1. Fw: Distance between 2 points
- Posted by Ralf Nieuwenhuijsen <nieuwen at XS4ALL.NL>
Sep 29, 1998
-
Last edited Sep 30, 1998
Oops, this was only sent to Patrat himself, so I forwarded it to the list.
(and yes, Patrat, I cleaned it up a bit)
>Hello,
>Does any1 know how to calulate the distance and angle from
>point A to point B given their coordinates.
>Thanks
So who's gonna supply us, with the variable dimension routines for angles ?
I know trig, but Im a bit confused about the angles and how sin () and cos()
work in Euphoria.
Any one has any faster suggestions for the sum function ?
Or shouldnt we start another speed/optimization thread ?
function sum (object x)
integer result
if sequence (x) then
result = 0
for index = 1 to length(x) do
result = result + sum (x[index])
end for
return result
else
return x
end for
end function
function distance (sequence a, sequence b)
return sqrt(sum(power(a - b,2)))
end function
-- Voila!
? distance ({ 2, 34, 5} , {3, 5434, 23))
? distance ({2,2} , {4, 4})
? distance ({ 23, 2534, 234, 23, 23}, {24, 23, 32}) -- Last two dimensions
are variable.
Ralf Nieuwenhuijsen
nieuwen at xs4all.nl
"They say truth will set you free, but they dont say how deep the cut will
be, suspicion and paranoia takes its place, and I cant hide!" -- Ten Foot
Pole
2. Re: Fw: Distance between 2 points
Ralf Nieuwenhuijsen wrote:
> >Hello,
> >Does any1 know how to calulate the distance and angle from
> >point A to point B given their coordinates.
> >Thanks
> function sum (object x)
> integer result
> if sequence (x) then
> result = 0
> for index = 1 to length(x) do
> result = result + sum (x[index])
> end for
> return result
> else return x
> end for
> end function
> function distance (sequence a, sequence b)
> return sqrt(sum(power(a - b,2)))
> end function
try this instead. 1 function call vs 3, should be worlds
faster for atoms and sequences both, i think...
function distance(object a, object b)
return sqrt( (a*a)+(b*b) )
end function
and something that should be benchmarked against the
above 2 distance() functions as well should be:
function distance(object a, object b)
return power( (a*a)+(b*b),0.5 )
end function
to see if power is faster than sqrt.
will work on angle later...
--Hawke'
p.s. see carl? i learning :>
3. Re: Fw: Distance between 2 points
- Posted by Ralf Nieuwenhuijsen <nieuwen at XS4ALL.NL>
Sep 30, 1998
-
Last edited Oct 01, 1998
>try this instead. 1 function call vs 3, should be worlds
Yes, I could inline it also. I just loven another thread on the sum ()
function, it will add atoms it can find.
I could just have used the '+' operator, since Patrat was prolly refering to
a 2D field.
However, it is cleaner to use recursive functions. And to be dimension free,
who knows how many dimensions we are gonna explore in the future ? Dont want
no Y2K typ of problem with my software when we reach 4K 
Just kidding, I
dunno why I do this things... impulses maybe ?
>faster for atoms and sequences both, i think...
For both atoms and sequences ? Actually I think with atoms, it will should
return the difference. pos (a - b)
That is also the problem with your function here, you forget to substract.
> function distance(object a, object b)
> return sqrt( (a*a)+(b*b) )
> end function
I think you meant this:
-- Assuming 2D world:
function distance (sequence a, sequence b)
return sqrt ( ((a[1] - b[1]) * (a[1] - b[1])) + ((a[2] - b[2]) * (a[2] -
b[2])) )
end function
-- Assuming 3D world:
function distance (sequence a, sequence b)
return sqrt ( ((a[1] - b[1]) * (a[1] - b[1])) + ((a[2] - b[2]) * (a[2] -
b[2])) + ((a[3] - b[3]) * (a[3] - b[3])) )
end function
-- Assuming 1D world
function distance (atom a, atom b)
return sqrt ( (a - b) * (a - b)) -- in other words: the positive
difference between 'a' and 'b'
end function
>and something that should be benchmarked against the
>above 2 distance() functions as well should be:
> function distance(object a, object b)
> return power( (a*a)+(b*b),0.5 )
> end function
It might be less precize, because the result will always be a float because
you use .5
I wonder when Robert is gonna make calculations with floats convert back to
machine integers when possible.
>to see if power is faster than sqrt.
