How to get the intersection points of an Arc and a line? - javascript

I need to place several number of line segments inside an arc, but to do that i need to have intersection points so that i can place the lines inside the arc perfectly;
I thought about a way to calculate the distance and check if it is less than radius, but the thing is i need to know the C,D & E points so that i can place the line segments, so i'm lost here, any one can help please?
EDIT
The radius is specified
Number of line segments may vary, but there are 2 lines at least
Lines start at starting border, end at the ending border; e.g: Start is C, end point is D
EDIT
In order to be clear about what i'm trying to do, i'm uploading another illustration;
I need to get the coordinates of [CD],[EI],[JK] lines,

Ok... Here we go. The following snippet should work for any arc (defined with an angle and a radius) and for any number of equally spaced segments you want.
Currently, it assumes that the arc is perfectly placed horizontally (like in your example), but it can be "easily" extended to allow translated/rotated arcs.
The getLinesCoords() function will return an object whose x and y properties contains arrays with the corresponding coordinates for each segment.
"y" coordinates are the "height" of the segment from the center (G in your image) and "x" are the start/end position, always from center (left/right depends on sign +/-).
If you have any question, please ask.
// *** ARC ***
const R = 100; // RADIUS
const PHI = 180; // ANGLE (DEG)
// *** LINES ***
const LINES = 10; // NUMBER OF LINES TO BE PLACED
// *** CALCULATIONS ***
const coords = getLinesCoords(R, PHI, LINES);
console.log(coords);
function getLinesCoords(radius, angle, linesNum) {
let i = 0;
let arcAvailHeight = 0;
let linesSep = 0;
let linesYCoords = [];
let linesXCoords = [];
let angleRad = angle * Math.PI / 180;
// GET AVAILABLE HEIGHT FOR PLACING LINES
arcAvailHeight = radius * (1 - Math.cos(angleRad / 2));
// GET LINES SEPARATION
linesSep = arcAvailHeight / (linesNum + 1);
// GET Y COORDINATES FOR LINES
for (i = 0; i < linesNum; i++) {
linesYCoords[i] = linesSep * (i + 1);
}
// GET CORRESPONDING X COORDINATES FOR LINES
linesYCoords.forEach((y) => {
linesXCoords.push(Math.sqrt(radius**2 - (radius * Math.cos(angleRad / 2) + y)**2));
});
return ({x: linesXCoords, y: linesYCoords});
}

Related

SAT Polygon Circle Collision - resolve the intersection in the direction of velocity & determine side of collision

Summary
This question is in JavaScript, but an answer in any language, pseudo-code, or just the maths would be great!
I have been trying to implement the Separating-Axis-Theorem to accomplish the following:
Detecting an intersection between a convex polygon and a circle.
Finding out a translation that can be applied to the circle to resolve the intersection, so that the circle is barely touching the polygon but no longer inside.
Determining the axis of the collision (details at the end of the question).
I have successfully completed the first bullet point and you can see my javascript code at the end of the question. I am having difficulties with the other parts.
Resolving the intersection
There are plenty of examples online on how to resolve the intersection in the direction with the smallest/shortest overlap of the circle. You can see in my code at the end that I already have this calculated.
However this does not suit my needs. I must resolve the collision in the opposite direction of the circle's trajectory (assume I already have the circle's trajectory and would like to pass it into my function as a unit-vector or angle, whichever suits).
You can see the difference between the shortest resolution and the intended resolution in the below image:
How can I calculate the minimum translation vector for resolving the intersection inside my test_CIRCLE_POLY function, but that is to be applied in a specific direction, the opposite of the circle's trajectory?
My ideas/attempts:
My first idea was to add an additional axis to the axes that must be tested in the SAT algorithm, and this axis would be perpendicular to the circle's trajectory. I would then resolve based on the overlap when projecting onto this axis. This would sort of work, but would resolve way to far in most situations. It won't result in the minimum translation. So this won't be satisfactory.
My second idea was to continue to use magnitude of the shortest overlap, but change the direction to be the opposite of the circle's trajectory. This looks promising, but there are probably many edge-cases that I haven't accounted for. Maybe this is a nice place to start.
Determining side/axis of collision
I've figured out a way to determine which sides of the polygon the circle is colliding with. For each tested axis of the polygon, I would simply check for overlap. If there is overlap, that side is colliding.
This solution will not be acceptable once again, as I would like to figure out only one side depending on the circle's trajectory.
My intended solution would tell me, in the example image below, that axis A is the axis of collision, and not axis B. This is because once the intersection is resolved, axis A is the axis corresponding to the side of the polygon that is just barely touching the circle.
