alan2here

I recently descovered a materal apparently called clinker. It's a deep purple color, a little reflective and has the apperance of ceramic, stone\brick and metal. It is irregular yet smooth and reminds me of expensive underused materals such as marble. It is however apparently free and considered always to be just a waste product.

A conventional house, made of bricks, wood, metal pipes etc... takes many professionals a long time to construct and so is expensive for labour costs and relies on there being people with these skills available. If the process was simplified but kept to resemble a conventional house, for example:

Bricks shaped like jigsaw pieces at either end requiring no cement.

Those bendy pipes made of many small segments like the sort you get for showers that screw on either end.

etc... such that the house could be built by anyone from simple instructions then do you think the extra cost of production would outweigh the labour cost saving?

I thought everything was like a float not an integer by default as it is algebra.

In this link. What do the #'s in the solution mean? Also there could be 0 to 4 solutions for given values of x, q, w, r, t and u. How do the equations provided return those up to 4 values?

Imagine standing on the equator (I forgot to mention that part) and you go east or west, your angle relative to the north pole is pi/2 and the same at all times. Now imagine standing anwhere and going north, again your angle to the north pole always stays at 0 untill you cross the north pole.

This time you are standing 10 meters from the north pole, you travel at an angle of pi/2 10 meters. You are now further away from the north pole but not as much further as if you have traveled directly away from it (angle of pi).

Your angle relative to the north pole is different from when you started, it is your final_angle. To be the same as the angle variable in this case the man would have had to move in a curve, but he didn't he moved in a straight line.

If your final angle is the facing direction of the man it is sort of irrelevant using the above equations. If he moves diagonally he is still facing north if he was already facing north. The only time he won't be facing north is when he is facing south when he crosses the pole?

Only on a map, not on a sphere or the "faulty_latitude_move" code would work. Continuing with the globe analogy he starts in England, turns to the north and moves some distance getting him close to the north pole, in this case his angle relative to the north pole would not change. If he were to move east or west putting him in the ocean or the direction of Russia the result would be the same.

However if he turns to face north east then moves you would not expect his angle relative to the north pole to stay the same. It's going to start at 1/8th of a turn clockwise and end up at 1/4 of a turn when he is north of Russia, imagine this happening on a globe.

How are you presenting this?

Very visually. It's going to be easy to see if it has worked.

inputs:
 sphere in space: x, y, z, radius
 man on sphere: latitude, longitude
 man's intention: angle, distance

faulty_latitude_move = sin(angle) * distance
faulty_longitude_move = cos(angle) * distance
latitude_move = arcsin(sin(angle) * sin(distance\radius))
longitude_move = arcsin(sin(angle) * sin(distance\radius))

final_latitude = latitude + latitude_move
final_longitude = longitude + longitude_move
final_z = z + (radius * sin(final_latitude))
final_x = x + (radius * cos(final_latitude) * cos(final_longitude))
final_y = y + (radius * cos(final_latitude) * sin(final_longitude))
final_angle = angle

outputs:
 man on sphere: final_latitude, final_longitude, final_angle
 man in space: final_z, final_x, final_y

Horray, see the old version labled faulty and the new version at the top.

Now only one more part to do:

final_angle = angle

Do 'a' and 'b' help the rest of the function at all?

btw the function should work for moving east around the sphere and north, it's only when moving diagonally that there may be a problem, which is that the man will curve towards the poles. The other problem is that final_angle will produce the wrong answer.

Corrected below.

inputs:
 sphere in space: x, y, z, radius
 man on sphere: latitude, longitude
 man's intention: angle, distance

a = arcsin(sin(angle) * sin(distance\radius))
b = arcsin(sin(angle) * sin(distance\radius))

final_latitude = latitude + (sin(angle) * distance)
final_longitude = longitude + (cos(angle) * distance)
final_z = z + (radius * sin(final_latitude))
final_x = x + (radius * cos(final_latitude) * cos(final_longitude))
final_y = y + (radius * cos(final_latitude) * sin(final_longitude))
final_angle = angle

outputs:
 man on sphere: final_latitude, final_longitude, final_angle
 man in space: final_z, final_x, final_y

angle is the angle the man is facing in radians relative to the north pole

final_angle is angle the man is facing in radians relative to the north pole after he has moved distance in angle direction

etc...

90 degrees is 1.57 radians, sin(1.57) = 1

Thanks, looks like I miscalculated that first time somehow.

I simplfied out the devide by 1.

How do I interface the parts together?