Something I dont expect. First of all, sqrt takes less arguments, and thus
needs less conversion. (if the argument passed is an float.. use the
sqrt_float routine, other wise the sqrt_sequence or the sqrt_integer
routine) .. for power I suspect there are internally 9 different routines.
(Robert, if i'm wrong please say so .. )
Off course a simple benchmark would tell wouldn't it ?
And now here's my fastest function I could come up with (allowing a number
of dimension ranging from 0 to eternity)
function distance (object a, object b)
if atom (a) and atom(b) then
if a > b then
return a-b
else
return b-a
end if
end if
a = power(a - b,2)
b = a[1]
for index = 2 to length(a) do
b = b + a[index]
end for
return sqrt (b)
end function
Now, how does this benchmark ?
Ralf
4. Re: Fw: Distance between 2 points
Ralf Nieuwenhuijsen wrote:
>...to be dimension free, who knows how many dimensions
>we are gonna explore in the future ?
agreed, wholeheartedly...
>That is also the problem with your function here,
>you forget to substract.
> function distance(object a, object b)
> return sqrt( (a*a)+(b*b) )
> end function
>I think you meant this:
actually, i was thinking of a pair of lists of
right triangle legs.... so i really shoulda
called it:
function hyp(object opp, object adj)
return sqrt( (opp*opp)+(adj*adj) )
end function
and then you throw a list of pairs of legs at it...
i figured doing it this way would make the
angle calc easier (as carl showed how to do that)
not only that, the theorem states:
c^2=a^2 + b^2 which is what i thought was getting
thrown at it in the first place...
using carl's and mine together:
function hyp2d(object opp, object adj)
return sqrt( (opp*opp)+(adj*adj) )
end function
function hyp3d(object x, object y, object z)
return sqrt( (x*x)+(y*y)+(z*z) )
end function
function FindDistanceAngle(sequence coor1, sequence coor2)
--accepts a pair of coordinate lists, in the form:
--{{x,y},{x,y}...{x,y}} OR {{x,y,z},{x,y,z}...{x,y,z}}
--and calculates the distance between them and the relative angle
--returns {distance,theta} where distance and theta
--are sequences equal to length(coor1) and theta is in radians.
--now, theta bears special note, that if it refers to a 2d
--world then theta is a sequence of atoms. if theta
--is to refer to a 3d world then theta is a sequence of
--sequences in the form {rotation,rise} where rotation
--is the angle along the x/y axis' and rise is the angle
--above or below that plane
integer len
sequence dist,theta
object opp,adj,x,y,z
len = length(coor1)
if len!=length(coor2) then
Error("unequal argument length ")
end if
dist=coor2-coor1
theta=repeat(0,len)
temp=dist[1]
if atom(temp) then
Error("got a 2 sided triangle handy???")
end if
if length(temp)=2 then
for i=1 to len do
temp=dist[i]
adj=temp[1] opp=temp[2]
--below is what i had in mind
dist[i] =hyp2d(opp,adj)
theta[i]=arctan(opp/adj)
end for
elsif length(temp)=3 then
for i=1 to len do
temp=dist[i]
x=temp[1] y=temp[2] z=temp[3]
dist[i] =hyp3d(x,y,z)
temp =arctan(y/x)
theta[i]={temp,arctan(z/temp)}
end for
else Error("trying to use the 4th dimension? ;)")
end if
return {dist,theta}
end function
>> function distance(object a, object b)
>> return power( (a*a)+(b*b),0.5 )
>> end function
>It might be less precize, because the result will always
>be a float because you use .5
yeah i was worried about that...
a 3-4-5 triangle becomes a 3-4-4.999999 triangle...
*blech*
>I wonder when Robert is gonna make calculations with
>floats convert back to machine integers when possible.
that could be handy...
>>to see if power is faster than sqrt.
>Something I dont expect.
ditto... but it would be interesting to see, no? :)
>First of all, sqrt takes less arguments, and thus
>needs less conversion.
ah, but was there actually another thread written for that,
OR!, does sqrt actually call power??? makes ya wonder eh? :)
> Off course a simple benchmark would tell wouldn't it ?
'tis why i suggested it compadre...
>And now here's my fastest function I could come
>up with (allowing a number of dimension ranging
>from 0 to eternity)
> function distance (object a, object b)
> if atom (a) and atom(b) then
> if a > b then
> return a-b
> else
> return b-a
> end if
> end if
--kill this below
> a = power(a - b,2)
--these two lines, instead of power(a-b,2) may
--be a lot faster... EU is optimized for them
a=a-b
a=a*a
--note: using a=(a-b)*(a-b) calculates (a-b) TWICE...
