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# Understanding 1d Perlin Noise

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#1
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Posted 08 August 2011 - 10:33 AM

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#2
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Posted 08 August 2011 - 10:37 AM

For your example, if the point at 0 is assigned 0.3 and the point at one is assigned at 0.9, then the point at 0.5 would be equal to 0.3 + 0.5*(0.9-0.3) or 0.6.

A more general explanation:

If the input coordinate is called P, you can calculate the two bounding integral boundaries by:

P1=floor(P) P2=P1+1

Then you call the function to obtain the values for P1 and P2. The interpolator is calculated as:

interp=P-P1

And the final interpolation (using a linear interpolation) is done as:

value=P1 + interp*(P2-P1)

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#3
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Posted 08 August 2011 - 02:34 PM

To look at it in a way that's more applicable to what I'm trying to do, if I have a map that's 1000 units wide would I start out by assigning a random value to points at a specific distance apart (let's say every 100 spaces), then use an interpolation function to find the value of every point between those (1 to 99, 101 to 199, etc), and then finally plug every single point (0 to 1000) into the main Perlin Noise function? The code structure would look something like the following:

- Assign a random variable every 100 units, starting with 0 and working toward 1000.
- Loop through every single point and give it a value based on the interpolation of the value immediately before and after it. Do this until the maximum width is reached.
- Plug every integer value (0 to 1000) into the main Perlin Noise function. The value returned by this replaces the previous value for this point.

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#4
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Posted 08 August 2011 - 02:52 PM

The typical means I use to fill an Array from a given Perlin function is as follows:

RegionWidth: The width of the 2D region of the function to map to the grid RegionHeight: The height of the 2D region of the function to map to the grid MapWidth: The width of the 2D grid MapHeight: The height of the 2D grid PerlinFunc: 2D Perlin noise function for x=0,MapWidth-1,1 do for y=0,MapHeight-1,1 do local nx=x/MapWidth local ny=y/MapWidth nx=nx*RegionWidth ny=ny*RegionHeight Array[x][y]=PerlinFunc:get(nx,ny) end end

Doing the mapping in this way allows me to easily specify a different size of area to map to the grid, and doesn't require multiple iterating of the grid to generate intermediate values.

I think that one of your misconception is that the values for the grid integer locations of the Perlin noise function need to be stored somewhere. They do not, but are simply calculated on the fly whenever they are needed. The Perlin noise function is basically a purely mathematical entity that doesn't store state, but merely calculates values on the fly as it is called.

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#5
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Posted 08 August 2011 - 03:51 PM

Your way seems to me to indicate a fundamental misunderstanding.

Yep, and that's putting it lightly haha!

Going off of your example, in the case of a 1d array, would the X input for the Perlin Noise function simply be its point in the array (0 to 1000)? If you're trying to create terrain for an entire map why would the region width be different from the map width (you may just be using the function differently than I am)?

Now what lead me to believe I needed to assign a value to each point along the way was the fact that in the example code on the page I linked above he has X as a float. I assumed this to be the random value from 0 to 1. Would it be more accurate then to say that the following is true, or is that random value assigned elsewhere in the code?

Array[x]=PerlinFunc:get(RandomNoiseFunc(nx))

If this is the case it brings me back to my earlier question since from my understanding at this point you need to run the Perlin Noise function for every single point on the X axis, and each of these points is given a unique value from 0 to 1. My terrain is going to be block based, so I'm concerned that by randomly assigning heights to every single point (representing the location of a block) it will vary wildly rather than be a smooth transition. Will the combined result of the different amplitudes/frequencies result in a smooth transition?

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#6
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Posted 08 August 2011 - 04:09 PM

By using a different region width, one that is much smaller than array width, you are in essences zooming into the function so that the features of it will be spread out across several array elements.

Also, your code:

Array[x] = Perlin:get(RandomNoiseFunc(nx))

shows that you still don't quite get it. A Perlin function is self-contained; the random noise generation is part of it.

for x=0,MapWidth-1,1 do local nx=x/MapWidth nx=nx*RegionWidth Array[x]=Perlin(nx) end

So if you had a RegionWidth of 3, in essence this would fill your array with what amounts to approximately three hills or valleys across. The reason the noise function input is a float is because the function is defined for any input point, not just integral input coordinates. Inputs that include a fractional part are evaluated by interpolating the enclosing integral-boundary values.

