I'm looking for an easy method to create a perfect circular or elliptical motion without using "Follow Path" modifier or curves, extra addons, or many keyframes.
Another option is using drivers.
Drivers create an animation fcurve. The name
frame is the scene current frame.
To the icospere in gif am typing a driver expression into property field. If it's not purple (a driver already starting with a hash
# indicates "make this a driver expression".
Most of the usual suspect math methods are available, including
In the location x
sin(frame / 24)
and in z
cos(frame / 24)
The smaller icosphere has x location driven by
sin((frame - 8) / 24) / 2
ie it is delayed by 8 frames and halved in amplitude._
In answer to How to do Trammel Of Archimedes Animation? I mention some other things to consider in expression eg
sin(2 * pi * frame / fps) where
fps is a driver variable with value of frame rate, this will rotate at 1 rev per second.
Any of phase, amplitude, offset etc can be taken from the value of another property, which itself could be animated.
Added three empties, the first is displayed as a circle, which at default is a unit circle Have chosen to use the $XZ$ plane to match this for demo purposes. As stipulated in question it's not a path, although as in trammel link, do like to use fixed offest alternative
The next two are parented to first (or if you prefer, made children of first) and one given the X driver, the other the Z
The unit circle in local space. Using object scale will change our world space,
the cubes in the GIF have copy location constraints, to get X from X empty and Y from Y empty. Using WORLD space
The parent empty circle displays the maximum path. Non uniformly Scaling the parent empty will make path elliptical
Can also make a cube path elliptical by setting a different constraint influence between X and Y.
Smaller cube has half influence for both, same again on smaller cube.
A fun example is a spriral ellliptical path by animating the growing scale.
This method uses only a single keyframe and the graph editor modifiers. It can be done using those 4 easy steps:
If you want your object to orbit around a specific axis, you add the modifiers to the other two axes.
To get a perfect circular motion, both modifiers must have the same values for Amplitude & Phase Multiple.
Both modifiers are the built-in function but one of them has to be sine and the other is cosine.
If the two modifiers have different Amplitude values, this will create an elliptical motion.
For video tutorial: https://youtu.be/lhag5CrSgUw