Green represents placeholder code that's not entirely accurate. Blue represents code that's not part of the forumuli yet.

sin_a = sin(angle) * sin(distance)

sin_b = sin(angle) * sin(distance)

a = 2 * pi * sin_a + arcsin(sin_a)

b = 2 * pi * sin_b + arcsin(sin_b)

o1 = a/2

o2 = b/2

final_latitude = latitude + (sin(angle) * distance)

final_longitude = longitude + (cos(angle) * distance)

final_z = radius * sin(final_latitude)

final_x = radius * cos(final_latitude) * cos(final_longitude)

final_y = radius * cos(final_latitude) * sin(final_longitude)

final_angle = angle

Cool, trigonometry is not that hard, the solution I preposed in my last post uses it.

travel 10000km 25º NE

 

you would use

 

[math]sin a = \frac{sin 25 \cdot sin 10000}{sin 90} [/math]

 

and

 

[math] sin b = \frac{sin 65 \cdot sin 10000}{sin 90} [/math]

 

[math] \theta_{radians} = a/earths radius [/math]

 

[math] \phi_{radians} = b/earths radius [/math]

 

where [math] \theta [/math] and [math] \phi [/math] are in radians

Ok, lets try that.

radius = 2

latitude = equator = 0.5

longitude = 0

angle = 0.3

distance = 1

I'm working in radians

[math]sin a = sin 0.3 * sin 1 = 0.25[/math]

[math]sin b = sin 1.27 * sin 1 = 0.8[/math]

Wolfram Alpha told me that "a = 2 * pi * n + sin^-1(n)".

sin^-1(n) = arcsin(n)?

a = 1.82

b = 5.95

o1 = 1.82/2 = 0.91

o2 = 5.95/2 = 2.98

convert to degrees and add to longitude and latitude accordingly and then convert spherical coordinates to Cartesian for Cartesian x, y, z values...

I'm not sure how to do this bit. Also nothing functions in degrees.

which values corispond to the required outputs?

Thanks. I may try Buttacups solution, it could take a long time to implement.

Buttacup, does your latest post add onto the post before or is it a solution in itself?

Whats the Euler rotational matrix, a component of Buttacups solution or a whole solution?

Will the following work instead?

final_latitude = latitude + (sin(angle) * distance)

final_longitude = longitude + (cos(angle) * distance)

final_z = radius * sin(final_latitude)

final_x = radius * cos(final_latitude) * cos(final_longitude)

final_y = radius * cos(final_latitude) * sin(final_longitude)

final_angle = angle

This should be unfortunatly distorted as it presumes the N, S, E & W you would get on a map not a sphere. But it seems braudly correct, provided the equation from here that it is based on is correct.

For this:

z (up & down) = r sin(lat)

x = r cos(lat) cos(long)

y = r cos(lat) sin(long)

(0, 0, 0) is not in the center of the sphere, it's on an edge? and the sphere moves left and right depending on the latitude chosen? Might not be a particularly hard problem but seems odd.

Merged post follows:

C++ std library. I have coded matrix multiplications before, but this is on just one point, in quite a simple way, where the point starts in a place defined by a simple equation before it is moved. I'm sure it doesn't need all the little things that could go wrong, take a long time to do, may require dependency on other libraries and are hard to test with matrix transformations. Especially as I don't have much of a maths background and this is the only part of the whole program that needs any part of the calculation.

If I must code up complicated matrix stuff is there one transformation matrix that I can use instead of 3 in a row?

Thanks, looks like it would make a good question for one studying maths, hard with various possible methods of doing it covering all sorts of stuff.

I can use this equation from the article to create a simple verstion of the problem.

x = r Cos[lat] Cos[long]

y = r Cos[lat] Sin[long]

z = r Sin[lat]

This will be ok as a placeholder for now. I tried to create a similar equation but it was not as good as my sphere was lemon shaped.

Merged post follows:

z = up\down

latitude of -pi*0.5 = north pole

latitude of pi*0.5 = south pole

longitude of 0 is the same place as a longitude of pi.

This is for a coding project I'm working on. It's not to do with men walking on spheres but it's a good way of describing the problem.

Can anyone produce equations, code or similar to calculate outputs for given inputs.

Inputs are:

x, y & z coordinates of the spheres center

radius of the sphere

start latitude coordinate of the man

start longitude coordinate of the man

start angle of the man

distance for the man to travel

For the following scenario:

Imagine the sphere centered at given coordinates where the north pole is at the topmost point. A man stands on the sphere at the latitude and longitude specified where 0 latitude is the north pole, 1 latitude is the south pole and ±1 longitude rotates east\west once around the sphere. He is facing a specified angle where 0 and 1 are facing the north pole, 0.5, 0.25 to the east, 0.75 to the west etc... he walks the specified distance around the sphere from his point of view in a straight line in the direction he is facing where 1 represents once round the largest part of the sphere such as the equator.

I then need to be able to get the following outputs:

final x, y & z coordinates of the man

final latitude coordinate of the man

final longitude coordinate of the man

final angle of the man

The mans start angle will definitely be equal to his final angle if the distance traveled is a multiple of 1 or he starts with a latitude of 0.5 and an angle of 0.25 or 0.75 representing him going either way round the equator. Similarly going away from or towards the poles without passing them would show no change in angle.

Four would be 100, or [math]a_2=1, a_{n \neq 2}=0[/math].

Thanks. It's a differnt way to describe binary numbers.

