/* This project was super interesting to me, since my last project in which I have made a coordinate system with non perpendicular axes. That lead me to experiment with higher dimensional coordinate systems. For example, I have made a 3d cube after that I implemented a 4d Hypercube!. That project was the most I'm proud of, because I've implemented it from scratch on my own. It is probably the most complicated project I've done. The topic is still on my mind, hence this code. When I heard of general relativity and the concept of space-time in which its curvature prodces gravity as we precieve it was mind blowing to me. After learned some of the physics and math of it, I thought of making a curved space but I had no idea what to do and how to do it. This idea of making a curved coordinate system was on my mind for the last 2-3 weeks , and suddenly today I had the general concept of how to do it. it took me around 7 hours to complete. */ let ampX = 20; let freqX = 1; let ampY = 10; let freqY = 10; function setup(){ createCanvas(windowWidth-20, windowHeight -50); } let p; function draw(){ background(10); translate(width/2, height/2); baseAxis(1); baseAxis(0); // draw point of origin strokeWeight(8); pt = plot(0, 0); point(pt[0], pt[1]); // draw the function beginShape(); for (let i = -100; i < 100; i+=1){ p = plot(i, f(i)); stroke(255, 0, 100); strokeWeight(2); noFill(); // vertex(p[0], p[1]); vertex(p[0], p[1]); } endShape(); // noLoop(); // change the frequence of the freqX += 1; freqY += 1; ampX /= 1.05; ampY /= 1.01; } // the function graphed on the x and y axes. function f(t){ // return 100*sin(t/20); return t; } // x-axis equation. (if you change the equation to return 0 you'll get a straight line) // i.e the cartesian coordinate system. function f_x(x, amp, freq){ return amp*Math.sin(x/freq); // return 0; } // y-axis equation. if you change it to instead return 0, you'll get back // to the normal cartesian plane. function f_y(y, amp, freq){ return amp*Math.cos(y/freq); // return 0; } // this is an important function to calculate the distance along a curved line. // this is used to find the distance from the origin to the given point on an axis. function arcLength(a, b, ind){ let sum = 0; let dx = 0.1; let sum2, dy; for (let i = a; i < b; i+= dx){ if (ind == 0){ dy = f_x(i + dx, ampX, freqX) - f_x(i, ampX, freqX) } else{ dy = f_y(i + dx, ampY, freqY) - f_y(i, ampY, freqY) } // arc length formula. sum2 = dx*dx + dy*dy sum += Math.sqrt(sum2); } return sum; } // this function draws the main axes of the plane used using the corresponding // mathematical function from above (f_x(x) and f_y(y)). function baseAxis(ind){ let len = 0; // let step = 50; // let r; // let n = 0; stroke(255); if (ind == 0){ len = width; for (i = -len/2; i < len/2; i += 1){ line(i, -f_x(i, ampX, freqX), i+1, -f_x(i+1, ampX, freqX)); // r = arc_x(-len/2 +n*step, i) // if (r > step && r < step + 1.5){ // n += 1; // push(); // strokeWeight(8); // stroke(255, 0, 100); // point(i, -f_x(i)) // // point(i, 0); // pop(); // } } } else{ len = height; for (i = -height/2; i < height/2; i += 1){ line(f_y(i, ampY, freqY), -i, f_y(i+1, ampY, freqY), -(i+1)); // r = arc_y(-len/2 +n*step, i) // if (r > step && r < step + 1){ // n += 1; // push(); // strokeWeight(8); // stroke(255, 0, 100); // point(f_y(i), i) // // point(i, 0); // pop(); // } } } } // to determine the location of the point on the curved space. function plot(x, y){ let dx = 1; let startX = 0, startY = 0; let endX, endY, Lx, Ly; let x_pos = [], y_pos = []; // push(); if (x < 0){ startX = x; endX = 0; } endX = width/2; for (let i = startX; i < endX; i += dx){ Lx = arcLength(startX, i, 0); if (Lx > x){ x_pos = [i, -f_x(i, ampX, freqX)]; break; } } if (y < 0){ startY = y; endY = 0; } endY = height/2; for (let i = startY; i < endY; i += dx){ Ly = arcLength(startY, i, 1); if (Ly > y){ y_pos = [f_y(i, ampY, freqY), -i]; break; } } return [x_pos[0]+y_pos[0], x_pos[1]+y_pos[1]]; }

Straight line on curved space