> b = a[1]
> for index = 2 to length(a) do
> b = b + a[index]
> end for
> return sqrt (b)
> end function
> Now, how does this benchmark ?
try both ways.. eh???
--Hawke'
5. Re: Fw: Distance between 2 points
[Code]
Pretty neat, Hawke.
I didnt mess with the angle, cuz I was wondering how I would do that for
multiple dimensions.
However you can easily calculate distance of two points defined in 5
dimensions.
And in one, is no problem also. Its just pos (a -b)
Could we get angles to work in any number of dimensions.
In theory, giving 2 dimensions, you will get {distance, angle}
Giving 3 dimensions should give you {distanze, xyangle, zangle }
And with 4 dimensions we should get {distanze, xyangle, zangle, more_angles}
And Hawke, I started to talk about sum ()
Lets try an optimization thread on sum ()
Here is the beginning function:
function sum (object x)
atom ret
if atom(x)
return x
elsis not length(x) then
return 0
end if
ret = sum (x[1])
for index = 2 to length(x) do
ret = ret + sum(x[index])
end for
return ret
end function
[Code snipped]
Oh yes, Hawke, I forgot all about that. You're right, I should have used the
a * a approuch.
Oh well, I think it will be the fastest.
The sum function above on the otherhand is just a simple one, it can be much
faster I think.
Ralf
6. Re: Fw: Distance between 2 points
On Thu, 1 Oct 1998, Ralf Nieuwenhuijsen wrote:
> And Hawke, I started to talk about sum ()
> Lets try an optimization thread on sum ()
> Here is the beginning function:
>
> function sum (object x)
> atom ret
> if atom(x)
> return x
> elsis not length(x) then
> return 0
> end if
>
> ret = sum (x[1])
> for index = 2 to length(x) do
> ret = ret + sum(x[index])
> end for
>
> return ret
> end function
Carl goes into overdrive:
function sum(object x)
--function product(object x)
atom out
out = 0
-- out = 1
if atom(x) then
return x
elsif length(x) then
for i = 1 to length(x) do
out = out + sum(x[i])
-- out = out * product(x[i])
end for
end if
return out
end function
1) Comments show changes for "product()" :)
2) This is best read in a syntax coloured environment ;)
However, that's a generic function.
For 1, 2 and 3d work it's simply (not much error checking though):
function sum(object x) -- x is 1, 2, or 3 in length() [or an atom]
x = x & {0,0}
return x[1] + x[2] + x[3]
end function
Happly codifilairingousness,
Carl
--
Carl R White
E-mail...: cyrek- at -bigfoot.com -- Remove the hyphens before mailing. Ta :)
Url......: http://www.bigfoot.com/~cyrek/
"Ykk rnyllaqur rgiokc cea nyemdok ymc giququezka caysgr." - B.Q.Vgesa
7. Re: Fw: Distance between 2 points
Ralf Nieuwenhuijsen wrote:
> Pretty neat, Hawke.
thanks. I 'preciate it.
>I didnt mess with the angle, cuz I was wondering
>how I would do that for multiple dimensions.
well, anything other than 2d or 3d rapidly becomes
metaphysical or undefined.
first, let's determine how many dimensions we actually
live within. let's divide the world into the smallest
practicle unit of space, and analyze that entity.
we can call it a particle. now a particle can easily
have a physical location that bears upon another
fixed particle location. that would give you your
delta {x,y,z} (Dxyz). good so far, right? well that
particle exists only during a specific time frame.
so there is another dimension of existence. time.
so now you have delta {time_born..time_died} (Dt)
well that's great you say, we have a time vector now.
here's the question that'll stump the big boys:
define the angle of vector Dt in relationship to
the angle of vector Dxyz.
now, there are a few more states of existence that
a particle may have. (oh,JOY!)
a particle can have spin (Ds),and that dimension
exists independently of Dxyz, but is dependent
upon Dt. so, define Ds as
delta{time_spin_began..time_spin_ceased,
speed&direction_of_rotation,
rotationalacceleration}
where direction is (-) for ccw and (+) for cw and speed
is rev/s and rotationalacceleration is +-rev/s^2.
now, tell me the angle of Ds in relation to Dxyz and Dt?
another plane of existence for a particle is temperature,
(Dk= delta kelvin) which is dependent upon Dt and Ds
(after all, the spin may be producing the Dk) but not
upon Dxyz.
there are several more:
1>a particle can be charged (+/-,amount)
2>a particle can be magnetized (amount)
3>a particle can be accelerating (+/-,amount)
4>a particle can be stable or decaying
and maybe not last, defnly not least:
5>a particle can be emitting (radioactive)
(typeofemission,amount)
so now we need definitions of vector angles for 1..5
as well, for a total of how many dimensions of
existence for a particle?
if you can determine the angular definitions for all
these states, we can include them in the program.
if you cannot, i suggest we leave it with 2d/3d.
