Study of electricity/electromagnetic effects in space

I think the terminology derives from something like this ( * == multiply, ^ == exponentiate ):

( Some 'independent' variable ) z = co-efficient_A * ( x^2 ) + co-efficient_B * ( y^2 ) + const

The independence of the left hand variable is by 'deeming': meaning because we placed it there and have mathematically 'phrased' it that way. Algebraic manipulation can lead to other forms with x = whatever, or y = whatever.

However if co-efficient_A = 0 then we have:

z = co-efficient_B * ( y^2 ) + const

which is a parabola ( similiarly if co-efficient_B = 0 ) or 'parabolic' if you like.

If both co-efficient_A = 0 AND co-efficient_B = 0 then we have:

z = const

which is a straight line, or 'linear' if you like.

Now when neither co-efficient_A = 0 NOR co-efficient_B = 0 we have some complexity.

If co-efficient_A and co-efficient_B are equal then we can fiddle-faddle the above to get:

z = something_constant * ( (x - offset_x)^2 + (y - offset_y)^2 )

which when plotted is circular, with some centre point. This is the ellipse with zero eccentricity, otherwise known as the circle. Remember 'completing the square' on quadratics? The something_constant yields radius, and the centre from the offsets.

If co-efficient_A and co-efficient_B are non-equal, but of the same sign ( both positive or both negative ) then we can get:

z = something_constant * (x - offset_x)^2 + different_constant * (y - offset_y)^2

which is a true ellipse, non-zero eccentricity, 'centred' as per the offsets and those constants determining the relative 'squashing' of the ellipse axes from circular.

Now for the final lunge, co-efficient_A and co-efficient_B are of different signs, you get:

z = something_constant * (x - offset_x)^2 - different_constant * (y - offset_y)^2

which is an hyperbola, or 'hyperbolic'.

So by ramping our co-efficients through the full range of their possible combinations we have created several sub-geometries from the single more generalised equation.

So all those geometric terms refer to those combinations of algebraic terms ie. what have you multiplied the quadratic ( squared ) terms by?

Cheers, Mike.

( edit ) It occurs that I might be a tad confusing for some, compared to the 'high school' versions, as I have gone 'above the plane' by introducing the third variable 'z' - but hang loose as my explanation is meant to be generic for many dimensions. The overall idea is 'flat' == linear, 'curved back in on itself' == circular/ellipsoidal, 'curved but open' == hyperbolic, 'curved but just open' == 'parabolic'. Parabolic is the cusp condition on the exact verge between open and closed.

( edit ) I've just re-read the thread entirely. I'd add that one should distinguish between descriptions of geometrical findings within the space ( 'observers eye view' ), and those from without ( 'higher dimensional Gods eye view' ). We do the former with experiments, but modelling with the latter. So what I've labelled as 'parabolic' by algebraic description ( ie. on the cusp between open/closed ) without could be measured as 'flat' by geometric observation within ( say by finding included angles in an extended triangle adding up to exactly 180 degrees ). That's the big challenge with higher dimensional stuff, trying to analogise properly...