We Keep Coding

Figuring out the value for PI

Let's say I have a function called bars()
bars () {
const bars = []
for (let i = 0; i < this.numberOfBars; i++) {
bars.push(Math.sqrt(this.numberOfBars * this.numberOfBars - i * i))
}
return bars
}
If I'm reducing the bars array to approximate PI, what should be on the right side of the arrow function?
PI = bars().reduce((a, b) =>
I tried adding the values and dividing by the number of bars, but I'm not getting anywhere near the approximation of Pi. I feel like there's a simple trick that I'm missing.

Your funcion seems to list lengths of "bars" in a quarter of a circle, so we have to add them all up (to have the area of the quarter of a circle), then multiply by 4 (because there is 4 quarter) and the divide by this.numberOfBars ^ 2 because area = π * r^2, but like we have to know the radius, it is better using a pure function :
// Your function rewritten as a pure one
const bars = numberOfBars => {
const bars = []
for (let i = 0; i < numberOfBars; i++) {
bars.push(Math.sqrt(numberOfBars * numberOfBars - i * i))
}
return bars
}
// Here we take 1000 bars as an example but in your case you replace it by this.numberOfBars
// Sum them all up, multiply by 4, divide by the square of the radius
const PI = bars(1000).reduce((g, c) => g + c) * 4 / Math.pow(1000, 2)
console.log(PI)

/** Approximates PI using geometry
* You get a better approximation using more bars and a smaller step size
*/
function approximatePI(numberOfBars, stepSize) {
const radius = numberOfBars * stepSize;
// Generate bars (areas of points on quarter circle)
let bars = [];
// You can think of i as some point along the x-axis
for (let i = 0; i < radius; i += stepSize) {
let height = Math.sqrt(radius*radius - i*i)
bars.push(height * stepSize);
}
// Add up all the areas of the bars
// (This is approximately the area of a quarter circle if stepSize is small enough)
const quarterArea = bars.reduce((a, b) => a + b);
// Calculate PI using area of circle formula
const PI = 4 * quarterArea / (radius*radius)
return PI;
}
console.log(`PI is approximately ${approximatePI(100_000, 0.001)}`);

There is no reason to push all terms to an array, then to reduce the array by addition. Just use an accumulator variable and add all terms to it.
Notice that the computation becomes less and less accurate the closer you get to the end of the radius. If you sum to half of the radius, you obtain r²(3√3+π)/24, from which you can draw π.
(Though in any case, this is one of the worst methods to evaluate π.)

Creating a Logarithm Scale with Min, Max and Discrete "steps"

I'm building a scale for generating colors for a map, where various ranges of numbers correspond to colors. In the scale below, the a, b, and c values (along with the min and max) form "steps" or "buckets" corresponding to colors. Anything between -1 and min will be colored "#ffeda0 in this example, and anything between min and a will be colored differently:
return [
"#e7e9e9",
-1,
"#ffeda0",
min,
"#fed976",
a
"#feb24c",
b
"#fd8d3c",
c
"#fc4e2a",
max
"#e31a1c"
];
This scale will always start an a minimum value, called min, and will end at a maximum value, max. I've currently got a simple linear scale like this:
const min = Math.min(data);
const range = Math.max(data) - min;
const step = range / 4;
return [
"#e7e9e9",
-1,
"#ffeda0",
min,
"#fed976",
min + step,
"#feb24c",
min + step * 2,
"#fd8d3c",
min + step * 3,
"#fc4e2a",
max,
"#e31a1c"
];
How could one write a function that, given a minimum, maximum, and the number of steps in a scale (in this case three) output the correct numbers for a logarithmic scale?

A logarithmic scale is linear in its logarithms. So:
const min = Math.min(data);
const max = Math.max(data);
const logmin = Math.log(min);
const logmax = Math.log(max);
const logrange = logmax - logmin;
const logstep = logrange / 4;
Now where the linear scale has
value = min + n * step;
the logarithmic scale has:
value = Math.exp(logmin + n * logstep);
This will work only if both min and max are positive, because you cannot take the logarithm of zero or a negative number and bcause in logarithmic scale, each value has a constant positive factor to the previous one. You can make this work for negative min and max by adjusting the signs.
You can do the calculations in other bases, for example 10:
const logmin = Math.log10(min);
const logmax = Math.log10(max);
value = Math.pow(10, logmin + n * logstep);
Alternatively, you can find the common factor from one step to the next by taking the n-th root of the quotient of the min and max values:
const min = Math.min(data);
const max = Math.max(data);
const fact = Math.pow(max / min, 1.0 / 4.0);
The values are then found by repeated multiplication:
value = prev_value * fact
or
value = xmin * Math.pow(fact, n);
Both methods might yield "untidy" numbers even if the scale should have nice round numbers, because the logarithms and n-th roots could produce inaccurate floating-point results. You might get better results if you pick a round factor and adjust your min/max values.