My ideas/attempts:
Currently I assume the axis of collision is that perpendicular to the MTV (minimum translation vector). This is currently incorrect, but should be the correct axis once I've updated the intersection resolution process in the first half of the question. So that part should be tackled first.
Alternatively I've considered creating a line from the circle's previous position and their current position + radius, and checking which sides intersect with this line. However, there's still ambiguity, because on occasion there will be more than one side intersecting with the line.
My code so far
function test_CIRCLE_POLY(circle, poly, circleTrajectory) {
// circleTrajectory is currently not being used
let axesToTest = [];
let shortestOverlap = +Infinity;
let shortestOverlapAxis;
// Figure out polygon axes that must be checked
for (let i = 0; i < poly.vertices.length; i++) {
let vertex1 = poly.vertices[i];
let vertex2 = poly.vertices[i + 1] || poly.vertices[0]; // neighbouring vertex
let axis = vertex1.sub(vertex2).perp_norm();
axesToTest.push(axis);
}
// Figure out circle axis that must be checked
let closestVertex;
let closestVertexDistSqr = +Infinity;
for (let vertex of poly.vertices) {
let distSqr = circle.center.sub(vertex).magSqr();
if (distSqr < closestVertexDistSqr) {
closestVertexDistSqr = distSqr;
closestVertex = vertex;
}
}
let axis = closestVertex.sub(circle.center).norm();
axesToTest.push(axis);
// Test for overlap
for (let axis of axesToTest) {
let circleProj = proj_CIRCLE(circle, axis);
let polyProj = proj_POLY(poly, axis);
let overlap = getLineOverlap(circleProj.min, circleProj.max, polyProj.min, polyProj.max);
if (overlap === 0) {
// guaranteed no intersection
return { intersecting: false };
}
if (Math.abs(overlap) < Math.abs(shortestOverlap)) {
shortestOverlap = overlap;
shortestOverlapAxis = axis;
}
}
return {
intersecting: true,
resolutionVector: shortestOverlapAxis.mul(-shortestOverlap),
// this resolution vector is not satisfactory, I need the shortest resolution with a given direction, which would be an angle passed into this function from the trajectory of the circle
collisionAxis: shortestOverlapAxis.perp(),
// this axis is incorrect, I need the axis to be based on the trajectory of the circle which I would pass into this function as an angle
};
}
function proj_POLY(poly, axis) {
let min = +Infinity;
let max = -Infinity;
for (let vertex of poly.vertices) {
let proj = vertex.projNorm_mag(axis);
min = Math.min(proj, min);
max = Math.max(proj, max);
}
return { min, max };
}
function proj_CIRCLE(circle, axis) {
let proj = circle.center.projNorm_mag(axis);
let min = proj - circle.radius;
let max = proj + circle.radius;
return { min, max };
}
// Check for overlap of two 1 dimensional lines
function getLineOverlap(min1, max1, min2, max2) {
let min = Math.max(min1, min2);
let max = Math.min(max1, max2);
// if negative, no overlap
let result = Math.max(max - min, 0);
// add positive/negative sign depending on direction of overlap
return result * ((min1 < min2) ? 1 : -1);
};
I am assuming the polygon is convex and that the circle is moving along a straight line (at least for a some possibly small interval of time) and is not following some curved trajectory. If it is following a curved trajectory, then things get harder. In the case of curved trajectories, the basic ideas could be kept, but the actual point of collision (the point of collision resolution for the circle) might be harder to calculate. Still, I am outlining an idea, which could be extended to that case too. Plus, it could be adopted as a main approach for collision detection between a circle and a convex polygon.
I have not considered all possible cases, which may include special or extreme situations, but at least it gives you a direction to explore.
Transform in your mind the collision between the circle and the polygon into a collision between the center of the circle (a point) and a version of the polygon thickened by the circle's radius r, i.e. (i) each edge of the polygon is offset (translated) outwards by radius r along a vector perpendicular to it and pointing outside of the polygon, (ii) the vertices become circular arcs of radius r, centered at the polygons vertices and connecting the endpoints of the appropriate neighboring offset edges (basically, put circles of radius r at the vertices of the polygon and take their convex hull).
Now, the current position of the circle's center is C = [ C[0], C[1] ] and it has been moving along a straight line with direction vector V = [ V[0], V[1] ] pointing along the direction of motion (or if you prefer, think of V as the velocity of the circle at the moment when you have detected the collision). Then, there is an axis (or let's say a ray - a directed half-line) defined by the vector equation X = C - t * V, where t >= 0 (this axis is pointing to the past trajectory). Basically, this is the half-line that passes through the center point C and is aligned with/parallel to the vector V. Now, the point of resolution, i.e. the point where you want to move your circle to is the point where the axis X = C - t * V intersects the boundary of the thickened polygon.
