How the Fast Animator Works

Introduction

Modelling the appearance of objects moving at relativistic speeds requires both the unusual colour effects, due to relativistic and mundane doppler shifting to be modelled, and also the geometric distortion due to the Lorenz contraction and the finite speed of light to be modelled.

This introduction looks at how the geometry of objects appears to an observer (or a camera), when the objects are whizzing by at relativistic speeds.

Lorenz Contraction

The Lorenz Contraction is the well known effect whereby objects moving swiftly in an observer's space are measured to contract in their direction of motion. (Things get flatter when they move quick.)

This is a Real effect - if you make simultaneous measurements of the back and front of a moving object with a stationary ruler, you will find the length of the object has shrunk.

The contraction is given by the Lorenz contraction formula:

                       /---------------
contraction =        \/ 1 - (V**2/C**2)

(where 'C' = speed of light,
       'V' = velocity of object
       '**2' =  'squared' )

Finite Speed of Light

In addition to the Lorenz contraction, if we are looking, or photographing, an object, we can see another effect. Imagine we are looking at a long train, travelling almost directly towards us. Light from the headlamp at the front of the train doesn't have as far to travel as light from the back of the train. So, light we see from the front of the train arrives earlier - we see a more recent view of the front of the train than the back. Since the train is travelling towards us, this more recent view must be closer - so the train appears to be stretched...

Now, if it wasn't for the Lorenz contraction, this would mean that fast moving objects would appear stretched and sheared - but the Lorenz contraction almost exactly balances out this effect - the shrinking and the stretching/shearing combine to look like the object is turning away from us (look at the buildings in the picture gallery). In addition to the turning however, there is a non-linear shear for objects not an infinite distance away.

Modelling the Shape of Objects

The upshot of all this is that if we wish to find where an object appears to be to an observer or a camera, we need to first apply the Lorenz contraction, and then, for every point on the object, work out where it appears to be to the observer, given its current velocity and position. The key equation turns out to be (hold onto your sliderule:)

-Given the current position (Pi = Px, Py, Pz) of a particle, and its velocity (Vi = Vx, Vy, Vz), this equation tells us how much time has passed since the light we are seeing now left the particle. (i.e. what the time difference in the observer's frame from the present, when the observer sees light from the particle, and the event of that light being emitted in the past.)

Knowing this, the apparent (visual) position of the point is simply:

Visual Pos = Actual_Pos - Delta_T * Velocity

(Where Delta_T is the quantity calculated above, the Actual_Pos is the current position of the particle, and the velocity is the velocity of the particle.)

All Clear Now?

What it all Means

- Simply that with one equation we can tell, given a particle's current position and velocity, where it appears to be. And since it's only one equation it's quite easy to program into a computer, and get reasonably swift animations out of it... However modelling colour is a different story...

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