Animating a fractal tree inside a Javascript Canvas

After I saw a video from the Coding Train on youtube about fractal trees, I tried to build one myself. Which worked great and I played with some variables to get different results.
I would love to see the tree moving like it got hit by some wind. I tried different approaches like rotating the branches a little bit or some minor physics implementations but that failed miserably.
So my question is: What would be the best approach to render a fractal tree and give it some sort of "life" like little shakes from wind.
Is there some sort of good reference ?
Do I need physics ? -> If so where do I have to look ?
If not -> How could I fake such an effect?
I am glad about every help I can get.
Source for the idea: https://www.youtube.com/watch?v=0jjeOYMjmDU

Tree in the wind.
The following are some short points re bending a branch in the wind. As the whole solution is complex you will have to get what you can from the code.
The code includes a seeded random number functions. A random recursive tree renderer, a poor quality random wind generator, all drawn on canvas using an animation loop.
Wind
To apply wind you need to add a bending force to each branch that is proportional to the angle of the branch to the wind.
So if you have a branch in direction dir and a wind in the direct wDir the amount of scaling the bending force needs is
var x = Math.cos(dir); // get normalize vector for the branch
var y = Math.sin(dir);
var wx = Math.cos(wDir); // get normalize vector for the wind
var wy = Math.sin(wDir);
var forceScale = x * wy - y * wx;
The length of the branch also effects the amount of force to include that you lengthen the vector of the branch to be proportional to its length
var x = Math.cos(dir) * length; // get normalize vector for the branch
var y = Math.sin(dir) * length;
var wx = Math.cos(wDir); // get normalize vector for the wind
var wy = Math.sin(wDir);
var forceScale = x * wy - y * wx;
Using this method ensures that the branches do not bend into the wind.
There is also the thickness of the branch, this is a polynomial relationship related to the cross sectional area. This is unknown so is scaled to the max thickness of the tree (an approximation that assumes the tree base can not bend, but the end branches can bend a lot.)
Then the elastic force of the bent branch will have a force that moves the branch back to its normal position. This acts like a spring and is very much the same as the wind force. As the computational and memory load would start to overwhelm the CPU we can cheat and use the wind to also recoil with a little bit of springiness.
And the tree.
The tree needs to be random, yet being fractal you don't want to store each branch. So you will also need a seeded random generator that can be reset at the start of each rendering pass. The tree is rendered randomly with each iteration but because the random numbers start at the same seed each time you get the same tree.
The example
Draws random tree and wind in gusts. Wind is random so tree may not move right away.
Click tree image to reseed the random seed value for the tree.
I did not watch the video, but these things are quite standard so the recursive function should not be to far removed from what you may have. I did see the youTube cover image and it looked like the tree had no randomness. To remove randomness set the leng, ang, width min, max to be the same. eg angMin = angMax = 0.4; will remove random branch angles.
The wind strength will max out to cyclone strength (hurricane for those in the US) to see the max effect.
There are a zillion magic numbers the most important are as constants with comments.
const ctx = canvas.getContext("2d");
// click function to reseed random tree
canvas.addEventListener("click",()=> {
treeSeed = Math.random() * 10000 | 0;
treeGrow = 0.1; // regrow tree
});
/* Seeded random functions
randSeed(int) int is a seed value
randSI() random integer 0 or 1
randSI(max) random integer from 0 <= random < max
randSI(min, max) random integer from min <= random < max
randS() like Math.random
randS(max) random float 0 <= random < max
randS(min, max) random float min <= random < max
*/
const seededRandom = (() => {
var seed = 1;
return { max : 2576436549074795, reseed (s) { seed = s }, random () { return seed = ((8765432352450986 * seed) + 8507698654323524) % this.max }}
})();
const randSeed = (seed) => seededRandom.reseed(seed|0);
const randSI = (min = 2, max = min + (min = 0)) => (seededRandom.random() % (max - min)) + min;
const randS = (min = 1, max = min + (min = 0)) => (seededRandom.random() / seededRandom.max) * (max - min) + min;
/* TREE CONSTANTS all angles in radians and lengths/widths are in pixels */
const angMin = 0.01; // branching angle min and max