So you have to check (1) first axis intersection for edges and then (2) axis intersection with circular arcs pertaining to the vertices of the original polygon.
Assume the polygon is given by an array of vertices P = [ P[0], P[1], ..., P[N], P[0] ] oriented counterclockwise.
(1) For each edge P[i-1]P[i] of the original polygon, relevant to your collision (these could be the two neighboring edges meeting at the vertex based on which the collision is detected, or it could be actually all edges in the case of the circle moving with very high speed and you have detected the collision very late, so that the actual collision did not even happen there, I leave this up to you, because you know better the details of your situation) do the following. You have as input data:
C = [ C[0], C[1] ]
V = [ V[0], V[1] ]
P[i-1] = [ P[i-1][0], P[i-1][1] ]
P[i] = [ P[i][0], P[i][1] ]
Do:
Normal = [ P[i-1][1] - P[i][1], P[i][0] - P[i-1][0] ];
Normal = Normal / sqrt((P[i-1][1] - P[i][1])^2 + ( P[i][0] - P[i-1][0] )^2);
// you may have calculated these already
Q_0[0] = P[i-1][0] + r*Normal[0];
Q_0[1] = P[i-1][1] + r*Normal[1];
Q_1[0] = P[i][0]+ r*Normal[0];
Q_1[1] = P[i][1]+ r*Normal[1];
Solve for s, t the linear system of equations (the equation for intersecting ):
Q_0[0] + s*(Q_1[0] - Q_0[0]) = C[0] - t*V[0];
Q_0[1] + s*(Q_1[1] - Q_0[1]) = C[1] - t*V[1];
if 0<= s <= 1 and t >= 0, you are done, and your point of resolution is
R[0] = C[0] - t*V[0];
R[1] = C[1] - t*V[1];
else
(2) For the each vertex P[i] relevant to your collision, do the following:
solve for t the quadratic equation (there is an explicit formula)
norm(P[i] - C + t*V )^2 = r^2
or expanded:
(V[0]^2 + V[1]^2) * t^2 + 2 * ( (P[i][0] - C[0])*V[0] + (P[i][1] - C[1])*V[1] )*t + ( P[i][0] - C[0])^2 + (P[i][1] - C[1])^2 ) - r^2 = 0
or if you prefer in a more code-like way:
a = V[0]^2 + V[1]^2;
b = (P[i][0] - C[0])*V[0] + (P[i][1] - C[1])*V[1];
c = (P[i][0] - C[0])^2 + (P[i][1] - C[1])^2 - r^2;
D = b^2 - a*c;
if D < 0 there is no collision with the vertex
i.e. no intersection between the line X = C - t*V
and the circle of radius r centered at P[i]
else
D = sqrt(D);
t1 = ( - b - D) / a;
t2 = ( - b + D) / a;
where t2 >= t1
Then your point of resolution is
R[0] = C[0] - t2*V[0];
R[1] = C[1] - t2*V[1];
Circle polygon intercept
If the ball is moving and if you can ensure that the ball always starts outside the polygon then the solution is rather simple.
We will call the ball and its movement the ball line. It starts at the ball's current location and end at the position the ball will be at the next frame.
To solve you find the nearest intercept to the start of the ball line.
There are two types of intercept.
Line segment (ball line) with Line segment (polygon edge)
Line segment (ball line) with circle (circle at each (convex only) polygon corner)
The example code has a Lines2 object that contains the two relevant intercept functions. The intercepts are returned as a Vec2containing two unit distances. The intercept functions are for lines (infinite length) not line sgements. If there is no intercept then the return is undefined.
For the line intercepts Line2.unitInterceptsLine(line, result = new Vec2()) the unit values (in result) are the unit distance along each line from the start. negative values are behind the start.
To take in account of the ball radius each polygon edge is offset the ball radius along its normal. It is important that the polygon edges have a consistent direction. In the example the normal is to the right of the line and the polygon points are in a clockwise direction.
For the line segment / circle intercepts Line2.unitInterceptsCircle(center, radius, result = new Vec2()) the unit values (in result) are the unit distance along the line where it intercepts the circle. result.x will always contain the closest intercept (assuming you start outside the circle). If there is an intercept there ways always be two, even if they are at the same point.
Example
The example contains all that is needed
The objects of interest are ball and poly
ball defines the ball and its movement. There is also code to draw it for the example
poly holds the points of the polygon. Converts the points to offset lines depending on the ball radius. It is optimized to that it only calculates the lines if the ball radius changes.
The function poly.movingBallIntercept is the function that does all the work. It take a ball object and an optional results vector.
