Firstly, I am afraid you are completely mistaken with "The Earth does technically "move away" from the observer exponentially... if it is round." NO it DOES not! Do you know what "exponentially means"? Obviously not. It means a variation with some constant, commonly e. So an exponential variation would have:
distance = (some constant) x e(height/constant).
Study up on your math a bit!
Elevation | Horz dist d = 3.57xh0.5 | Horz (Exp) |
0 m | 0.0 km | 0.7 km |
2 m | 5.0 km | 5.0 km |
5 m | 8.0 km | 101.4 km |
7 m | 9.4 km | 749.3 km |
10 m | 11.3 km | 15050.1 km |
An exponential variation and a square root variation are completely different animals (or whatever - they are as different as chalk and cheese!)
I'm so glad that Orbisect brought this thread back up. I almost missed how you tried to school me on math and exponential equations and completely blew it.
Now let me help you out rabinoz, I know you haven't been to school in about 40 or 50 years, and maybe things have changed since the 50's.
You really do have a sense of humour with your "Now let me help you out rabinoz", or more like a blown up sense of you own self-importance.And yes, I know a "square root variation" can be turned into a quadratic variation by changing the independent variable!
You're a bit out there as it happens, by a few decades, but things like this haven't changed in centuries!
A lot of "higher maths" has, but not this elementary stuff!In any case, after school, it was university, then teaching in electrical engineering. Things like logarithmic and exponential variations are rather vital in that field. In fact much of the work involves complex (as in complex numbers etc, you know the
x + iy stuff), plus all the stuff solving higher order polynomials (comes into finding the poles and zeros of linear network transfer functions) and numerical optimisation.
First the difference between something that is linear and exponential.
A linear equation represents a line that travels along a slope in a straight flat line.
Example of a straight line formula
Example of a straight line plotted on a Cartesian coordinate system:
Get on with it! Of course I know the elementary stuff.
An exponential function describes a line that travels in an increasingly curved line:
Graphed:
Yes, no problem with that! But the critical thing is that to be an "exponential variation"
the exponent must contain the independent variable! As does your equation
f(x) = bx.
Actually though it's more than an "travels in an increasingly curved line", a cubic equation does that, but an exponential variation "travels in an
exponentially increasingly curved line".
All the derivatives of the simple exponential function are themselves exponentials. I do hope you know what all this means!
A parametric equation, is one that can describe a curved line, it can be parabola, a circle, or countless other odd lines.
Example of the equation for the circle:
A circle on the same type of graph:
Sure, I guess any equation can be written in parametric form, and most can be written in non-parametric form, but often these have multi-valued solutions as the case for a circle where:
y = (r2 - x2)1/2. For each value of
x in the range
-r <= x <= +r there are 2 values of
y.
As you can see the equation for a straight line and something else is obviously very different. One has exponents, and one doesn't. In case you don't remember what an exponent is: y2<-- this is a exponent. Different types of lines will have different exponents, it's not always squared. But in our case, a curved line equation deals with squared variables. This is why you can "square" that cute little equation you gave and it will still technically resolve into the "8 inches per mile squared for a 6' tall observer" little ditty you see floating around.
Now you are starting to get screwed up. Just because an equation has exponents does not make the
variation exponential! A quadratic equation such as
f(x) = ax2 + bx + c contains an exponent, but since it is a constant exponent (2)
it does not represent an exponential variation. Technically, the horizon on a curved Earth would move away from you exponentially, in the sense that it doesn't move away from you in a linear fashion.
No, just because a variation is not linear does not make it an exponential variation. A quadratic variation is not linear but it not an exponential variation either.
So, "technically, the horizon on a curved Earth" definitely does not "move away from you exponentially". The variation can be
approximated by a quadratic variation (like the old 8" x miles
2 not bad to even hundreds of miles).
This variation
contains an exponent, 2, but is not an
exponential variation Maybe you should study up on your math a bit. Break out that trust abacus you used to use in Senior High.
Sure, I could be a little rusty on Laplace transforms, the more difficult trigonometric identities and complicated integration,
but trivial stuff like this is child's play. And, no I never used an abacus, though I gather an expert can do wonders on them. In high school we had to put up with log and trig tables. Then onto slide-rules (which really are a mechanical version of log tables) for multiplication, division and exponentiation and the good old "Mark I brain" for addition and subtraction and graph paper for plotting. When the HP-35 calculator was released everyone thought it was "magic", but I waited till 1973 (I happened to be at UCB and HP that year) and got an early HP-45 and have stuck with them ever since, through the HP-65, HP-41c and finally an HP-48 (which I no longer use). Of course since 1962 with the installation of the old GE-225 computer and later computers repetitive stuff was done of these bigger machines.
Now, strangely enough I do most calculation in Excel on a Windows PC, though an HP-41c emulator on Android phone and tablet is great for quick little sums - has all the nice functions for trig and all that, plus a bit of storage -
it does make a nice replacement for you little abacus!No, I for this stuff I really don't think I need much extra study, but I sure get bogged down on GR and stuff like that! I even struggle a bit with spherical geometry.
But, you most certainly
"should study up on your math a bit". Maybe try this - seems about you level:
Math Videos for Grade 8 Algebra 1, Exponential and Linear Functions Even Wikipedia would set you straight:
Exponential growth Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression.
The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is xt = x0(1+r)t where x0 is the value of x at time 0. For example, with a growth rate of r = 5% = 0.05, going from any integer value of time to the next integer causes x at the second time to be 5% larger than what it was at the previous time. Since the time variable, which is the input to this function, occurs as the exponent, this is an this is an exponential function.
From Wikipedia, the free encyclopedia, Exponential_growth
| Spacer | The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth. (red) linear growth (blue) cubic growth (green) exponential growth |
Hope I didn't make too many little typos in this!
But it's getting a bit bit late and you know how these
poor geriatrics need their sleep, especially after using
pick and shovel to fix up the wrecked garden we have!