Mr Smiley

Look at Mr Smiley: 🙄

Now some would simply say he's mad, but in fact Mr Smiley is always looking at his imaginary friend Pete, who exists in 3D (Mr Smiley has no trouble with imagined depth perception). Being a smiley, Mr Smiley has perfect eyesight, and keeps Pete in focus at all times; being imaginary, Pete can ignore relativity, and he tends to move towards and away from Mr Smiley at very high speeds, and is capable of teleportation. As you might have guessed, Pete's movements are periodic.

Other things to note about Mr Smiley:
His eyes are spherical, but the rest of Mr Smiley's body is flat, on a plane running through the centre of his eyes;
His eyes have black spots on the back of them, at opposite points to the pupils;
He can rotate his eyes to any angle at all, and at any speed (again, no relativity);

Now describe (in approximate terms) the motions of Pete, as imagined by Mr Smiley.

Note: this problem was conceived of at about 1 am, after some time looking at previous puzzles. I think it shows. 😵

Originally posted by Acolyte
Look at Mr Smiley: 🙄

Now some would simply say he's mad, but in fact Mr Smiley is always looking at his imaginary friend Pete, who exists in 3D (Mr Smiley has no trouble with imagined depth perception). Being a smiley, Mr Smiley has p ...[text shortened]... after some time looking at previous puzzles. I think it shows. 😵

Originally posted by econundrum
*gawks in silent perplexity*

Then states aloud: "what the he!!?"

Originally posted by DumDum81
I aggree, except I would say-WHAT THE @!#*ing H#%@ IS THAT!!!😠

Originally posted by Acolyte
Look at Mr Smiley: 🙄

Now some would simply say he's mad, but in fact Mr Smiley is always looking at his imaginary friend Pete, who exists in 3D (Mr Smiley has no trouble with imagined depth at about 1 am, after some time looking at previous puzzles. I think it shows. 😵

Originally posted by fearlessleader
draw line A connecting the centers of m smiley's eyes. draw line B on m smiley's plane perpendicular to line A. creat plane C perpendicular to smiley's plane intercecting at line B. on place C, draw line D such that the angle formd by lines B and D is 1/infinity. Pete starts infinetly far from Smiley on line D, and moves parrilel to line B, w/ vilo ...[text shortened]... then he dose the exact same thing he just did, but on the other side of smiley's plane. repeat.

Originally posted by fearlessleader
draw line A connecting the centers of m smiley's eyes. draw line B on m smiley's plane perpendicular to line A. creat plane C perpendicular to smiley's plane intercecting at line B. on place C, draw line D such that the angle formd by lines B and D is 1/infinity. Pete starts infinetly far from Smiley on line D, and moves parrilel to line B, w/ vilo ...[text shortened]... then he dose the exact same thing he just did, but on the other side of smiley's plane. repeat.

Originally posted by Acolyte
Pretty much. Stricly speaking, Mr Smiley's eyes never point directly upwards (if you look at where the centre of the pupils move), so Pete moves in something like a steep hyperbola, rather than a pair of lines.