Here’s how:

1. Stars at the Ecliptic Pole: For stars located at the ecliptic pole (90° latitude), the parallax path forms a circle. This is because the star’s position relative to the Earth’s orbit remains perpendicular throughout the year.

2. Stars at Intermediate Latitudes: As you move away from the ecliptic pole, the parallax path becomes an ellipse. The semi-major axis of the ellipse remains constant, but the semi-minor axis decreases as the ecliptic latitude decreases.

3.  Stars on the Ecliptic Plane: For stars located on the ecliptic plane (0° latitude), the parallax path flattens into a straight line. This is because the star’s position aligns with the plane of Earth’s orbit, causing the parallax motion to be linear.

In summary, the parallax path transitions from a circle to an ellipse to a straight line as the ecliptic latitude of the star decreases from 90° to 0°.