const angMax= 0.6;
const lengMin = 0.8; // length reduction per branch min and max
const lengMax = 0.9;
const widthMin = 0.6; // width reduction per branch min max
const widthMax = 0.8;
const trunkMin = 6; // trunk base width ,min and max
const trunkMax = 10;
const maxBranches = 200; // max number of branches
const windX = -1; // wind direction vector
const windY = 0;
const bendability = 8; // greater than 1. The bigger this number the more the thin branches will bend first
// the canvas height you are scaling up or down to a different sized canvas
const windStrength = 0.01 * bendability * ((200 ** 2) / (canvas.height ** 2)); // wind strength
// The wind is used to simulate branch spring back the following
// two number control that. Note that the sum on the two following should
// be below 1 or the function will oscillate out of control
const windBendRectSpeed = 0.01; // how fast the tree reacts to the wing
const windBranchSpring = 0.98; // the amount and speed of the branch spring back
const gustProbability = 1/100; // how often there is a gust of wind
// Values trying to have a gusty wind effect
var windCycle = 0;
var windCycleGust = 0;
var windCycleGustTime = 0;
var currentWind = 0;
var windFollow = 0;
var windActual = 0;
// The seed value for the tree
var treeSeed = Math.random() * 10000 | 0;
// Vars to build tree with
var branchCount = 0;
var maxTrunk = 0;
var treeGrow = 0.01; // this value should not be zero
// Starts a new tree
function drawTree(seed) {
branchCount = 0;
treeGrow += 0.02;
randSeed(seed);
maxTrunk = randSI(trunkMin, trunkMax);
drawBranch(canvas.width / 2, canvas.height, -Math.PI / 2, canvas.height / 5, maxTrunk);
}
// Recusive tree
function drawBranch(x, y, dir, leng, width) {
branchCount ++;
const treeGrowVal = (treeGrow > 1 ? 1 : treeGrow < 0.1 ? 0.1 : treeGrow) ** 2 ;
// get wind bending force and turn branch direction
const xx = Math.cos(dir) * leng * treeGrowVal;
const yy = Math.sin(dir) * leng * treeGrowVal;
const windSideWayForce = windX * yy - windY * xx;
// change direction by addition based on the wind and scale to
// (windStrength * windActual) the wind force
// ((1 - width / maxTrunk) ** bendability) the amount of bending due to branch thickness
// windSideWayForce the force depending on the branch angle to the wind
dir += (windStrength * windActual) * ((1 - width / maxTrunk) ** bendability) * windSideWayForce;
// draw the branch
ctx.lineWidth = width;
ctx.beginPath();
ctx.lineTo(x, y);
x += Math.cos(dir) * leng * treeGrowVal;
y += Math.sin(dir) * leng * treeGrowVal;
ctx.lineTo(x, y);
ctx.stroke();
// if not to thing, not to short and not to many
if (branchCount < maxBranches && leng > 5 && width > 1) {
// to stop recusive bias (due to branch count limit)
// random select direction of first recusive bend
const rDir = randSI() ? -1 : 1;
treeGrow -= 0.2;
drawBranch(
x,y,
dir + randS(angMin, angMax) * rDir,
leng * randS(lengMin, lengMax),
width * randS(widthMin, widthMax)
);
// bend next branch the other way
drawBranch(
x,y,
dir + randS(angMin, angMax) * -rDir,
leng * randS(lengMin, lengMax),
width * randS(widthMin, widthMax)
);
treeGrow += 0.2;
}
}
// Dont ask this is a quick try at wind gusts
// Wind needs a spacial component this sim does not include that.
function updateWind() {
if (Math.random() < gustProbability) {
windCycleGustTime = (Math.random() * 10 + 1) | 0;
}
if (windCycleGustTime > 0) {
windCycleGustTime --;
windCycleGust += windCycleGustTime/20
} else {
windCycleGust *= 0.99;
}
windCycle += windCycleGust;
currentWind = (Math.sin(windCycle/40) * 0.6 + 0.4) ** 2;
currentWind = currentWind < 0 ? 0 : currentWind;
windFollow += (currentWind - windActual) * windBendRectSpeed;
windFollow *= windBranchSpring ;
windActual += windFollow;
}
requestAnimationFrame(update);
function update() {
ctx.clearRect(0,0,canvas.width,canvas.height);
updateWind();
drawTree(treeSeed);
requestAnimationFrame(update);
}
body {
font-family : arial;
}
<canvas id="canvas" width="250" heigth="200"></canvas>
Click tree to reseed.
Update
I just noticed that the wind and branch length are absolute thus drawing the tree on a larger canvas will create a bending force too great and the branches will bend past the wind vector.
To scale the sim up either do it via a global scale transform, or reduce the windStrength constant to some smaller value. You will have to play with the value as its a 2nd order polynomial relation. My guess is multiply it with (200 ** 2) / (canvas.height ** 2) where the 200 is the size of the example canvas and the canvas.height is the new canvas size.
I have added the calculations to the example, but its not perfect so when you scale you will have to change the value windStrength (the first number) down or up if the bending is too far or not enough.

