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:

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:

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:

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...

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

I understand better now, how different values for the coefficients lead to different geometric possibilities, and hence the measurement of them is useful for evaluating different cosmological models (as with Omega, the ratio of the average mass density of the universe to the critical density, which is the amount of mass required to 'close' the universe or halt the expansion of it).

## I think the terminology

)

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...

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Thanks Mike! I understand

)

Thanks Mike!

I understand better now, how different values for the coefficients lead to different geometric possibilities, and hence the measurement of them is useful for evaluating different cosmological models (as with Omega, the ratio of the average mass density of the universe to the critical density, which is the amount of mass required to 'close' the universe or halt the expansion of it).