Merged post follows:

Just in case the OP still isn't clear on this, the expression of any natural number in any base [imath]b[/imath] expression in the form [imath]\sum_{n=0}^{\infty} a_n b^n \;| a_n \in [0\dots b-1][/imath].

Thats a verry confusing looking equation.

For instance, 5427 = 5x10³ + 4x10² + 2x10¹ + 7x10⁰.

Yes, this makes sence.

Thanks. I see now how the equation works now (b = base)

A = 542 = A₂*b² + A₁*b¹ + A₀*b⁰.

542 = [math]\sum[/math]A_nbⁿ

And the binomial co-efficients [imath]\binom{n}{k}= \,^{n}C_k \,=\frac{n!}{k!\,(n-k)!}[/imath] is the count of the amount of ways that, for instance, [imath]k[/imath] out of [imath]n[/imath] boxes can be ticked.

So an equation calculates how many differnt K of N combinations exist is. [imath]\frac{n!}{k!\,(n-k)!}[/imath]

edit: seems to work.

It's a step in that direction, however. Another step in that direction will be when they design "new" DNA (eg get a cell line to evolve or removing extra genes), and then they could synthesize this "new" DNA and insert it into a cell to make a "new" life form.

That can already be done. With functions smaller than genes, short sequences of base pairs put together to create a DNA strand that describes a functioning lifeform and implanted into a cell that is designed to loose it's identity and become as described by any DNA that it encounters.

Any (natural) number x can be written as [math]x = a_0 2^0 + a_1 2^1 + a_2 2^2 + a_3 2^3 + \dots = \sum_n a_n 2^n[/math]

I see, binomial coefficients? I've never come across this before. How would the number 4 be represented in this notation?

Merged post follows:

(from the thread title I thought you might be talking about digitally balanced numbers, which you might find interesting)

Thanks. This means that the central row of each one shows digitally balanced numbers, with varying degrees of imbalance towards the top and bottom.

This was fairly obviously as a result of you choosing numbers of the form [imath]\sum_n 1\cdot 2^n[/imath]

I think the [imath]\cdot[/imath] operator means multiply?

Note how there is a large frequency of primes in the second to last row, but it's not all but one.

It's every other one instead. Seems I was wrong, it does appear as if there is some pattern of prime distributions though. Seems to much of a coincidence.

Wolfram alpha shows that several rows including the middle one (digitally balanced numbers) produce a fractal like graph.

I'm confused by this.

Various TED talks, the video "DNA Hacking" as well as at least one "Futures in Biotech" episode all sugest that the following things have been possible for some time.

Reading DNA into a computer file.

Creating DNA from a computer file.

Implanting DNA into a cell that has been designed to morph to match the DNA.

The sending of samples to be read, DNA or a cell made from DNA of your choosing being sent by mail order.

Storing sequences as a "component" and combining them to abstract other components and more ambitious functionality.

The abbilty to look up any of these components online and get there base pair sequences.

Performing more complicated queries on archives of base pair series from animals including humans.

Projects to refactor functions found in DNA so that for example two functions don't overlap and to document them.

etc...

Which leaves me wondering is this somewhat old news that journalists have only just noticed? Or there is something new that was just to technical to elaborate on much.

Despite the talk of human error and terror etc... that the BBC seems to have made up it seems positive to inform the public of progress.

I should imagine that a building could be constructed like this to be extremely tall. Although given the angles of the cables they would have to be extremely long.

Given that the diameter of the earth is 12756.2km and the angles and heights of the cables proportional to the height of the building, one could work out the tallest possible building of this design.

The 2nd row shows powers of two due to construction: It contains all values [math]\sum_n a_n 2^n[/math]

The 2nd row is an ordered set where [math]s_{n}[/math] = [math]2^{n}[/math].

How is it the sum of something times by [math]2^{n}[/math] where n is the sum counter? Isn't that going to return a number?

for which exactly one [math]a_n[/math] is one and the others are zero. I must admit that I do not find it particularly interesting that you see an approximate pattern in the distribution of the primes. I think you could see an approximate prime pattern in any random arrangement of the numbers from 1 to 15.

Are you saying that for subsequent elements in the sequence this observation of primes wouldn't hold true?

Take the binary numbers of a cirtain length, for example.

00

01

10

11

Sort them into how many 1's they have.

00

01, 10

11

Write them in a more familiar form.

0

1, 2

3

And you have a pattern of numbers, which is more interesting with larger examples.

0

1

0

1, 2

3

0

1, 2, 4

3, 5, 6

7

0

1, 2, 4, 8

3, 5, 6, 9, 10, 12

7, 11, 13, 14

15

The first and last rows contain one element.

The 2nd row shows powers of 2.

More interestingly in all cases all but the last element of the 2nd to last row are prime.

Something reminded me of this remarkable building recently.

http://farm2.static.flickr.com/1053/1488454529_cc202cbd73.jpg

It must have an aspect ratio of a million to 1, with weight reasonably distributed along it, including at the top, and yet it stands.