Non-uniform (biased towards a value) random numbers in JavaScript

I am aware that random integers can be generated in JavaScript like this:
function getRandomInt(min, max) {
return Math.floor(Math.random() * (max - min + 1)) + min;
}
But what I want is a set of random numbers that are biased towards a specific value.
As an example, if my specific center value is 200, I want a set of random numbers that has a large range, but mostly around 200. Hopefully there will be a function like
biasedRandom(center, biasedness)

It sounds like a Gaussian distribution might be about right here.
This stackoverflow post describes how to produce something that is Gaussian in shape. We can then scale and shift the distribution by two factors;
The mean (200 in this case) which is where the distribution is centred
The variance which gives control over the width of the distribution
I have included a histogram of the generated numbers (using plotly) in my example so you can easily see how varying these two parameters v and mean affects the numbers generated. In your real code you would not need to include the plotly library.
// Standard Normal variate using Box-Muller transform.
function randn_bm() {
var u = 0, v = 0;
while(u === 0) u = Math.random(); //Converting [0,1) to (0,1)
while(v === 0) v = Math.random();
return Math.sqrt( -2.0 * Math.log( u ) ) * Math.cos( 2.0 * Math.PI * v );
}
//generate array
// number of points
let n = 50;
// variance factor
let v = 1;
// mean
let mean = 200;
let numbers = []
for (let i=0; i<n;i++){
numbers.push(randn_bm())
}
// scale and shift
numbers = numbers.map( function (number){ return number*v + mean})
// THIS PURELY FOR PLOTTING
var trace = {
x: numbers,
type: 'histogram',
};
var data = [trace];
Plotly.newPlot('myDiv', data);
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
<div id="myDiv"></div>

Getting frequency from a pixel value for Web Audio API EQ

I'm creating an visual EQ using the Web Audio API and canvas. I am plotting frequencies in a logarithmic graph on a HTML5 Canvas using the following function:
function frequencyToPixel(freq){
var min = Math.log(graph.min)/Math.log(10)
, max = Math.log(graph.max)/Math.log(10)
, range = max-min
, pixel = (Math.log(freq)/Math.log(10) - min) / range * canvas.width;
return pixel;
}
I would like to also reverse this equation to get a function that returns the frequency that a particular pixel relates to. At the moment I'm using the following function but it's not producing the desired result. For example if I input 1000 into the above function it returns 434.93. I should therefore be able to put 434.93 into the below equation to return 1000;
function pixelToFrequency(pixel){
var min = Math.log(graph.min)/Math.log(10)
, max = Math.log(graph.max)/Math.log(10)
, range = max-min
, x = (pixel * canvas.width * range) + min
, frequency = Math.pow(10, x);
return frequency;
}
It's likely i've rearranged the equation in the wrong way so any help would be much appreciated.

The function below works out the frequency for a given pixel on the logarithmic-scaled canvas frequency graph:
function pixelToFrequency(pixel){
var min = Math.log(app.graph.min)/Math.log(10)
, max = Math.log(app.graph.max)/Math.log(10)
, range = max-min
, frequency = Math.pow(10, pixel * (range / canvas.width) + min)
return frequency;
}

pixel = (Math.log(freq)/Math.log(10) - min) / range * canvas.width
Try this?
freq = E((Math.log(10) * (pixel * (range / canvas.width)))+ Math.log(graph.min))
See if that helps at all. Where E is the Euler's Number raised to the parameters in parenthesis. It might be equivalent to what you have there, I didn't look to closely too be honest. It's worth a shot though.