It returns the position as a Vec2 of the ball if it contacts the polygon.
It does this by finding the smallest unit distance to the offset lines, and point (as circle) and uses that unit distance to position the result.
Note that if the ball is inside the polygon the intercepts with the corners is reversed. The function Line2.unitInterceptsCircle does provide 2 unit distance where the line enters and exits the circle. However you need to know if you are inside or outside to know which one to use. The example assumes you are outside the polygon.
Instructions
Move the mouse to change the balls path.
Click to set the balls starting position.
Math.EPSILON = 1e-6;
Math.isSmall = val => Math.abs(val) < Math.EPSILON;
Math.isUnit = u => !(u < 0 || u > 1);
Math.TAU = Math.PI * 2;
/* export {Vec2, Line2} */ // this should be a module
var temp;
function Vec2(x = 0, y = (temp = x, x === 0 ? (x = 0 , 0) : (x = x.x, temp.y))) {
this.x = x;
this.y = y;
}
Vec2.prototype = {
init(x, y = (temp = x, x = x.x, temp.y)) { this.x = x; this.y = y; return this }, // assumes x is a Vec2 if y is undefined
copy() { return new Vec2(this) },
equal(v) { return (this.x - v.x) === 0 && (this.y - v.y) === 0 },
isUnits() { return Math.isUnit(this.x) && Math.isUnit(this.y) },
add(v, res = this) { res.x = this.x + v.x; res.y = this.y + v.y; return res },
sub(v, res = this) { res.x = this.x - v.x; res.y = this.y - v.y; return res },
scale(val, res = this) { res.x = this.x * val; res.y = this.y * val; return res },
invScale(val, res = this) { res.x = this.x / val; res.y = this.y / val; return res },
dot(v) { return this.x * v.x + this.y * v.y },
uDot(v, div) { return (this.x * v.x + this.y * v.y) / div },
cross(v) { return this.x * v.y - this.y * v.x },
uCross(v, div) { return (this.x * v.y - this.y * v.x) / div },
get length() { return this.lengthSqr ** 0.5 },
set length(l) { this.scale(l / this.length) },
get lengthSqr() { return this.x * this.x + this.y * this.y },
rot90CW(res = this) {
const y = this.x;
res.x = -this.y;
res.y = y;
return res;
},
};
const wV1 = new Vec2(), wV2 = new Vec2(), wV3 = new Vec2(); // pre allocated work vectors used by Line2 functions
function Line2(p1 = new Vec2(), p2 = (temp = p1, p1 = p1.p1 ? p1.p1 : p1, temp.p2 ? temp.p2 : new Vec2())) {
this.p1 = p1;
this.p2 = p2;
}
Line2.prototype = {
init(p1, p2 = (temp = p1, p1 = p1.p1, temp.p2)) { this.p1.init(p1); this.p2.init(p2) },
copy() { return new Line2(this) },
asVec(res = new Vec2()) { return this.p2.sub(this.p1, res) },
unitDistOn(u, res = new Vec2()) { return this.p2.sub(this.p1, res).scale(u).add(this.p1) },
translate(vec, res = this) {
this.p1.add(vec, res.p1);
this.p2.add(vec, res.p2);
return res;
},
translateNormal(amount, res = this) {
this.asVec(wV1).rot90CW().length = -amount;
this.translate(wV1, res);
return res;
},
unitInterceptsLine(line, res = new Vec2()) { // segments
this.asVec(wV1);
line.asVec(wV2);
const c = wV1.cross(wV2);
if (Math.isSmall(c)) { return }
wV3.init(this.p1).sub(line.p1);
res.init(wV1.uCross(wV3, c), wV2.uCross(wV3, c));
return res;
},
unitInterceptsCircle(point, radius, res = new Vec2()) {
this.asVec(wV1);
var b = -2 * this.p1.sub(point, wV2).dot(wV1);
const c = 2 * wV1.lengthSqr;
const d = (b * b - 2 * c * (wV2.lengthSqr - radius * radius)) ** 0.5
if (isNaN(d)) { return }
return res.init((b - d) / c, (b + d) / c);
},
};
/* END of file */ // Vec2 and Line2 module
/* import {vec2, Line2} from "whateverfilename.jsm" */ // Should import vec2 and line2
const POLY_SCALE = 0.5;
const ball = {
pos: new Vec2(-150,0),
delta: new Vec2(10, 10),
radius: 20,
drawPath(ctx) {
ctx.beginPath();
ctx.arc(this.pos.x, this.pos.y, this.radius, 0, Math.TAU);
ctx.stroke();
},
}
const poly = {
bRadius: 0,
lines: [],
set ballRadius(radius) {
const len = this.points.length
this.bRadius = ball.radius;
i = 0;
while (i < len) {
let line = this.lines[i];
if (line) { line.init(this.points[i], this.points[(i + 1) % len]) }
