61–70 of 144 posts
http://www.sliderules.info/a-to-z/log-tables.htm
I did get taught them in a mech engineering class - this was just at the inflection point (late 70's) when everyone switched from slide rules to calculators
https://www.researchgate.net/publication/276103942_REVIEW_of...
Anyone can explain the meaning behind this: > When the Onion imitates real life, it's usually tragic. linking to an article titled "Man Puts Glass Of Water On Bedside Table In Case He Needs To Make Huge Mess In Middle Of Night"? Is the word "tragic" in the OP article here used jokingly, or is it a reference to an actually tragic event I don't know about?
A good reason to put a glass of water on bedside table is because you might be thirsty in the middle of the night. You might accidentally knock the glass over and cause a huge spill instead. So "in case he causes an accident" is silly.
If in real life someone knocked over a glass of water that they put on their bedside table, it's just like the joke, and "tragic". (And an example of life imitating art).
Earlier quoted context omitted.
I think the point of these was for when they read low-precision values out of a printed table and then multiplied them using slide rules. Even though the article doesn't say it, I can't think of any other practical reason to multiply by adding logarithms.
Generally, slide rules were only good for rough calculations (3 significant digits if you were lucky), but they did contain multiple scales providing multiplication, division, exponential, log10, ln, roots, sin, cos, tan, and hyperbolic trig functions. Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. When more precision was needed, I and my fellow engineers would break out…
Huh? It's trivial to make a slide rule that you can add and subtract on, you just rule linear scales on it.
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Prosthaphaeresis works because an angle measure inherently is a type of logarithm. It’s the logarithm of a rotation, instead of the logarithm of a scale. Like other kinds of logarithms, this converts multiplication (composition of rotations) to addition (addition of angle measures). If you want to multiply two numbers, you can treat them as rotations, take the logarithm of each (i.e. find the angle measure), add the…
Is there a more general definition of logarithms than being the inverse of the exponential function? Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" [0], which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm. Maybe something to do with Euler's formula? [0] https://mathw…
This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".
exp(ix) = cos(x) + i*sin(x)
[cos(x)]^2 + [sin(x)]^2 = 1
i*i = -1
Any complex number can be written as an exponent like rexp(ix), where x is the angle and r is the distance from the origin in the complex plane. Using this understanding, we can interpret complex number multiplication as a rotation in the complex plane:
(a * exp(ix)) * (b * exp(iy)) = ab * exp(i (x+y))
exp(i*2x) = cos(2x) + i * sin(2x)
exp(i*2x) = [exp(i*x)]^2
= [cos(x) + i*sin(x)]^2
= [cos(x)]^2 + (i*i)*[sin(x)]^2 + 2i * cos(x) * sin(x)
= [cos(x)]^2 - [sin(x)]^2 + 2i * cos(x) * sin(x)
cos(2x) + i * sin(2x) = [cos(x)]^2 - [sin(x)]^2 + 2i * cos(x) * sin(x)
cos(2x) = [cos(x)]^2 - [sin(x)]^2
sin(2x) = 2 * cos(x) * sin(x)
atan2() is incredibly useful in game development. You can use it to determine the angle between objects, calculate the angle of a 2d vector, and other useful things. https://ryankubik.com/blog/atan2/
there is also sincos(), which is useful when you need both, such as building matrices. https://man7.org/linux/man-pages/man3/sincos.3.html https://home.deec.uc.pt/~jlobo/tc/opcode_f.html
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Yup, I felt incredibly stupid about six months ago finding out about atan2 for the first time, given the number of times I've made stupid errors using atan, and then correcting those errors by manually checking what quadrant I was in.
Same. My favorite "looks simple but is hard" task is "given object position and angle and target position return +1 if the object should turn clockwise and -1 if the object should turn counterclockwise to face the target". I've got it subtly wrong so many times and using atan2 only solves half the problems. Now I just copypaste it from my previous game :)
Let me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes…
"i know two things:
[cos(x)]^2 + [sin(x)]^2 = 1
i*i = -1
but can re-derive all the rest"
haversine is incredibly useful when you're doing quick distance calculations between points on the Earth though...