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#7
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Posted 08 August 2011 - 08:15 PM

In the following function, what is the purpose of changing the values into integers?

function InterpolatedNoise_1 (float x) integer_X = int(x) fractional_X = x - integer_X v1 = SmoothedNoise1(integer_X) v2 = SmoothedNoise1(integer_X + 1) return Interpolate(v1, v2, fractional_X) end function

In the random noise function, does it matter what the seed is? In particular, I'm not familiar with the meaning of this bit of code: x = (x<<13) ^ x; Is it just converting the X value somehow, or does it relate to setting the seed? In my tests when I set the seed to the X value that gets plugged into the function the RegionWidth has an effect on the final output, but when I set it to use a random seed the output is *completely* random, with each point not appearing to relate to the others. I've uploaded a screenshot that has a comparison of the two. The top screenshot is using a random seed, and the bottoms screenshot is using the X input as the seed, with the RegionWidth set to 50 (as an aside, when I have it at 3 it's more or less a strait line; is this to be expected?). The problem with this is that every single time I run the program the values are all the same.

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#8
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Posted 08 August 2011 - 09:58 PM

The pieces seem to be fitting together finally thanks to your explanation; I actually have it at a point where I can use the data it outputs to create the blocks. However, there are a few things that just don't seem to be lining up for me still.

In the following function, what is the purpose of changing the values into integers?function InterpolatedNoise_1 (float x) integer_X = int(x) fractional_X = x - integer_X v1 = SmoothedNoise1(integer_X) v2 = SmoothedNoise1(integer_X + 1) return Interpolate(v1, v2, fractional_X) end function

The point is that pseudo-random values are generated for integral coordinate locations. What the above function does is it takes an arbitrary input point and finds the enclosing integer coordinates. For each of those coordinates, a pseudo-random value is calculated. Then the fractional part is used to interpolate those positions.

So imagine that you call the function with x=5.5; taking the int of x gives us the value of 5, so we know that the coordinates X1=5 and X2=6 are the integer coordinates that enclose 5.5. Now, we feed these values of X1 and X2 to the SmoothedNoise function, which generates a pseudo-random value based on the coordinate. For example sake, assume that SmoothedNoise(5) returns 0.26 and SmoothedNoise(6) returns -0.95. We calculate the interpolant as the fractional part of x. This is simply done by subtracting X1 from x, since X1 is the nearest integer that is less than x.

**interpolant = x - X1**. The result with x=5.5 and X1=5 is

**interpolant=0.5**. Now, we use this interpolant to interpolate between the values of 0.26 and -0.95:

**value = 0.26 + interpolant * (-0.95 - 0.26) = 0.345**. So the value of the function at x=5.5 is equal to 0.345.

In the random noise function, does it matter what the seed is? In particular, I'm not familiar with the meaning of this bit of code: x = (x<<13) ^ x; Is it just converting the X value somehow, or does it relate to setting the seed? In my tests when I set the seed to the X value that gets plugged into the function the RegionWidth has an effect on the final output, but when I set it to use a random seed the output is *completely* random, with each point not appearing to relate to the others. I've uploaded a screenshot that has a comparison of the two. The top screenshot is using a random seed, and the bottoms screenshot is using the X input as the seed, with the RegionWidth set to 50 (as an aside, when I have it at 3 it's more or less a strait line; is this to be expected?). The problem with this is that every single time I run the program the values are all the same.

All of the stuff that you see in the noise function is just a series of operations intended to

**hash**the input integer coordinate to a pseudo-random value. A pseudo-random value is a value that is seemingly random, and apparently has no perceptible correlation to the value it is generated from, and yet is repeatable; ie, everytime the function is called with a given input it will provide the same output. The noise function provided there just does some magical trickery, including a multiplication by 8192 (which is what << 2

^{13}does ) then XORs the result with the original value of x. Then it does some additional trickery. The actual mechanics of this really aren't important; there are a thousand ways that you could perform this hash. The result is what is important: a seemingly random, repeatable value. The random-part of pseudo-random is what ensures that subsequent values (ie the function at x=5 and the function at x=6) have no relation to one another and thus will generate no visible patterns in the output.

What provides the "magic" of this technique is the deterministic aspect of it: that plugging in the coordinate to the noise function nets you the same output each time. Imagine the chaos that results if you evaluate the point x=4.5 where X1=4 and X2=5, and the randomized output of Noise(X2=5) was 0.26. But then you evaluated the point 5.5 where X1=5 and X2=6, but this time the value of Noise(X1=5) outputted 0.45 instead of 0.26 as it did initially. There would be not continuity, and the final output of the function would be chaos; or, basically, what you got in your top screenshot.

