Generating a triangle wave in code
Now for something between maths and programming.
Recently, for Advent of Code 2017, on day 13, I had to simulate a periodic triangular function (AKA triangle wave) in Java. Since I had trouble finding an online resource detailing the thought process from a mathematical function to code, I wrote mine. It is also partly documented in my solution.
Constraints
All I could tell from the problem was that I needed to generate a triangular time series T that would start from T(0)=0, increase linearly to a maximum image T(D)=D1, then decrease linearly back to T(2*D)=0, and start over again.
Thought process

First of all, in order to draw plots for the series we will generate, we will apply D=3.

We start from the simplest function,
f1(t)=t
, which will give us a linearly increasing function. 
From there,
f2(t)=f1(t)%((D1)*2)
will give us a sawtooth wave, i.e a function that increases from 0 to (D1)*1 and starts again from 0. The range and cycle size are now therefore correct. 
f3(t)=f2(t)(D1)
offsets the sawtooth function by (D1) on the yaxis, making it increase from (D1) to D1 and starting again from (D1). That way, half of each period is negative and half is positive. 
f4(t)=abs(f3(t))
mirrors the negative half of each period around the xaxis, making the function decrease from D1 to 0 and then increase back to D1. The shape is now correct. Now the only issue is that the function starts at f4(0)=D1 instead of f4(0)=0. 
f5(t)=f4(t+D1)
offsetsf4
by D1, making it correctly increase from 0 to D1, then decrease back to 1, and start again from 0.
We now put develop f5
, resulting in the final function f(t)=abs((t+D1)%((D1)*2)(D1))
, with the following plot:
So for example, I could write the following Java 8 code to print a transposed plot of the function:
1 
int D=3; 
Producing the following output:






And so onâ€¦
What we can learn from this
This may seem like a simple exercise, but I think this is a good illustration of how we can solve problems that seem theoretical. That is the essence of Computeraided software engineering (CASE).
Note: The plots in this article were generated using GNU Octave. The final plot was created with the command t=0:1:40; D=3; plot(t, abs(mod(t+D1,(D1)*2)(D1)))
, along with some cosmetic improvements.