>However you can easily calculate distance of two
>points defined in 5 dimensions.
>And in one, is no problem also. Its just pos (a -b)
*distance*, yes. *angle*, no.
so i left out the definition of FindDistanceAngle
for 1d, since it doesnt describe a right-triangle.
we could define it as {pos(a-b),0} if you like.
for 5d, yes, distance=no prob. *angle*=BIG prob.
>Could we get angles to work in any number of dimensions.
>In theory, giving 2 dimensions, you will get {distance,angle}
>Giving 3 dimensions should give you {distanze,xyangle,zangle}
>And with 4 dimensions we should get
> {distanze,xyangle,zangle,more_angles}
define more_angles, as per above???
> Oh yes, Hawke, I forgot all about that.
'tis ok :) you've corrected me, politely, many times
when i fergot sumfin... i call 'em 'brainfarts'. :)
speaking of forgetting:
<code snippet, previous post>
if length(temp)=2 then
for i=1 to len do
temp=dist[i]
adj=temp[1] opp=temp[2]
--below is what i had in mind
dist[i] =hyp2d(opp,adj)
--***look here:
theta[i]=arctan(opp/adj)
end for
ummmmm what should we do if adj is 0?
if it's zero, theta is *not* zero, in theory,
theta is *undefined*, is it not? :>
*thmoke! thmoke! i thee thmoke!*
--Hawke'
8. Re: Fw: Distance between 2 points
- Posted by Ralf Nieuwenhuijsen <nieuwen at XS4ALL.NL>
Oct 01, 1998
-
Last edited Oct 02, 1998
>if you cannot, i suggest we leave it with 2d/3d.
Well, at least time (4D) is a nice dimension to include. Distance is usually
not really *that* important. Its also, how many time you have to get there
ANd I think, time as 4th (or actually as first) dimension is pretty
accepted, after that it becomes even more speculative.. (maybe, within a
game, multiple ways a game could go, an enemy moving forward in some
dimension, whereby he responds differently to things.
>>However you can easily calculate distance of two
>>points defined in 5 dimensions.
>>And in one, is no problem also. Its just pos (a -b)
>*distance*, yes. *angle*, no.
Angle, either 0 or 180 for 1D. Is it moving forward or backward ?
This would make it for 1D equal to:
{ compare (a,b) * (b - a), 180 + compare (a, b) * 180 }
Or:
{ pos (a-b), (pos (a-b) = a - b) * 180 }
Hmm, im on a roll, I feel.
>so i left out the definition of FindDistanceAngle
>for 1d, since it doesnt describe a right-triangle.
What do right-triangles have to do with this ?
In practicle theory 12323123^0 = 0, however consistency (and much smarter
people as well make us use 12312329^0 = 1
However, real logic applied on this example (a power of zero) would give
zero.
Same goes for the distance and angle in the 1st dimension.
You cannot really set the angle in practicle logic, but if you are
consistent, you can. It still has a choice of direction, and therefor an
angle of direction
>>Could we get angles to work in any number of dimensions.
>>In theory, giving 2 dimensions, you will get {distance,angle}
>>Giving 3 dimensions should give you {distanze,xyangle,zangle}
>>And with 4 dimensions we should get
>> {distanze,xyangle,zangle,more_angles}
>define more_angles, as per above???
>> Oh yes, Hawke, I forgot all about that.
>'tis ok :) you've corrected me, politely, many times
>when i fergot sumfin... i call 'em 'brainfarts'. :)
Good name, smelly though.
>ummmmm what should we do if adj is 0?
>if it's zero, theta is *not* zero, in theory,
>theta is *undefined*, is it not? :>
Angle would be 0 or 180 wouldnt it be ?
