I have a line that goes from points A to B; I have (x,y) of both points. I also have a rectangle that\'s centered at B and the width and height of the rectangle.
I need to find the point in the line that intersects the rectangle. Is there a formula that gives me the (x,y) of that point?
You might want to check out Graphics Gems - this is a classic set of routines for graphics and includes many of the algorithms required. Although it\'s in C and slightly dated the algorithms still sparkle and it should be trivial to transfer to other languages.
For your current problem the just create the four lines for the rectangle and see which intersect your given line.
The point A is always outside of the rectangle and the point B is always at the center of the rectangle
Assuming the rectangle is axis-aligned, this makes things pretty simple:
The slope of the line is s = (Ay - By)/(Ax - Bx).
Once you know the edge it intersects you know one coordinate: x = Bx ± w/2 or y = By ± h/2 depending on which edge you hit. The other coordinate is given by y = By + s * w/2 or x = Bx + (h/2)/s.
/**
* Finds the intersection point between
* * the rectangle
* with parallel sides to the x and y axes
* * the half-line pointing towards (x,y)
* originating from the middle of the rectangle
*
* Note: the function works given min[XY] <= max[XY],
* even though minY may not be the \"top\" of the rectangle
* because the coordinate system is flipped.
* Note: if the input is inside the rectangle,
* the line segment wouldn\'t have an intersection with the rectangle,
* but the projected half-line does.
* Warning: passing in the middle of the rectangle will return the midpoint itself
* there are infinitely many half-lines projected in all directions,
* so let\'s just shortcut to midpoint (GIGO).
*
* @param x:Number x coordinate of point to build the half-line from
* @param y:Number y coordinate of point to build the half-line from
* @param minX:Number the \"left\" side of the rectangle
* @param minY:Number the \"top\" side of the rectangle
* @param maxX:Number the \"right\" side of the rectangle
* @param maxY:Number the \"bottom\" side of the rectangle
* @param validate:boolean (optional) whether to treat point inside the rect as error
* @return an object with x and y members for the intersection
* @throws if validate == true and (x,y) is inside the rectangle
* @author TWiStErRob
* @licence Dual CC0/WTFPL/Unlicence, whatever floats your boat
* @see <a href=\"http://stackoverflow.com/a/31254199/253468\">source</a>
* @see <a href=\"http://stackoverflow.com/a/18292964/253468\">based on</a>
*/
function pointOnRect(x, y, minX, minY, maxX, maxY, validate) {
//assert minX <= maxX;
//assert minY <= maxY;
if (validate && (minX < x && x < maxX) && (minY < y && y < maxY))
throw \"Point \" + [x,y] + \"cannot be inside \"
+ \"the rectangle: \" + [minX, minY] + \" - \" + [maxX, maxY] + \".\";
var midX = (minX + maxX) / 2;
var midY = (minY + maxY) / 2;
// if (midX - x == 0) -> m == ±Inf -> minYx/maxYx == x (because value / ±Inf = ±0)
var m = (midY - y) / (midX - x);
if (x <= midX) { // check \"left\" side
var minXy = m * (minX - x) + y;
if (minY <= minXy && minXy <= maxY)
return {x: minX, y: minXy};
}
if (x >= midX) { // check \"right\" side
var maxXy = m * (maxX - x) + y;
if (minY <= maxXy && maxXy <= maxY)
return {x: maxX, y: maxXy};
}
if (y <= midY) { // check \"top\" side
var minYx = (minY - y) / m + x;
if (minX <= minYx && minYx <= maxX)
return {x: minYx, y: minY};
}
if (y >= midY) { // check \"bottom\" side
var maxYx = (maxY - y) / m + x;
if (minX <= maxYx && maxYx <= maxX)
return {x: maxYx, y: maxY};
}
// edge case when finding midpoint intersection: m = 0/0 = NaN
if (x === midX && y === midY) return {x: x, y: y};
// Should never happen :) If it does, please tell me!