else { line = new Line2(new Vec2(this.points[i]), new Vec2(this.points[(i + 1) % len])) }
this.lines[i++] = line.translateNormal(radius);
}
this.lines.length = i;
},
points: [
new Vec2(-200, -150).scale(POLY_SCALE),
new Vec2(200, -100).scale(POLY_SCALE),
new Vec2(100, 0).scale(POLY_SCALE),
new Vec2(200, 100).scale(POLY_SCALE),
new Vec2(-200, 75).scale(POLY_SCALE),
new Vec2(-150, -50).scale(POLY_SCALE),
],
drawBallLines(ctx) {
if (this.lines.length) {
const r = this.bRadius;
ctx.beginPath();
for (const l of this.lines) {
ctx.moveTo(l.p1.x, l.p1.y);
ctx.lineTo(l.p2.x, l.p2.y);
}
for (const p of this.points) {
ctx.moveTo(p.x + r, p.y);
ctx.arc(p.x, p.y, r, 0, Math.TAU);
}
ctx.stroke()
}
},
drawPath(ctx) {
ctx.beginPath();
for (const p of this.points) { ctx.lineTo(p.x, p.y) }
ctx.closePath();
ctx.stroke();
},
movingBallIntercept(ball, res = new Vec2()) {
if (this.bRadius !== ball.radius) { this.ballRadius = ball.radius }
var i = 0, nearest = Infinity, nearestGeom, units = new Vec2();
const ballT = new Line2(ball.pos, ball.pos.add(ball.delta, new Vec2()));
for (const p of this.points) {
const res = ballT.unitInterceptsCircle(p, ball.radius, units);
if (res && units.x < nearest && Math.isUnit(units.x)) { // assumes ball started outside poly so only need first point
nearest = units.x;
nearestGeom = ballT;
}
}
for (const line of this.lines) {
const res = line.unitInterceptsLine(ballT, units);
if (res && units.x < nearest && units.isUnits()) { // first unit.x is for unit dist on line
nearest = units.x;
nearestGeom = ballT;
}
}
if (nearestGeom) { return ballT.unitDistOn(nearest, res) }
return;
},
}
const ctx = canvas.getContext("2d");
var w = canvas.width, cw = w / 2;
var h = canvas.height, ch = h / 2
requestAnimationFrame(mainLoop);
// line and point for displaying mouse interaction. point holds the result if any
const line = new Line2(ball.pos, ball.pos.add(ball.delta, new Vec2())), point = new Vec2();
function mainLoop() {
ctx.setTransform(1,0,0,1,0,0); // reset transform
if(w !== innerWidth || h !== innerHeight){
cw = (w = canvas.width = innerWidth) / 2;
ch = (h = canvas.height = innerHeight) / 2;
}else{
ctx.clearRect(0,0,w,h);
}
ctx.setTransform(1,0,0,1,cw,ch); // center to canvas
if (mouse.button) { ball.pos.init(mouse.x - cw, mouse.y - ch) }
line.p2.init(mouse.x - cw, mouse.y - ch);
line.p2.sub(line.p1, ball.delta);
ctx.lineWidth = 1;
ctx.strokeStyle = "#000"
poly.drawPath(ctx)
ctx.strokeStyle = "#F804"
poly.drawBallLines(ctx);
ctx.strokeStyle = "#F00"
ctx.beginPath();
ctx.arc(ball.pos.x, ball.pos.y, ball.radius, 0, Math.TAU);
ctx.moveTo(line.p1.x, line.p1.y);
ctx.lineTo(line.p2.x, line.p2.y);
ctx.stroke();
ctx.strokeStyle = "#00f"
ctx.lineWidth = 2;
ctx.beginPath();
if (poly.movingBallIntercept(ball, point)) {
ctx.arc(point.x, point.y, ball.radius, 0, Math.TAU);
} else {
ctx.arc(line.p2.x, line.p2.y, ball.radius, 0, Math.TAU);
}
ctx.stroke();
requestAnimationFrame(mainLoop);
}
const mouse = {x:0, y:0, button: false};
function mouseEvents(e) {
const bounds = canvas.getBoundingClientRect();
mouse.x = e.pageX - bounds.left - scrollX;
mouse.y = e.pageY - bounds.top - scrollY;
mouse.button = e.type === "mousedown" ? true : e.type === "mouseup" ? false : mouse.button;
}
["mousedown","mouseup","mousemove"].forEach(name => document.addEventListener(name,mouseEvents));
#canvas {
position: absolute;
top: 0px;
left: 0px;
}
<canvas id="canvas"></canvas>
Click to position ball. Move mouse to test trajectory
Vec2 and Line2
To make it easier a vector library will help. For the example I wrote a quick Vec2 and Line2 object (Note only functions used in the example have been tested, Note The object are designed for performance, inexperienced coders should avoid using these objects and opt for a more standard vector and line library)
It's probably not what you're looking for, but here's a way to do it (if you're not looking for perfect precision) :
You can try to approximate the position instead of calculating it.