Well, here is my new relation (between two points) function:
function relat (object a, object b)
sequence angles
atom dist
if atom (a) and atom(b) then
if a >= b then
return { a-b, 0}
else
return { b-a, 1.5708}
-- Any1 wants to replace 1.5708 with something more precize ?
end if
end if
a = a - b
angles = {arctan(a[1]/a[2])}
b = a * a
dist = b[1]
for index = 2 to length(a) do
dist = dist + b[index]
angles = append(angles, arctan(sqrt(dist), b[index])
end for
return prepend(angles, sqrt(dist))
end function
-- Voila (wow je parle francais! Or not.. haha..oh well .. )
Have fun all,
Ralf
9. Re: Fw: Distance between 2 points
----Original Message Follows----
>well, anything other than 2d or 3d rapidly becomes
>metaphysical or undefined.
Actually, I'd say it becomes 'abstract'. It's just as valid
algebraicly, but doesn't really mean anything to anyone living in 3
dimensions of space. In fact, a lot of the stuff mentioned below
(temperature, electrical charge) doesn't have anything to do with a
location in space at all.
>here's the question that'll stump the big boys:
> define the angle of vector Dt in relationship to
> the angle of vector Dxyz.
It's the same relationship that Dz has to Dxy. Again, we can't really
visualize this, but it makes sense.
>now, there are a few more states of existence that
>a particle may have. (oh,JOY!)
>a particle can have spin (Ds),and that dimension
>exists independently of Dxyz, but is dependent
>upon Dt. so, define Ds as
> delta{time_spin_began..time_spin_ceased,
> speed&direction_of_rotation,
> rotationalacceleration}
I don't think that you'd want to call this a new dimension. Have to
check with my calc books, but I think the best way to talk about this
would be in some sort of a paramaterization. I think that's how it's
usually dealt with.
> 1>a particle can be charged (+/-,amount)
> 2>a particle can be magnetized (amount)
> 3>a particle can be accelerating (+/-,amount)
> 4>a particle can be stable or decaying
>and maybe not last, defnly not least:
> 5>a particle can be emitting (radioactive)
> (typeofemission,amount)
>so now we need definitions of vector angles for 1..5
>as well, for a total of how many dimensions of
>existence for a particle?
>if you can determine the angular definitions for all
>these states, we can include them in the program.
>if you cannot, i suggest we leave it with 2d/3d.
Any time you're talking about an angle, you're dealing with something
that exists in a certain plane (2d). We've already said (I think
someone did), that to represent an angle in 3-space, we need two
figures. Well, for each new dimension, we just add another angle. A
computer can handle 5 dimensions as easily as 2 or 3, even though we
can't (by visualization).
>>However you can easily calculate distance of two
>>points defined in 5 dimensions.
>>And in one, is no problem also. Its just pos (a -b)
>*distance*, yes. *angle*, no.
>for 5d, yes, distance=no prob. *angle*=BIG prob.
All you're doing for distances is breaking down the components in the
proper axes and computing. That's all you need to do for the angles.
>>Could we get angles to work in any number of dimensions.
>>In theory, giving 2 dimensions, you will get {distance,angle}
>>Giving 3 dimensions should give you {distanze,xyangle,zangle}
>>And with 4 dimensions we should get
>> {distanze,xyangle,zangle,more_angles}
>define more_angles, as per above???
Well, I don't know why it might be useful to know the angle between
something's location and its temperature, say, but that's not to say
that it's difficult (or impossible) to calculate.
But I bet if you dug around in some psychology journals, you'd find a
lot of studies that used many dimensions. Stuff like personality tests
and the like. Could be all sorts of metrics where this type of analysis
might make sense.
There's my two cents. :)
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10. Re: Fw: Distance between 2 points
>ANd I think, time as 4th (or actually as first) dimension is pretty
>accepted, after that it becomes even more speculative..
Are you sure you aren't Albert Einstein?
Did you know that...
...If you were to travel on a ship at the speed of light, your twin
brother stayed on earth, and 90 years had gone by, you're brother would
be dead and you would be the same age as when you left.
___________________________
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Web users: <A HREF=mailto:"matt1421 at juno.com">matt1421 at juno.com</A> or <A
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11. Re: Fw: Distance between 2 points
- Posted by Hawke <mdeland at NWINFO.NET>
Oct 01, 1998
-
Last edited Oct 02, 1998
Ralf Nieuwenhuijsen wrote:
> What do right-triangles have to do with this ?
errr.... everything... your using the pythagorean
theorem to determine a hypoteneuse(sp?) of the legs
of a right triangle to determine distance and
angle... ummm... where do you think the arctan()
part of this came from??? *shrug*
>>ummmmm what should we do if adj is 0?