throw \"Cannot find intersection for \" + [x,y]
+ \" inside rectangle \" + [minX, minY] + \" - \" + [maxX, maxY] + \".\";
}
(function tests() {
var left = 100, right = 200, top = 50, bottom = 150; // a square, really
var hMiddle = (left + right) / 2, vMiddle = (top + bottom) / 2;
function intersectTestRect(x, y) { return pointOnRect(x,y, left,top, right,bottom, true); }
function intersectTestRectNoValidation(x, y) { return pointOnRect(x,y, left,top, right,bottom, false); }
function checkTestRect(x, y) { return function() { return pointOnRect(x,y, left,top, right,bottom, true); }; }
QUnit.test(\"intersects left side\", function(assert) {
var leftOfRect = 0, closerLeftOfRect = 25;
assert.deepEqual(intersectTestRect(leftOfRect, 25), {x:left, y:75}, \"point above top\");
assert.deepEqual(intersectTestRect(closerLeftOfRect, top), {x:left, y:80}, \"point in line with top\");
assert.deepEqual(intersectTestRect(leftOfRect, 70), {x:left, y:90}, \"point above middle\");
assert.deepEqual(intersectTestRect(leftOfRect, vMiddle), {x:left, y:100}, \"point exact middle\");
assert.deepEqual(intersectTestRect(leftOfRect, 130), {x:left, y:110}, \"point below middle\");
assert.deepEqual(intersectTestRect(closerLeftOfRect, bottom), {x:left, y:120}, \"point in line with bottom\");
assert.deepEqual(intersectTestRect(leftOfRect, 175), {x:left, y:125}, \"point below bottom\");
});
QUnit.test(\"intersects right side\", function(assert) {
var rightOfRect = 300, closerRightOfRect = 250;
assert.deepEqual(intersectTestRect(rightOfRect, 25), {x:right, y:75}, \"point above top\");
assert.deepEqual(intersectTestRect(closerRightOfRect, top), {x:right, y:75}, \"point in line with top\");
assert.deepEqual(intersectTestRect(rightOfRect, 70), {x:right, y:90}, \"point above middle\");
assert.deepEqual(intersectTestRect(rightOfRect, vMiddle), {x:right, y:100}, \"point exact middle\");
assert.deepEqual(intersectTestRect(rightOfRect, 130), {x:right, y:110}, \"point below middle\");
assert.deepEqual(intersectTestRect(closerRightOfRect, bottom), {x:right, y:125}, \"point in line with bottom\");
assert.deepEqual(intersectTestRect(rightOfRect, 175), {x:right, y:125}, \"point below bottom\");
});
QUnit.test(\"intersects top side\", function(assert) {
var aboveRect = 0;
assert.deepEqual(intersectTestRect(80, aboveRect), {x:115, y:top}, \"point left of left\");
assert.deepEqual(intersectTestRect(left, aboveRect), {x:125, y:top}, \"point in line with left\");
assert.deepEqual(intersectTestRect(120, aboveRect), {x:135, y:top}, \"point left of middle\");
assert.deepEqual(intersectTestRect(hMiddle, aboveRect), {x:150, y:top}, \"point exact middle\");
assert.deepEqual(intersectTestRect(180, aboveRect), {x:165, y:top}, \"point right of middle\");
assert.deepEqual(intersectTestRect(right, aboveRect), {x:175, y:top}, \"point in line with right\");
assert.deepEqual(intersectTestRect(220, aboveRect), {x:185, y:top}, \"point right of right\");
});
QUnit.test(\"intersects bottom side\", function(assert) {
var belowRect = 200;
assert.deepEqual(intersectTestRect(80, belowRect), {x:115, y:bottom}, \"point left of left\");
assert.deepEqual(intersectTestRect(left, belowRect), {x:125, y:bottom}, \"point in line with left\");
assert.deepEqual(intersectTestRect(120, belowRect), {x:135, y:bottom}, \"point left of middle\");
assert.deepEqual(intersectTestRect(hMiddle, belowRect), {x:150, y:bottom}, \"point exact middle\");
assert.deepEqual(intersectTestRect(180, belowRect), {x:165, y:bottom}, \"point right of middle\");
assert.deepEqual(intersectTestRect(right, belowRect), {x:175, y:bottom}, \"point in line with right\");
assert.deepEqual(intersectTestRect(220, belowRect), {x:185, y:bottom}, \"point right of right\");
});
QUnit.test(\"intersects a corner\", function(assert) {
assert.deepEqual(intersectTestRect(left-50, top-50), {x:left, y:top}, \"intersection line aligned with top-left corner\");
assert.deepEqual(intersectTestRect(right+50, top-50), {x:right, y:top}, \"intersection line aligned with top-right corner\");
assert.deepEqual(intersectTestRect(left-50, bottom+50), {x:left, y:bottom}, \"intersection line aligned with bottom-left corner\");
assert.deepEqual(intersectTestRect(right+50, bottom+50), {x:right, y:bottom}, \"intersection line aligned with bottom-right corner\");
});
QUnit.test(\"on the corners\", function(assert) {
assert.deepEqual(intersectTestRect(left, top), {x:left, y:top}, \"top-left corner\");
assert.deepEqual(intersectTestRect(right, top), {x:right, y:top}, \"top-right corner\");
assert.deepEqual(intersectTestRect(right, bottom), {x:right, y:bottom}, \"bottom-right corner\");
assert.deepEqual(intersectTestRect(left, bottom), {x:left, y:bottom}, \"bottom-left corner\");
});
QUnit.test(\"on the edges\", function(assert) {
assert.deepEqual(intersectTestRect(hMiddle, top), {x:hMiddle, y:top}, \"top edge\");
assert.deepEqual(intersectTestRect(right, vMiddle), {x:right, y:vMiddle}, \"right edge\");
assert.deepEqual(intersectTestRect(hMiddle, bottom), {x:hMiddle, y:bottom}, \"bottom edge\");
assert.deepEqual(intersectTestRect(left, vMiddle), {x:left, y:vMiddle}, \"left edge\");
});
QUnit.test(\"validates inputs\", function(assert) {
assert.throws(checkTestRect(hMiddle, vMiddle), /cannot be inside/, \"center\");
assert.throws(checkTestRect(hMiddle-10, vMiddle-10), /cannot be inside/, \"top left of center\");
assert.throws(checkTestRect(hMiddle-10, vMiddle), /cannot be inside/, \"left of center\");
assert.throws(checkTestRect(hMiddle-10, vMiddle+10), /cannot be inside/, \"bottom left of center\");
assert.throws(checkTestRect(hMiddle, vMiddle-10), /cannot be inside/, \"above center\");
assert.throws(checkTestRect(hMiddle, vMiddle), /cannot be inside/, \"center\");
assert.throws(checkTestRect(hMiddle, vMiddle+10), /cannot be inside/, \"below center\");
assert.throws(checkTestRect(hMiddle+10, vMiddle-10), /cannot be inside/, \"top right of center\");
assert.throws(checkTestRect(hMiddle+10, vMiddle), /cannot be inside/, \"right of center\");
assert.throws(checkTestRect(hMiddle+10, vMiddle+10), /cannot be inside/, \"bottom right of center\");
assert.throws(checkTestRect(left+10, vMiddle-10), /cannot be inside/, \"right of left edge\");
assert.throws(checkTestRect(left+10, vMiddle), /cannot be inside/, \"right of left edge\");
assert.throws(checkTestRect(left+10, vMiddle+10), /cannot be inside/, \"right of left edge\");
assert.throws(checkTestRect(right-10, vMiddle-10), /cannot be inside/, \"left of right edge\");
assert.throws(checkTestRect(right-10, vMiddle), /cannot be inside/, \"left of right edge\");
assert.throws(checkTestRect(right-10, vMiddle+10), /cannot be inside/, \"left of right edge\");
assert.throws(checkTestRect(hMiddle-10, top+10), /cannot be inside/, \"below top edge\");
assert.throws(checkTestRect(hMiddle, top+10), /cannot be inside/, \"below top edge\");
assert.throws(checkTestRect(hMiddle+10, top+10), /cannot be inside/, \"below top edge\");
assert.throws(checkTestRect(hMiddle-10, bottom-10), /cannot be inside/, \"above bottom edge\");
assert.throws(checkTestRect(hMiddle, bottom-10), /cannot be inside/, \"above bottom edge\");
assert.throws(checkTestRect(hMiddle+10, bottom-10), /cannot be inside/, \"above bottom edge\");
});
QUnit.test(\"doesn\'t validate inputs\", function(assert) {
assert.deepEqual(intersectTestRectNoValidation(hMiddle-10, vMiddle-10), {x:left, y:top}, \"top left of center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle-10, vMiddle), {x:left, y:vMiddle}, \"left of center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle-10, vMiddle+10), {x:left, y:bottom}, \"bottom left of center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle, vMiddle-10), {x:hMiddle, y:top}, \"above center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle, vMiddle), {x:hMiddle, y:vMiddle}, \"center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle, vMiddle+10), {x:hMiddle, y:bottom}, \"below center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle+10, vMiddle-10), {x:right, y:top}, \"top right of center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle+10, vMiddle), {x:right, y:vMiddle}, \"right of center\");
assert.deepEqual(intersectTestRectNoValidation(hMiddle+10, vMiddle+10), {x:right, y:bottom}, \"bottom right of center\");
});
})();<link href=\"https://code.jquery.com/qunit/qunit-2.3.2.css\" rel=\"stylesheet\"/>
<script src=\"https://code.jquery.com/qunit/qunit-2.3.2.js\"></script>
<div id=\"qunit\"></div>Here is a solution in Java that returns true if a line segment (the first 4 parameters) intersects an axis aligned rectangle (the last 4 parameters). It would be trivial to return the intersection point instead of a boolean. It works by first checking if completely outside, else using the line equation y=m*x+b. We know the lines that make up the rectangle are axis aligned, so the checks are easy.
public boolean aabbContainsSegment (float x1, float y1, float x2, float y2, float minX, float minY, float maxX, float maxY) {
// Completely outside.
if ((x1 <= minX && x2 <= minX) || (y1 <= minY && y2 <= minY) || (x1 >= maxX && x2 >= maxX) || (y1 >= maxY && y2 >= maxY))
return false;
float m = (y2 - y1) / (x2 - x1);
float y = m * (minX - x1) + y1;
if (y > minY && y < maxY) return true;
y = m * (maxX - x1) + y1;
if (y > minY && y < maxY) return true;
float x = (minY - y1) / m + x1;
if (x > minX && x < maxX) return true;
x = (maxY - y1) / m + x1;
if (x > minX && x < maxX) return true;
return false;
}
It is possible to shortcut if the start or end of the segment is inside the rectangle, but probably it is better to just do the math, which will always return true if either or both segment ends are inside. If you want the shortcut anyway, insert the code below after the \"completely outside\" check.
// Start or end inside.
if ((x1 > minX && x1 < maxX && y1 > minY && y1 < maxY) || (x2 > minX && x2 < maxX && y2 > minY && y2 < maxY)) return true;
I\'ll not give you a program to do that, but here is how you can do it:
I am not a math fan nor do I particularly enjoy translating stuff from other languages if others have already done so, so whenever I complete a boring translation task, I add it to the article that led me to the code. To prevent anyone doing double work.
So if you want to have this intersection code in C#, have a look here http://dotnetbyexample.blogspot.nl/2013/09/utility-classes-to-check-if-lines-andor.html
Another option that you can consider especially if you are planning on testing many lines with the same rectangle is to transform your coordinate system to have the axes align with diagonals of the rectangle. Then since your line or ray starts at the center of the rectangle you can determine the angle then you can tell which segment it will intersect by the angle (i.e. <90deg seg 1, 90deg< <180deg seg 2 etc...). Then of course you have to transform back to the original coordinate system
Although this seems like more work the transformation matrix and its inverse can be calculated once and then reused. This also extends to higher dimensional rectangles more easily where you would have to consider quadrants and intersections with faces in 3D and so on.
I don\'t know if this is the best way, but what you could do is to figure out the proportion of the line that is inside the rectangle. You can get that from the width of the rectangle and the difference between the x coordinates of A and B (or height and y coordinates; based on the width and height you can check which case applies, and the other case will be on the extension of a side of the rectangle). When you have this, just take that proportion of the vector from B to A and you have your intersection point\'s coordinates.
Here is a slightly verbose method that returns the intersection intervals between an (infinite) line and a rectangle using only basic math:
// Line2 - 2D line with origin (= offset from 0,0) and direction
// Rectangle2 - 2D rectangle by min and max points
// Contacts - Stores entry and exit times of a line through a convex shape
Contacts findContacts(const Line2 &line, const Rectangle2 &rect) {
Contacts contacts;
// If the line is not parallel to the Y axis, find out when it will cross
// the limits of the rectangle horizontally
if(line.Direction.X != 0.0f) {
float leftTouch = (rect.Min.X - line.Origin.X) / line.Direction.X;
float rightTouch = (rect.Max.X - line.Origin.X) / line.Direction.X;
contacts.Entry = std::fmin(leftTouch, rightTouch);
contacts.Exit = std::fmax(leftTouch, rightTouch);
} else if((line.Offset.X < rect.Min.X) || (line.Offset.X >= rect.Max.X)) {
return Contacts::None; // Rectangle missed by vertical line
}
// If the line is not parallel to the X axis, find out when it will cross
// the limits of the rectangle vertically
if(line.Direction.Y != 0.0f) {
float topTouch = (rectangle.Min.Y - line.Offset.Y) / line.Direction.Y;
float bottomTouch = (rectangle.Max.Y - line.Offset.Y) / line.Direction.Y;
// If the line is parallel to the Y axis (and it goes through
// the rectangle), only the Y axis needs to be taken into account.
if(line.Direction.X == 0.0f) {
contacts.Entry = std::fmin(topTouch, bottomTouch);
contacts.Exit = std::fmax(topTouch, bottomTouch);
} else {
float verticalEntry = std::fmin(topTouch, bottomTouch);
float verticalExit = std::fmax(topTouch, bottomTouch);
// If the line already left the rectangle on one axis before entering it
// on the other, it has missed the rectangle.
if((verticalExit < contacts.Entry) || (contacts.Exit < verticalEntry)) {
return Contacts::None;
}
// Restrict the intervals from the X axis of the rectangle to where
// the line is also within the limits of the rectangle on the Y axis
contacts.Entry = std::fmax(verticalEntry, contacts.Entry);
contacts.Exit = std::fmin(verticalExit, contacts.Exit);
}
} else if((line.Offset.Y < rect.Min.Y) || (line.Offset.Y > rect.Max.Y)) {
return Contacts::None; // Rectangle missed by horizontal line
}
return contacts;
}
This approach offers a high degree of numerical stability (the intervals are, in all cases, the result of a single subtraction and division) but involves some branching.
For a line segment (with start and end points), you\'d need to provide the segment\'s start point as the origin and for the direction, end - start. Calculating the coordinates of the two intersections is a simple as entryPoint = origin + direction * contacts.Entry and exitPoint = origin + direction * contacts.Exit.
Here is a solution that works for me. I assume that the rect is aligned to the axes.
Data:
// Center of the Rectangle
let Cx: number
let Cy: number
// Width
let w: number
// Height
let h: number
// Other Point
let Ax: number
let Ay: number
Now translate point A by the center of the rectangle so the rect is centered in O(0,0) and consider the problem in the first quarter (i.e. x > 0 and y > 0).
// Coordinates Translated
let Px = Math.abs(Ax - Cx)
let Py = Math.abs(Ay - Cy)
// Slope of line from Point P to Center
let Pm = Py / Px
// Slope of rectangle Diagonal
let Rm = h / w
// If the point is inside the rectangle, return the center
let res: [number, number] = [0, 0]
// Check if the point is inside and if so do not calculate
if (!(Px < w / 2 && Py < h / 2)) {
// Calculate point in first quarter: Px >= 0 && Py >= 0
if (Pm <= Rm) {
res[0] = w / 2
res[1] = (w * Pm) / 2
} else {
res[0] = h / (Pm * 2)
res[1] = h / 2
}
// Set original sign
if (Ax - Cx < 0) res[0] *= -1
if (Ay - Cy < 0) res[1] *= -1
}
// Translate back
return [res[0] + Cx, res[1] + Cy]
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