I found this excellent question and answer which starts with x/y (plus the center x/y and degrees/radians) and calculates the rotated-to x'/y'. This calculation works perfectly, but I would like to run it in the opposite direction; starting with x'/y' and degrees/radians, I would like to calculate the originating x/y and the center x/y.
(x', y') = new position
(xc, yc) = center point things rotate around
(x, y) = initial point
theta = counterclockwise rotation in radians (radians = degrees * Pi / 180)
dx = x - xc
dy = y - yc
x' = xc + dx cos(theta) - dy sin(theta)
y' = yc + dx sin(theta) + dy cos(theta)
Or, in JavaScript/jQuery:
XYRotatesTo = function($element, iDegrees, iX, iY, iCenterXPercent, iCenterYPercent) {
var oPos = $element.position(),
iCenterX = ($element.outerWidth() * iCenterXPercent / 100),
iCenterY = ($element.outerHeight() * iCenterYPercent / 100),
iRadians = (iDegrees * Math.PI / 180),
iDX = (oPos.left - iCenterX),
iDY = (oPos.top - iCenterY)
;
return {
x: iCenterX + (iDX * Math.cos(iRadians)) - (iDY * Math.sin(iRadians)),
y: iCenterY + (iDX * Math.sin(iRadians)) + (iDY * Math.cos(iRadians))
};
};
The math/code above solves for the situation in Figure A; it calculates the position of the destination x'/y' (green circle) based on the known values for x/y (red circle), the center x/y (blue star) and the degrees/radians.
But I need math/code to solve for Figure B; where I can find not only the destination x/y (green circle), but also the destination center x/y (green star) from the known values of the starting x/y (grey circle, though probably not needed), the destination x'/y' (red circle) and the degrees/radians.
The code above will solve for the destination x/y (green circle) via iDegrees * -1 (thanks to #andrew cooke's answer which has since been removed by him), but in order to do that I need to feed into it the location of the destination center x/y (green star), and that is the calculations I'm currently missing, as you can see in Diagram C, below:
So... how do I find the coordinates ?/? (green star) given n, A (angle) and x'/y' (red circle)?
You're trying to find an inverse transformation. You start with the composition of two linear transformations, a translation T and a rotation R. You apply R first to a vector x and T second, so the expression is y = TRx. To solve the inverse problem you need the inverse of TR, written (TR)-1, which is equal to R-1T-1. The inverse of the rotation R is just the rotation by the negative of the angle (which you mention). The inverse of the translation is, similarly, the original translation multiplied by -1. So your answer is x = R-1T-1y.
In your present situation, you're given the rotation by means of its angle, but you'll need to compute the translation. You'll need the grey circle, which you didn't think you would need. Apply the rotation R (not its inverse) to the gray circle. Subtract this point from the red circle. This is the original translation T. Reverse the sign to get T-1.
Related
I'm writing a simple computer animation, which is a line that rotates around a fixed point at the center of that line. The amount of rotation is based on a gradient noise algorithm (OpenSimplex noise). The line has an origin [x,y] and a nr of the animation frame. These three values plugged into OpenSimplex noise give a rotation value. This part is working perfectly.
The problem is I want to make the line appear to follow the mouse cursor, depending on how far the mouse cursor is from the line. The cursor has coordinates [mx, my] (which change for every frame of animation). I can easily rotate the line and point straight towards the cursor. But I'm having difficulties factoring in the distance. To clarify; the line is rotation on the gradient noise and the mouse cursor alters that rotation to make the line (at [x, y]) point at [mx, my].
Also, the line has an 180 degree identity, so the closest end should point towards the mouse.
Basically what I'm doing now is taking "rotation line" plus "rotation mouse". If it is between 90 and 270 deg the back of the line is closest to the cursor, otherwise the front (for simplicity this is not included in the example code below). So I then take the difference, factor in the distance and substract or add it to the rotation of the line. And this works fairly well except for some artifacts.
let r = OpenSimplexNoise(x, y, frame); // gives current original rotation
let frame = 68; // whichever frame
let x = 60; // some fixed coordinate of line
let y = 60; // some fixed coordinate of line
let mouseX = 10; // changes when the mouse moves
let mouseY = 10; // changes when the mouse moves
let mouseRadius = 200;
let width = 100;
let height = 1;
function distance (x, y, cx, cy) {
return Math.sqrt((x - cx) * (x - cx) + (y - cy) * (y - cy));
}
function angle (x1, y1, x2, y2) {
let dx = x1 - x2;
let dy = y1 - y2;
return 360 + (Math.atan2(dy, dx) * 180 / Math.PI);
}
if (distance(x, y, mouseX, mouseY) <= mouseRadius) {
let dist = distance(x, y, mouseX, mouseY);
let mouseR = angle(x, y, mouseX, mouseY) % 360;
let near = (mouseRadius - dist) / mouseRadius;
let far = 1 - near;
r = (r * far + near * mouseR) % 360;
}
// r now includes mouse
A live version:
https://jsfiddle.net/Ruudt/56pk2wd1/1/
The problem lies in the cases where the mouse passes from being left to right of perpendicular to the (original rotation) line. Here the calculation will nominate the other end as "closests", then calculate the distance and apply this to the rotation. This results in the line jumping from pointing slightly left of the cursor to right of the cursor (or vice versa).