The way you set up your code has a big advantage : You have the last position of the circle before the collision. Thanks to that, you can just "iterate" through the trajectory and try to find a position that is closest to the intersection position.
I'll assume you already have a function that tells you if a circle is intersecting with the polygon.
Code (C++) :
// What we need :
Vector startPos; // Last position of the circle before the collision
Vector currentPos; // Current, unwanted position
Vector dir; // Direction (a unit vector) of the circle's velocity
float distance = compute_distance(startPos, currentPos); // The distance from startPos to currentPos.
Polygon polygon; // The polygon
Circle circle; // The circle.
unsigned int iterations_count = 10; // The number of iterations that will be done. The higher this number, the more precise the resolution.
// The algorithm :
float currentDistance = distance / 2.f; // We start at the half of the distance.
Circle temp_copy; // A copy of the real circle to "play" with.
for (int i = 0; i < iterations_count; ++i) {
temp_copy.pos = startPos + currentDistance * dir;
if (checkForCollision(temp_copy, polygon)) {
currentDistance -= currentDistance / 2.f; // We go towards startPos by the half of the current distance.
}
else {
currentDistance += currentDistance / 2.f; // We go towards currentPos by the half of the current distance.
}
}
// currentDistance now contains the distance between startPos and the intersection point
// And this is where you should place your circle :
Vector intersectionPoint = startPos + currentDistance * dir;
I haven't tested this code so I hope there's no big mistake in there. It's also not optimized and there are a few problems with this approach (the intersection point could end up inside the polygon) so it still needs to be improved but I think you get the idea.
The other (big, depending on what you're doing) problem with this is that it's an approximation and not a perfect answer.
Hope this helps !
I'm not sure if I understood the scenario correctly, but an efficient straight forward use case would be, to only use a square bounding box of your circle first, calculating intersection of that square with your polygone is extremely fast, much much faster, than using the circle. Once you detect an intersection of that square and the polygone, you need to think or to write which precision is mostly suitable for your scenarion. If you need a better precision, than at this state, you can go on as this from here:
From the 90° angle of your sqare intersection, you draw a 45° degree line until it touches your circle, at this point, where it touches, you draw a new square, but this time, the square is embedded into the circle, let it run now, until this new square intersects the polygon, once it intersects, you have a guaranteed circle intersection. Depending on your needed precision, you can simply play around like this.
I'm not sure what your next problem is from here? If it has to be only the inverse of the circles trajectory, than it simply must be that reverse, I'm really not sure what I'm missing here.

Spline offset in three.js

I have to make 3d text from font glyphs. Yes, I know that I can use TextGeometry, but I need to draw this manually because I need to do offset on font splines.
At this moment I have splines with their points and I can draw letters.
From points I know: previousPoint, currentPoint and nextPoint and I need to compute bisector between previous and next points and I have no idea to do that.
Or if is another way to move spline points outer of initial position to do offset.
My idea:
Thank you!
EDIT:
With yours answers I obtained correct values for each splines from font, but only at 'o' and '0' I have a problem.
This method draw a weird tangent in bottom of the letter and I don't know to resolve this problem..
here is the result
Anybody know how to resolve this?
EDIT 2:
Finally I finished my project. And this is the final product ( .stl exporter )
final offset
Thank you for yours answers!
There are the result from: x = (prev_x + next_x) / 2 and y = (prev_y + next_y) / 2
wrong result
desired result
Here is my code where let points is all the points from the path:
getPathPoints(path) {
let points = path.getPoints();
console.log(points)
for (let i = 0; i < points.length; i++) {
let A = points.extended(i - 1); // previousPoint => where extends is a custom array prototype
let B = points.extended(i); // currentPoint
let C = points.extended(i + 1); // nextPoint
let x = (A.x + C.x) / 2;
let y = (A.y + C.y) / 2;
let bisector = new THREE.Vector2(x,y);
console.log(bisector);
}
}
What splines describe your glyphs?
I know that TTF fonts use quadratic Bezier curves. For Bezier direction vector in starting and ending points has direction onto control point. So difference
S = ControlPoint[1] - ControlPoint[0]
represents direction in the starting point, difference
E = ControlPoint[1] - ControlPoint[2]
represents direction in the ending point.
