Mark left by laser pulse on spaceship Bob is travelling in a spaceship which is 10 meters long at uniform speed of 0.6c relative to Michael who is located on earth. Bob's spaceship is length contracted and it looks 8 meters long in Michael's frame of reference. By principle of symmetry and reciprocity, Bob notices that everything in Michael's frame of reference is also length contracted.
Both Bob and Michael are equipped with laser guns which emit cylindrical laser pulses in respective frames of reference. Bob would see Michael's laser gun length contracted and its bore appears to be an ellipse rather than circular. We assume that the laser pulse remains collimated and when it hits the spaceship, it leaves a mark on its surface. 
Michael fires his laser gun. He sees a circular or cylindrical pulse moving toward the spaceship. Now, here is the troubling point. The pulse hits spaceship and leaves a mark. Would that mark appear to be an ellipse to Michael? If it does appear to be an ellipse to Michael, how could he explain it to himself that his circular pulse left an elliptical mark on the spaceship?
 A: Note that a "laser pulse" is a three-dimensional object. I'm assuming that your pulse is brief enough that the light pulse (in the rest frame of the emitter) has an aspect ratio like a coin, rather than a longer pulse from the same aperture where the volume of light would be shaped more like a cigar or an uncooked spaghetti strand.
I won't solve the problem for you entirely, because of our guidelines on homework-ish questions.  But I'll tell you the shape of the solution.  You have four events that have to be located in spacetime:


*

*The laser pulse is emitted from the forward ("bow," to a sailor) end of Michael's gun.

*The laser pulse is emitted from the aft ("astern"?) end of Michael's gun.

*The edge from #1 makes a mark towards the stern of Bob's spacecraft.

*The edge from #2 makes a mark towards the bow of Bob's spacecraft.


Both Michael and Bob agree that the ships are parallel.  In Michael's reference frame, events 1,2 are simultaneous, and events 3,4 are simultaneous.  But in Bob's reference frame, the two emission events are not simultaneous, because "simultaneous" only happens in one reference frame. Likewise, in Bob's reference frame, there is a non-zero delay between events 3,4.
You have to figure out what this delay is, and how it is related to the length contraction across the aperture of the laser emitter, in order to figure out the aspect ratio of the mark made by Michael's laser on Bob's ship.  Then you have an additional coordinate transformation to decide whether Michael sees the mark on Bob's ship as a circle or as an ellipse (and if as an ellipse, which way the long axis goes).
A: Actually @Rob provides good guidelines. However, we can do well without any intricate diagrams but some other details must be considered. The mark may be $\gamma$ times contracted, or $\gamma$ times stretched or even of the same round shape, that depend on which angle the “shooter” inclines his laser gun.
Let's consider the following scenarios:
The light pulse was received when Michael and Bob were at points of closest approach.
Let's say that Michael is “at rest” and fires laser pulse towards Bob at right angle towards direction of Bob’s motion. Micheal must do that some time before Bob's arrival to their (frame independent) points of closest approach since it takes time for the light to cover distance between spaceships. The edges from the laser gun simultaneously depart in the Michael’s frame and simultaneously (in Michael’s frame) hit Bob’s spacecraft. Due – to Lorentz contraction of the Bob’s spacecraft the mark would appear $\gamma$ times stretched in the Bob's frame. Michael can see himself the stretched mark if Bob finally stops.
Good to note, that laser pulse that was emitted at right angle in Michael’s frame due to aberration of light would approach Bob at oblique angle “from the front”.
Bob may give (at least) twofold explanation of the “stretched” mark:
1)  I was moving in the Michael’s frame and my spaceship was Lorentz- contracted; that’s why I see the mark is $\gamma$ times stretched;
2)  Michael was moving in my frame and the edges of the mark left his laser pointer at different moments and from different distance; that’s why I see the mark as stretched.
The light pulse was emitted when Michael and Bob were at points of closest approach.
Let's consider another setup - Michael considers that he is moving in the frame of the Bob himself and emits laser pulse with the aim to hit Bob. In this case, when both Michael and Bob are at (frame – independent) points of closest approach Michael fires his laser (so as to take into account aberration of light) backward and this laser pulse would approach Bob at right angle. In this case the mark would appear $\gamma$ times contracted.
This picture may also have twofold explanation and both Michael and Bob may speculate about simultaneity of events. 
Speaking about simultaneity I mean that every participant employs standard (Einstein’s) definition of simultaneity in his own "rest" frame.
It should be noted, that non - standard definition of simultaneity is also - self consistent.
One can draw some parallels between your problem and relativistic photography; for some details please look here or there
Simplified analysis is here