Is there a way to fix this?
I've made an image to illustrate the situation.
The red line represents the line using only the rotation of the gradient noise
The black line is the line that also includes mouse position
the blue arc is the mouse rotation value (right end is origin)
line rotation:
I'm trying to use the answer provided here: Intersection of two Moving Objects with Latitude/Longitude Coordinates
But I have some questions..
What is this angle:
var angle = Math.PI + dir - target.dir
I was thinking that the angle that should be used in the law of cosines is already "alpha or target.dir".. What is that line doing? Also in these two steps:
var x = target.x + target.vel * time * Math.cos(target.dir);
var y = target.y + target.vel * time * Math.sin(target.dir);
Shouldn't the code be using the angle between x- or y-axis and the target velocity vector? Why is the author using alpha here?
What is this angle:
var angle = Math.PI + dir - target.dir
The variable named angle is indeed the angle alpha. Because the direction dir is the direction from chaser to target, and we need it the other way round for this calculation, we add π to it before we subtract target.dir.
Maybe using the word angle as a variable name was a bit vague; I'll change it to alpha, the name I used for this angle in the images.
Shouldn't the code be using the angle between x- or y-axis and the target velocity vector? Why is the author using alpha here?
var x = target.x + target.vel * time * Math.cos(target.dir);
var y = target.y + target.vel * time * Math.sin(target.dir);
We are indeed using target.dir, which is the direction of the target, i.e. the angle between the x-axis and the target vector, to calculate the coordinates of the interception point, and not the angle alpha.
I'm making a game in javascript, where an object is supposed to bounce from walls. I really tried to get it to work myself, but it never works correctly.
Let's say theres a ball bouncing inside this cage (blue = 30°, brown = 60°);
The ball's coordinates are known. The angle of movement is known. The point of collision (P) coordinates are known. The angle of the wall is known. The ball's position is updating it's coordinates inside a setInterval function using this function:
function findNewPoint(x, y, angle, distance) {
var result = {};
result.x =(Math.cos(angle * Math.PI / 180) * distance + x);
result.y = (Math.sin(angle * Math.PI / 180) * distance + y);
return result;
So, upon colliding, there should be a function that properly changes the ball's angle. It's a very complicated problem it seems, because even if I know that the wall is 30°, its important to know from what side the ball is colliding into it. I tried using the "Reflection across a line in the plane" formula and also some vectors, but it never worked out for me. I'm not expecting a complete answer with code, if someone could suggest in what way this should be programmed, it would help aswell.
Edit:
Thanks for your tips guys, I realized what was causing the most confustion; if I select an angle on the canvas with my mouse, the starting coordinate(0,0) is in the bottom left corner. But since the canvas' starting coordinate is in the top left corner, this has to be considered.
Basically using this formula for calculating the angle:
function angle(cx, cy, ex, ey) {
var dy = ey - cy;
var dx = ex - cx;
var theta = Math.atan2(dy, dx);
theta *= 180 / Math.PI;
return theta;
}
if the ball moved from (50,50) to (100,100), the angle would be -45.
Now, this angle changes in the following way when hitting walls:
If im honest, I got these out of trial and error, am not really understanding why exactly 60 and 120.
It is not wise to use angle for moving ball and calculate Cos/Sin again and again. Instead use unit velocity direction vector with components vx, vy like this:
new_x = old_x + vx * Velocity_Magnitude * Time_Interval
Note that vx = Cos(angle), vy = Sin(angle), but with direction approach you seldom need to use trigonometric functions.
Tilted wall with angle Fi has normal
nx = -Sin(Fi)
ny = Cos(Fi)
To find reflection , you need to calculate dot product of velocity and normal
dot = vx * nx + vy * ny
Velocity after reflection transforms:
vnewx = v.x - 2 * dot * n.x
vnewy = v.y - 2 * dot * n.y
Use these values for further moving
(note that you can use both internal and external normal direction, because direction flip changes both components, and sign of 2 * dot * n.x remains the same)
Examples:
horizontal moving right
vx=1, vy=0
30 degrees wall has normal
nx=-1/2, ny=Sqrt(3)/2
dot = -1/2
vnewx = 1 - 2 * (-1/2) * (-1/2) = 1/2
vnewy = 0 - 2 * (-1/2) * Sqrt(3)/2 = Sqrt(3)/2
(velocity direction angle becomes 60 degrees)
horizontal moving left
vx=-1, vy=0
330 degrees wall (left bottom corner) has normal
nx=1/2, ny=Sqrt(3)/2
dot = -1/2
vnewx = -1 - 2 * (-1/2) * (1/2) = -1/2
vnewy = 0 - 2 * (-1/2) * (Sqrt(3)/2) = Sqrt(3)/2
(velocity direction angle becomes 120 degrees)
Here is a function that returns the angle of reflection given an angle of incidence and a surface angle (in degrees). It also ensures that the returned angle is between 0 and 359 degrees.
function angleReflect(incidenceAngle, surfaceAngle){
var a = surfaceAngle * 2 - incidenceAngle;
return a >= 360 ? a - 360 : a < 0 ? a + 360 : a;
}
Here's a demonstration, where the blue line is the angle of incidence, the purple line is the angle of reflection, and the black line is the surface.
