What is the shape of planetary orbits?
Planetary orbits are approximately elliptical.
In the 17th century Johannes Kepler showed that planetary orbits are ellipses. Newton's laws of motion confirmed this.
Modelling planetary orbits as ellipses is quite accurate. In fact NASA publish the orbital parameters which define the ellipses for the orbits of the planets.
In actual fact, none of the planets' orbits are truly elliptical. The gravitational effects of the other planets constantly perturbs the orbits.
The orbit of the Earth is a good example; The time of perihelion, when the Earth is closest to the Sun should be about the same every year. It is getting later by about a day every 70 years due to precession, but it should follow a pattern. In fact perihelion can occur any time between the 2nd and 5th January and can vary by several days from year to year. The reason for this is the gravitational effects of the other planets cause the earth's orbit to change continuously.