Now, if you do want to be able to provide a random seed to change the behavior of the function, you have to make it a part of the

**IntNoise**function, and

*fold*it into the hash somehow. And, of course, you have to ensure that you always provide the same seed for a given mapping or evaluation of the function across a map or array; if the seed changes all the time, then the result would be the same kind of chaos as using a random input to IntNoise.

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#9
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Posted 09 August 2011 - 01:48 PM

A huge thank you for all your help! I like understanding the code I put into my program and your thorough explanations (and patience) got me to that point

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#10
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Posted 09 August 2011 - 04:50 PM

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#12
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Posted 10 August 2011 - 08:28 PM

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#14
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Posted 11 August 2011 - 10:52 PM

Code that calls the function.

FOR var_PositionX = 0 TO var_WorldGen_WorldWidth - 1 var_X# = var_PositionX / var_WorldGen_WorldWidth# var_X# = var_X# * var_WorldGen_Regions arr_WorldGen_WidthData(var_PositionX) = var_WorldGen_Dirt_BorderStart + ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0) - (var_WorldGen_HeightVariation / 2)) NEXT var_PositionX

Perlin Noise code.

FUNCTION func_PerlinNoise1D_Combined(var_X#) var_Total# = 0 FOR var_Octave = 0 TO glob_PerlinNoise_OctaveMax var_Frequency = 2 ^ var_Octave var_Amplitude# = glob_PerlinNoise_Persistance# ^ var_Octave var_Total# = var_Total# + func_PerlinNoise1D_Interpolate(var_X# * var_Frequency) * var_Amplitude# NEXT var_Octave ENDFUNCTION var_Total# FUNCTION func_PerlinNoise1D_Interpolate(var_X#) var_X_Integer = INT(var_X#) var_X_Fraction# = var_X# - var_X_Integer var_SmoothNoise_1# = func_PerlinNoise1D_Smooth(var_X_Integer) var_SmoothNoise_2# = func_PerlinNoise1D_Smooth(var_X_Integer + 1) var_Interpolate# = func_PerlinNoise1D_InterpolateCosine(var_SmoothNoise_1#, var_SmoothNoise_2#, var_X_Fraction#) ENDFUNCTION var_Interpolate# FUNCTION func_PerlinNoise1D_InterpolateCosine(var_SmoothNoise_1#, var_SmoothNoise_2#, var_X_Fraction#) var_Value1# = var_X_Fraction# * 3.1415927 var_Value2# = (1 - COS(var_Value1#)) * 0.5 var_InterpolateCosine# = var_SmoothNoise_1# * (1 - var_Value2#) + var_SmoothNoise_2# * var_Value2# ENDFUNCTION var_InterpolateCosine# FUNCTION func_PerlinNoise1D_Smooth(var_X) var_SmoothNoise# = func_PerlinNoise1D_Random(var_X) / 2.0 + func_PerlinNoise1D_Random(var_X - 1) / 4.0 + func_PerlinNoise1D_Random(var_X + 1) / 4.0 ENDFUNCTION var_SmoothNoise# FUNCTION func_PerlinNoise1D_Random(var_X) var_X = (var_X << 13) ~~ var_X var_Random# = (1.0 - ((var_X * (glob_WorldGen_Seed * var_X * var_X * 15731 + 789221) + 1376312589) && 0x7fffffff) / 1073741824.0) ENDFUNCTION var_Random#

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#15
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Posted 12 August 2011 - 09:09 PM

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#16
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Posted 13 August 2011 - 11:47 AM

arr_WorldGen_WidthData(var_PositionX) = var_WorldGen_Dirt_BorderStart + ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0) - (var_WorldGen_HeightVariation / 2))

To break it down a bit, there are three main values being combined in the code, but only this one is relevant to my problem: ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0)". Here you see the value "var_WorldGen_HeightVariation" appear; this is the maximum amount of blocks I want the border to be able to vary, which is set at 0.05% of the total map height. The Perlin Noise function is called, and immediately after being called the value it returns (again, -1.0 to 1.0) is increased by 1.0. This gets multiplied by "var_WorldGen_HeightVariation" and then divided by 2.0. I'm making it sound more complicated then it really is, since It's pretty much just cross multiplication.

Hopefully I answered your question, but if not let me know. Off to work for now, but I'll be able to elaborate further after I'm back.

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#17
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Posted 13 August 2011 - 01:16 PM

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#18
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Posted 14 August 2011 - 04:45 PM