Normalize these vectors for two neighbour curves and add them - now you have bisector vector.
Bisector = E(i).Normalized + S(i+1).Normalized

Isometric tilemap using canvas (with click detection)

I am currently developing a game, which requires a map consisting of various tile images. I managed to make them display correctly (see second image) but I am now unsure of how to calculate the clicked tile from the mouse position.
Are there any existing libraries for this purpose?
Please also note, that the tile images aren't drawn perfectly "corner-facing-camera", they are slightly rotated clockwise.
Isometric Transformations
Define a projection
Isometric display is the same as standard display, the only thing that has changed is the direction of the x and y axis. Normally the x axis is defined as (1,0) one unit across and zero down and the y axis is (0,1) zero units across and one down. For isometric (strictly speaking your image is a dimetric projection) you will have something like x axis (0.5,1) and y axis (-1,0.5)
The Matrix
From this you can create a rendering matrix with 6 values Two each for both axes and two for the origin, which I will ignore for now (the origin) and just use the 4 for the axis and assume that the origin is always at 0,0
var dimetricMatrix = [0.5,1.0,-1,0.5]; // x and y axis
Matrix transformation
From that you can get a point on the display that matches a given isometric coordinate. Lets say the blocks are 200 by 200 pixels and that you address each block by the block x and y. Thus the block in the bottom of your image is at x = 2 and y = 1 (the first top block is x = 0, y = 0)
Using the matrix we can get the pixel location of the block
var blockW = 200;
var blockH = 200;
var locX = 2;
var locY = 1;
function getLoc(x,y){
var xx,yy; // intermediate results
var m = dimetricMatrix; // short cut to make code readable
x *= blockW; // scale up
y *= blockH;
// now move along the projection x axis
xx = x * m[0];
yy = x * m[1];
// then add the distance along the y axis
xx += y * m[2];
yy += y * m[3];
return {x : xx, y : yy};
}
Befoer I move on you can see that I have scaled the x and y by the block size. We can simplify the above code and include the scale 200,200 in the matrix
var xAxis = [0.5, 1.0];
var yAxis = [-1, 0.5];
var blockW = 200;
var blockH = 200;
// now create the matrix and scale the x and y axis
var dimetricMatrix = [
xAxis[0] * blockW,
xAxis[1] * blockW,
yAxis[0] * blockH,
yAxis[1] * blockH,
]; // x and y axis
The matrix holds the scale in the x and y axis so that the two numbers for x axis tell us the direction and length of a transformed unit.
Simplify function
And redo the getLoc function for speed and efficiency
function transformPoint(point,matrix,result){
if(result === undefined){
result = {};
}
// now move along the projection x axis
result.x = point.x * matrix[0] + point.y * matrix[2];
result.y = point.x * matrix[1] + point.y * matrix[3];
return result;
}
So pass a point and get a transformed point back. The result argument allows you to pass an existing point and that saves having to allocate a new point if you are doing it often.
var point = {x : 2, y : 1};
var screen = transformPoint(point,dimetricMatrix);
// result is the screen location of the block
// next time
screen = transformPoint(point,dimetricMatrix,screen); // pass the screen obj
// to avoid those too
// GC hits that kill
// game frame rates
Inverting the Matrix
All that is handy but you need the reverse of what we just did. Luckily the way matrices work allows us to reverse the process by inverting the matrix.
function invertMatrix(matrix){
var m = matrix; // shortcut to make code readable
var rm = [0,0,0,0]; // resulting matrix
// get the cross product of the x and y axis. It is the area of the rectangle made by the
// two axis
var cross = m[0] * m[3] - m[1] * m[2]; // I call it the cross but most will call
// it the determinate (I think that cross
// product is more suited to geometry while
// determinate is for maths geeks)
rm[0] = m[3] / cross; // invert both axis and unscale (if cross is 1 then nothing)
rm[1] = -m[1] / cross;
rm[2] = -m[2] / cross;
rm[3] = m[0] / cross;
return rm;
}
Now we can invert our matrix
var dimetricMatrixInv = invertMatrix(dimetricMatrix); // get the invers
And now that we have the inverse matrix we can use the transform function to convert from a screen location to a block location
var screen = {x : 100, y : 200};
var blockLoc = transformPoint(screen, dimetricMatrixInv );
// result is the location of the block
The Matrix for rendering
For a bit of magic the transformation matrix dimetricMatrix can also be used by the 2D canvas, but you need to add the origin.
var m = dimetricMatrix;
ctx.setTransform(m[0], m[1], m[2], m[3], 0, 0); // assume origin at 0,0
Now you can draw a box around the block with
ctx.strokeRect(2,1,1,1); // 3rd by 2nd block 1 by 1 block wide.
