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►Full answer (spoilers)
The path traced out by the tip of the shadow can be...
1) A circle (if you are on one of the poles during their summer)
2) An ellipse (if you are in the Arctic or Antarctic circle, but not on either pole, during a day where the sun doesn't set)
3) A line (this happens if you are anywhere on Earth besides the poles, during either the Spring or Autumn equinox)
4) A parabola (this happens if you are in/on the Arctic or Antarctic circle during a day where the sun BARELY sets, meaning it sets and rises in the exact same location. This happens twice a year for any point in those circles. If you are ON one of the circles, then it will happen during the Summer or Winter solstice for the Arctic and Antarctic circles respectively).
5) A hyperbola (If you aren't in one of the places listed above on those specific days, then you'll get this shape. This is what happens at most places on Earth throughout most of the year).

►Resources
Interactive tool: https://www.geogebra.org/m/AHEgBVUd
Video showing linear path on Equinox:    • Equinox 20th March 2021 - simple sund...  

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