Conceptual Clarity on frameCount and rotate function in p5.js - processing

Question

I was doing one of the examples on the p5.js website - https://p5js.org/examples/form-regular-polygon.html. I got really confused by the way rotate function worked in that code . IN the below function if I just pass rotate(frameCount) , in the browser it shows me rotating two triangles intersected within forming a star , but as soon as I divide the frameCount it disappears. Also the equation used in the code - can some one give the mathematical intuition on how we reached to this point.
let sx = x + cos(a) * radius;
let sy = y + sin(a) * radius;
push();
translate(width * 0.2, height * 0.5);
rotate(frameCount / 50);
polygon(0,0,82,3);
pop();

Answer 1

Regarding "two triangles intersected within forming a star":
By default, the rotate function takes radians. When you do rotate(frameCount), you are increasing the angle by 1 radian at each frame. One radian equals about 57 degrees, so your triangle would rotate about 57 degrees at each frame. At frame 3, the triangle would have rotated about 120 degrees, and it would roughly overlap with the triangle at frame 1. Similarly, the triangle at frame 4 would roughly overlap with the triangle at frame 2.
The "two triangles" you are seeing is just two groups of triangles, one group being triangles at frame 1, 3, 5... and another group being triangles at frame 2, 4, 6...
That is why you should divide frameCount by some number if you would like to obtain a rather continuous rotation. Alternatively, you could also set angleMode to DEGREES. In that case, you don't have to divide frameCount anymore because at each frame the triangle would only rotate 1 degree instead of 1 radian.
Regarding the math formula:
In fact, the function used in that example should be called regularPolygon instead of polygon because it only draws regular polygons.
Now, how do you draw a regular polygon? You know the distance from each vertex to the center is a constant number. In this example, that number is the radius variable. And you know if you use polar coordinates with the center of the polygon as the origin point, the angle difference between every two adjacent vertices is also a constant number. In this example, that number is the angle variable.
More precisely, the polar coordinates of the vertices should take the form of:
v1 = (radius, 0)
v2 = (radius, angle)
v3 = (radius, angle*2)
...
Convert them to cartesian coordinates, you would obtain something like:
v1 = (cos(0) * radius, sin(0) * radius)
v2 = (cos(angle) * radius, sin(angle) * radius)
v3 = (cos(angle*2) * radius, sin(angle*2) * radius)
...
But what if the center of the polygon is not the origin point, but point (x, y), as in the example? Now the cartesian coordinates of the vertices become:
v1 = (x + cos(0) * radius, y + sin(0) * radius)
v2 = (x + cos(angle) * radius, y + sin(angle) * radius)
v3 = (x + cos(angle*2) * radius, y + sin(angle*2) * radius)
So when you do:
for (let a = 0; a < TWO_PI; a += angle) {
let sx = x + cos(a) * radius;
let sy = y + sin(a) * radius;
vertex(sx, sy);
}
You are really drawing the vertices v1, v2, v3....