The origin
I have left out the origin in all the above, I will leave that up to you to find as there is a trillion pages online about matrices as all 2D and 3D rendering use them and getting a good deep knowledge of them is important if you wish to get into computer visualization.

Canvas Graph plotting data incorrectly

I have made a simple graph in a canvas but am having difficulty with two issues.
The first issue is setting the vertical axis with an appropriate scale automatically with enough room for each data value in an array. Ideally i'd like the numbers to be more rounded to the nearest million or thousand etc depending on it's actual value ranges rather than a value like 33145 as the first scale line.
Currently one value is too high for the scale and is not being drawn on the canvas because it is out of bounds.
The second issue, is the points don't seem to be plotting in their correct location, which I am unsure where my mistake was.
I made a JSFiddle as for the most part it might be a bit confusing without seeing it in action:
http://jsfiddle.net/ezttywzr/
This is how i plot my data and draw my vertical axis:
Vertical Axis:
var x = 0,
y,
range = data.max() - data.min(),
valueStep = range / 10,
// get width of largest number
margin = 3 + ctx.measureText(data.min() + (valueStep*10)).width,
pixelStep = (graph.height-40) / 10,
verticalP = pixelStep,
output;
// draw left hand values
for(var i = 0; i < 11; i++){
output = data.min() + (valueStep*i);
y = graph.height-20 - (verticalP + i*pixelStep);
ctx.fillText(output,x,y+6);
ctx.beginPath();
ctx.moveTo(margin, y);
ctx.lineTo(x2,y);
ctx.stroke();
}
Data Plotting:
var y = graph.height,
x = margin,
pos,
valueStep = (graph.width-(margin*2)) / data.length,
pixelRange = graph.height-20,
pp = range / pixelRange;
for(var i = 0; i < data.length; i++){
x += valueStep;
pos = x - (valueStep/2);
ctx.beginPath();
ctx.moveTo(x, graph.height-20);
ctx.lineTo(x, graph.height);
ctx.stroke();
ctx.fillText('Week '+(i+1),pos-(ctx.measureText('Week '+(i+1)).width/2),y);
ctx.beginPath();
ctx.arc(pos,(graph.height-20)-(verticalP+(data[i]/pp)),2,0,2*Math.PI);
ctx.stroke();
ctx.fill();
}
Nice job so far.
I made a few changes: http://jsfiddle.net/ezttywzr/2/
To get the scale I used
STEP = data.max() / NUM_HORIZONTAL_LINES
Where NUM_HORIZONTAL_LINES is the number of horizontal lines you want above the x-axis. In this case I used 10.
This means the first line will be 1 * STEP, the second will be 2 * STEP, the third will be 3 * STEP and so on..
This scale is convenient because it guarantees that the max value fits on the graph. In fact, the max value is on the top line because of the way we defined the scale.
Once we have our scale it's easy to calculate the position of the points relative to the x-axis. It's simply:
(PIXELS_PER_STEP / STEP) * VALUE
To go a step further you can do some math to round the top point of the graph up and pick a scale with that has nice round numbers.

Javascript: Find point on perpendicular line always the same distance away

I'm trying to find a point that is equal distance away from the middle of a perpendicular line. I want to use this point to create a Bézier curve using the start and end points, and this other point I'm trying to find.
I've calculated the perpendicular line, and I can plot points on that line, but the problem is that depending on the angle of the line, the points get further away or closer to the original line, and I want to be able to calculate it so it's always X units away.
Take a look at this JSFiddle which shows the original line, with some points plotted along the perpendicular line:
http://jsfiddle.net/eLxcB/1/.
If you change the start and end points, you can see these plotted points getting closer together or further away.
How do I get them to be uniformly the same distance apart from each other no matter what the angle is?
Code snippit below:
// Start and end points
var startX = 120
var startY = 150
var endX = 180
var endY = 130
// Calculate how far above or below the control point should be
var centrePointX = ((startX + endX) / 2);
var centrePointY = ((startY + endY) / 2);
// Calculate slopes and Y intersects
var lineSlope = (endY - startY) / (endX - startX);
var perpendicularSlope = -1 / lineSlope;
var yIntersect = centrePointY - (centrePointX * perpendicularSlope);
// Draw a line between the two original points
R.path('M '+startX+' '+startY+', L '+endX+' '+endY);
Generally you can get the coordinates of a normal of a line like this:
P1 = {r * cos(a) + Cx, -r * sin(a) + Cy},
P2 = {-r * cos(a) + Cx, r * sin(a) + Cy}.
A demo applying this to your case at jsFiddle.

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