diff --git a/content/00_randomness.html b/content/00_randomness.html index e461715..8e4b22a 100644 --- a/content/00_randomness.html +++ b/content/00_randomness.html @@ -301,19 +301,6 @@ if (num < 0.6) { 
let h = random(200, 300);

Is this an accurate algorithm for creating a population of monkey heights? Think of a crowded sidewalk in New York City. Pick any person off the street, and it may appear that their height is random. Nevertheless, it’s not the kind of random that random() produces by default. People’s heights aren’t uniformly distributed; there are many more people of about average height than there are very tall or very short ones. To accurately reflect this population, random heights close to the mean (another word for average) should be more likely to be chosen, while outlying heights (very short or very tall) should be rarer.

That’s exactly how a normal distribution (sometimes called a Gaussian distribution, after mathematician Carl Friedrich Gauss) works. A graph of this distribution is informally known as a bell curve. The curve is generated by a mathematical function that defines the probability of any given value occurring as a function of the mean (often written as μ, the Greek letter mu) and standard deviation (σ, the Greek letter sigma).

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Figure 0.2: Two example bell curves of a normal distribution, with a low (left) and high (right) standard deviation
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In the case of height values from 200 to 300, you probably have an intuitive sense of the mean (average) as 250. However, what if I were to say that the standard deviation is 3? Or 15? What does this mean for the numbers? The graphs depicted in Figure 0.2 should give you a hint. On the left is a distribution with a very low standard deviation, with the majority of the values piling up around the mean (they don’t deviate much from the standard). The version on the right has a higher standard deviation, so the values are more evenly spread out from the average (they deviate more).

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The numbers work out as follows: given a population, 68 percent of its members will have values in the range of one standard deviation from the mean, 95 percent within two standard deviations, and 99.7 percent within three standard deviations. Given a standard deviation of 5 pixels, only 0.3 percent of the monkey heights will be less than 235 pixels (three standard deviations below the mean of 250) or greater than 265 pixels (three standard deviations above the mean of 250). Meanwhile, 68 percent of the monkey heights will be from 245 to 255 pixels.

Calculating Mean and Standard Deviation

Consider a class of 10 students who receive the following scores (out of 100) on a test: 85, 82, 88, 86, 85, 93, 98, 40, 73, and 83.

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The standard deviation is the square root of the variance: 15.13.

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In the case of height values from 200 to 300, you probably have an intuitive sense of the mean (average) as 250. However, what if I were to say that the standard deviation is 3? Or 15? What does this mean for the numbers? The graphs depicted in Figure 0.2 should give you a hint. On the left is a distribution with a very low standard deviation, with the majority of the values piling up around the mean (they don’t deviate much from the standard). The version on the right has a higher standard deviation, so the values are more evenly spread out from the average (they deviate more).

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Figure 0.2: Two example bell curves of a normal distribution, with a low (left) and high (right) standard deviation
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The numbers work out as follows: given a population, 68 percent of its members will have values in the range of one standard deviation from the mean, 95 percent within two standard deviations, and 99.7 percent within three standard deviations. Given a standard deviation of 5 pixels, only 0.3 percent of the monkey heights will be less than 235 pixels (three standard deviations below the mean of 250) or greater than 265 pixels (three standard deviations above the mean of 250). Meanwhile, 68 percent of the monkey heights will be from 245 to 255 pixels.

Luckily, to use a normal distribution of random numbers in a p5.js sketch, you don’t have to do any of these calculations manually. Instead, the randomGaussian() function takes care of the math and returns random numbers with a normal distribution:

function draw() {
   // Ask for a Gaussian random number.
diff --git a/content/02_forces.html b/content/02_forces.html
index f1c4ede..4e4b602 100644
--- a/content/02_forces.html
+++ b/content/02_forces.html
@@ -488,12 +488,12 @@ friction.mult(frictionMag);

Air and Fluid Resistance

Friction also occurs when a body passes through a liquid or gas. The resulting force has many names, all really meaning the same thing: viscous force, drag force, air resistance, or fluid resistance (see Figure 2.4).

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The effect of a drag force is ultimately the same as the effect in our previous friction examples: the object slows down. The exact behavior and calculation of a drag force is a bit different, however. Here’s the formula:

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\vec{F_d} = - \frac{1}{2}\rho{v}^2 A C_d\hat{v}
Figure 2.4: A drag force (air or fluid resistance) is proportional to the speed of an object and its surface area pointing in the opposite direction of the object’s velocity.
Figure 2.4: A drag force (air or fluid resistance) is proportional to the speed of an object and its surface area pointing in the opposite direction of the object’s velocity.
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The effect of a drag force is ultimately the same as the effect in our previous friction examples: the object slows down. The exact behavior and calculation of a drag force is a bit different, however. Here’s the formula:

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\vec{F_d} = - \frac{1}{2}\rho{v}^2 A C_d\hat{v}

Let me break this down to see what’s really necessary for an effective simulation in p5.js, making a simpler formula in the process: