AFTER A FRIEND OF MINE HAD THE IDEA OF VISUALIZING THE DERIVATIVES OF A TWO-VARIABLE IMPLICIT FUNCTION BY GRAPHING THE DERIVATIVE ON THE Z-AXIS, I DECIDED THAT TAKING ON THE CHALLENGE TO MAKE THE IDEA INTO REALITY WOULD BE A GREAT WAY FOR ME TO EXPAND MY PROGRAMMING KNOWLEDGE BEYOND INTEGRATED SYSTEMS AND 2D INTERFACES.
​
I USED THE UNITY ENGINE TO BUILD A SYSTEM FROM THE GROUND UP WHICH COULD DYNAMICALLY SOLVE AND RENDER IMPLICIT FUNCTIONS. THIS WAS A FAR MORE DIFFICULT CHALLENGE THAN I INITIALLY ANTICIPATED, BECAUSE IT REQUIRED HIGHLY OPTIMIZED DYNAMIC CODE EXECUTION TO SOLVE FUNCTIONS WHICH COULD NOT BE ANALYZED ALGEBRAICALLY. BY USING VARIOUS LIBRARIES AND ANALYTICAL TECHNIQUES, I WAS ABLE TO SOLVE THE DERIVATIVES AT AN ACCEPTABLE RATE. I THEN BEGAN WORK ON RENDERING THE SOLUTIONS. AN INITIAL 'DUMB' VERSION GENERATED THE APPEARANCE OF LINES BY SIMPLY RENDERING THOUSANDS OF SPHERES AT EVERY SOLUTION POINT. I DECIDED TO GO FURTHER AND ANALYTICALLY DETECT AND FOLLOW SOLUTION PATHS FROM THE 3D POINT CLOUD OF SOLUTIONS, ALLOWING THE 3D ENGINE TO BE LEVERAGED MUCH MORE EFFECTIVELY FOR TRUE LINE RENDERING. THIS REQUIRED MANY ITERATIONS TO DETECT EDGE CASES AND OPTIMIZE CALCULATION SPEED, BUT IT ULTIMATELY RESULTED IN A MUCH BETTER LOOKING AND FAR FASTER RENDER, TAKING LESS THAN A SECOND TO BOTH SOLVE AND RENDER THE IMAGES ABOVE!