# Find the curvature of a curve with a kissing circle using Processing

Taichi Kato

2021-06-03

To try to programatically solve this problem, I first set up a p5.js environment where I can plot a curve.

``````fr = 10
currentPoint=15
function setup() {
createCanvas(400, 400);
background(220);
frameRate(fr);
}
isDrawing = false;
points = [];
var e;
function draw() {
if(isDrawing){
points.push([mouseX, mouseY, points.length]);
ellipse(mouseX, mouseY, 2, 2);
if(points.length > 1){
line(mouseX, mouseY, points[points.length - 2], points[points.length - 2]);
}
}
if(points.length === currentPoint){
fill(255,123,0);
ellipse(points[currentPoint-2], points[currentPoint-2], 5, 5)
fill(255,0,0);
ellipse(points[currentPoint-1], points[currentPoint-1], 5, 5)
fill(255,123,0);
ellipse(points[currentPoint-3], points[currentPoint-3], 5, 5)
}
}
function mousePressed() {
if (isDrawing) {
isDrawing = false;
console.log(points)
} else {
isDrawing = true;
}
}
function keyPressed() {
if (keyCode === UP_ARROW) {
currentPoint += 1;
} else if (keyCode === DOWN_ARROW) {
currentPoint -= 1;
}

console.log('Current point: ', currentPoint);
}``````

Here, I have highlighted three points in red, which are the points I will use to create the kissing circle. Assuming that the three points do not lie on a straight line, we can find an unique circle which passes through them. We can use that circle, and its radius to approximate the curvature of the curve at the point. I came across a stack exchange post which was basically tacking the same problem. However, to try to create a system which solves the system of equations which are provided here seemed rather difficult, given that they are non-linear. I tried to analytically solve the equation for r to create a function r(p1, p2, p3). However, I was not able to do this by hand.  (blah blah, I gave up on solving this.)

I then went back to thinking about ways in which I could find the radius, given the three points. I figured that I could create my own system of equations, such that it is easier to find the radius. Since all points lie on the circle, we could use the equation for a circle to create a set of equations which represents this circle. This was definitely a step in the right direction, but this is still non-linear, so its difficult to solve. I could of course use numerical methods like Monte Carlo to approximate the values, but I figured it shouldn’t be too hard to find a way to turn this into a linear system of equations.

I was able to reduce it to a linear equation by creating three new variables A, B and C, which I can use to solve for `x_m`, `r` and `y_m` later on. Finally, I believe I can use these matrices to create a program to find a circle, and a curvature in the end.