>>if it's zero, theta is *not* zero, in theory,
>>theta is *undefined*, is it not? :>
> Angle would be 0 or 180 wouldnt it be ?
>snip
> a = a - b
> angles = {arctan(a[1]/a[2])} ***look here
> b = a * a
ummmm this is what i was trying to point out to ya :)
if in your function, a[2] is 0, which in my function
would have been adj(as above), then division by
0 is *undefined* and you can read that as **CRASH**.
and the adj (a[2]) can most definitely turn up
on occassion as 0, which would describe a straight
line vector along one of your axis.
so i ask again, if adj (a[2]) is 0, what would you
like done with it, and the subsequent calculation
of theta, which at that point is ALSO _undefined_...
(if division by zero is undefined, then any result
based upon that calcuation is undefined)
see, theta being zero is *already* defined if opp (a[1])
is 0 as you would get 0/num which is 0, so that would
describe one of your two vectors along the respective
axis that you have sent to the function.
you cannot use 180 since that would indicate (-) direction
along the axis we define if either opp (a[1]) is 0
(giving you 0/adj) or theta actually winds up as 0.
since 0 and 180 are taken, and we are looking at a
truly _undefined_ value in the first place...
things get ugly, which is why i phrased my question
the way i did in the first place :)
see? :)
--Hawke' the headache giver :)
12. Re: Fw: Distance between 2 points
My two cents on higher than 3 dimensions . . . (Don't worry, I don't
intend on a whole bunch of posts on this subject.)
>>well, anything other than 2d or 3d rapidly becomes
>>metaphysical or undefined.
>
>Actually, I'd say it becomes 'abstract'. It's just as valid
>algebraicly, but doesn't really mean anything to anyone living in 3
>dimensions of space. In fact, a lot of the stuff mentioned below
>(temperature, electrical charge) doesn't have anything to do with a
>location in space at all.
<snip>
>Any time you're talking about an angle, you're dealing with something
>that exists in a certain plane (2d). We've already said (I think
>someone did), that to represent an angle in 3-space, we need two
>figures. Well, for each new dimension, we just add another angle. A
>computer can handle 5 dimensions as easily as 2 or 3, even though we
>can't (by visualization).
As far as what the 4th diminsion would be (and beyond), maybe this would
help. This is how I see it (I could be wrong with 4d and above, but it
make sense to me! :)
1d = Something like morse code: --- --- - --- - - --- -- -
2d = Lots of something like morse code stacked together to make a
picture.
(With holes in it)
3d = Lots of pictures stacked together to make reality.
4d = Lots of realities stacked together. I guess this could be either
time
or parallel universes. (ie forward/back in time as another way of
moving.)
5d = Well, if time travel becomes possible, then this would be parallel
universes. You can travel back/forward in time in them, and between
universes. (Com'n, haven't you ever seen Sliders or an episode of
Star
Trek dealing with 'em? :) (Lots of timelines stacked together)
6d = Uhhhh... Lots of parallel universe groups stacked together.
(Perpendicular
universes?
7d = Lots of whatever 6d is stacked together.
And so on.
Well, I've got a big enough headache now . . . :)
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13. Re: Fw: Distance between 2 points
Robert B Pilkington wrote:
> 1d = Something like morse code: --- --- - --- - - --- -- -
> 2d = Lots of something like morse code stacked together
> 3d = Lots of pictures stacked together to make reality.
> 4d = Lots of realities stacked together.
let's go ahead and say that 4d is time...
then time, as per you, would be visualized as lots of
"rubik's cubes" that were lined up in a row. like a
child lining up a row of toy blocks. this could work.
each block represented a reality at a moment in time,
and from one point within one block to another point
in another block we could define both angle and distance.
> 5d = then this would be parallel universes.
this could be row's of 4d blocks stacked line by line on
top of each other... that could work...
this is about the point your head starts to hurt...
> And so on.
using your methodolgy, it becomes easier to visualize
'the problem'. after all, isn't coding really nothing
more than typing, once 'the problem' is both visualized
and solved??? implementation is near nada from that point.
thanks for your 'analogy', it helped me understand what
y'all were after in determining distance & angle for
any dimensional limits. a generic algorithm for distance
in any dimension has been provided, but, i'm still having
trouble working out theta for any dimension...
gonna go take a big big aspirin before i try :)
--Hawke'