If you're assuming that the ball behaves like light bouncing off a mirror, then the angle of incidence equals the angle of reflection.
So your board is 30° from 0° (straight up). The means the normal (perpendicular to the board at the point the ball hits ) is 300°. Say the ball arrives from 280°, it must leave at 320° as the difference between the angle of incidence and the normal and the angle of reflection and the normal must be equal.
I'm trying to draw the perimeter of a circle depending on the angle inputed by the user. The angle determines the perimeter completion : 360° being the full circle, 180 half of the circle, and so on.
My problem is : given the radius, the angle and the center coordinates of the circle, how can I dynamically compute the path of the perimeter ?
I know it's probably basic math but everything I tried so far didn't work.
Here is my fiddle : https://jsfiddle.net/Hal_9100/L311qq88/
My problem is finding the right formula for the x and y coordinates of the path :
var x = i * (radius * Math.cos(angle)) + centerX;
var y = i * (radius * Math.sin(angle)) + centerY;
Am I going all wrong here ?
Here is an example of what I'm trying to do : please note that only the black perimeter should be drawn : I just used the dotted red lines to give a visual example of how the perimeter should be drawn depending on the value given by the user.
Yes, the problem is your maths. Here is the correct way to calculate the x,y coordinate pairs (note that the iteration is from zero to the required angle, not from zero to the radius):
for (var i = 0; i <= angle; i++) {
var x = (radius * Math.cos((i-90)*Math.PI/180)) + centerX;
var y = (radius * Math.sin((i-90)*Math.PI/180)) + centerY;
Your fiddle works fine if you substitute these three lines.
I am struggling with connecting two circles with a line. I am using the famo.us library.
DEMO on Codepen
a.k.a. "Two balls, one line."
The Problem
Angle and length of the line are correct, but the position is wrong.
First attempt
The important part should be lines 114-116:
connection.origin = [.5, .5];
connection.align = [.5, .5];
connection.body.setPosition([
Math.min(sourcePos.x, targetPos.x),
Math.min(sourcePos.y, targetPos.y)
]);
Appearently i am doing something wrong with the math. Playing around with those values gives me all kinds of results, but nothing is close to correct.
Intended solution
(1) The minimal solution would be to connect the centres of the circles with the line.
(2) The better solution would be a line that is only touching the surface of both circles instead of going to the center.
(3) The ideal solution would have arrows on each end of the line to look like a directed graph.
This fixes it :
connection.body.setPosition([
sourcePos.x * Math.cos(angle) + sourcePos.y * Math.sin(angle),
sourcePos.x * Math.sin(-angle)+ sourcePos.y * Math.cos(angle)
]);
Your segment is defined by its extrimity in sourceand the angle and distance to target, thus you have to set its origin to be that of source
The rotation seems to not only rotate the object, but also rotate the coordinates around the origin, so I rotated them by -angle to compensate.
There might be a more famo.usesque way to do it (maybe you can get it to rotate before setting the position, or have the position be 0,0 and add the coordinates as a translation in the transformation).
To get your better solution, still with mostly math, you may keep the same code but
with r the radius of the source ball, remove [r * distX / distance, r * distY / distance] to the coordinates of the segment, to put it in contact with the outer part of the ball
remove both balls' radius from the distance
With that, we get :
var distX = sourcePos.x - targetPos.x;
var distY = sourcePos.y - targetPos.y;
var norm = Math.sqrt(distX * distX + distY * distY);
var distance = norm - (source.size[0]+target.size[0])/2;
var angle = -Math.atan2(-distY, distX);
connection.angle = angle;
connection.size = [distance, 2, 0];
connection.align = [.5, .5];
connection.origin = [.5, .5];
var posX = sourcePos.x - source.size[0]/2 * (distX / norm);
var posY = sourcePos.y - source.size[0]/2 * (distY / norm);
connection.body.setPosition([
posX * Math.cos(angle) + posY * Math.sin(angle),
posX * Math.sin(-angle)+ posY * Math.cos(angle)
]);
result on this fork : http://codepen.io/anon/pen/qEjPLg
I think the fact that the line length is off when the balls go fast is a timing issue. Most probably you compute the segment's length and position at a moment when the ball's centres are not yet updated for that frame.