I'm trying to move an arrow, which could be rotated, in a straight line. I'm having some difficulty coming up with the correct formula to use. I know it should probably involve sine and cosine, but I've tried various configurations and haven't been able to get something that works.
Here's a picture of my scene with the arrow and bow
rotateNumber is an integer like -1 (for 1 left rotation), 0 (no rotation), 1 (1 right rotation), etc.
rotateAngle is 10 degrees by default.
Here's the code to move the arrow:
if (arrowMoving) {
var rAngle = rotateAngle * rotateNumber;
var angleInRad = rAngle * (Math.PI/180);
var stepSize = 1/20;
arrowX += stepSize * Math.cos(angleInRad);
arrowY += stepSize * Math.sin(angleInRad);
DrawArrowTranslate(arrowX, arrowY);
requestAnimFrame(render);
} else {
DrawArrow();
arrowX = 0;
arrowY = 0;
}
Here's the code to draw and translate the arrow:
function DrawArrowTranslate(tx, ty) {
modelViewStack.push(modelViewMatrix);
/*
var s = scale4(0.3, -0.7, 1);
var t = translate(0, -4, 0);
*/
var s = scale4(0.3, -0.7, 1);
var t = translate(0, -5, 0);
var t2 = translate(0 + tx, 1 + ty, 0)
// rotate takes angle in degrees
var rAngle = rotateAngle;
var r = rotate(rAngle, 0, 0, 1);
var m = mult(t, r);
var m = mult(m, t2);
modelViewMatrix = mat4();
modelViewMatrix = mult(modelViewMatrix, m);
modelViewMatrix = mult(modelViewMatrix, s);
/*
// update bounding box
arrowBoundingBox.translate(0, -5);
arrowBoundingBox.rotate(rAngle);
arrowBoundingBox.translate(0, 1);
arrowBoundingBox.scale(0.3, -0.7);
*/
gl.uniformMatrix4fv(modelViewMatrixLoc, false, flatten(modelViewMatrix));
gl.drawArrays( gl.LINE_STRIP, 1833, 4);
gl.drawArrays( gl.LINE_STRIP, 1837, 4);
modelViewMatrix = modelViewStack.pop();
}
Your code looks quite correct, but you should eliminate the use of the rotateNumber. You can just use positive and negative angles for rotation instead, eliminating what I imagine is the cause of error here.
Sin and Cos can certainly handle angles of any magnitude positive, negative, or zero.
Good luck!
I figured out the problem. I was translating after rotating when I needed to translate before rotating as the rotation was messing up the translation.
Related
I'd like to programmatically draw a shape like this where there is an underlying spiral and equally spaced objects along it, placed tangent to the spiral as shown in this sketch:
I found an example of how to determine equally spaced points along the spiral here and am now trying to place hemispheres along the spiral. However, I'm not sure how to calculate the angle the shape needs to be rotated.
This is what I have so far (viewable here):
var totalSegments = 235,hw = 320,hh = 240,segments;
var len = 15;
points = [];
function setup(){
createCanvas(640,480);
smooth();
colorMode(HSB,255,100,100);
stroke(0);
noFill();
//println("move cursor vertically");
}
function draw(){
background(0);
translate(hw,hh);
segments = floor(totalSegments);
points = getTheodorus(segments,len);
angles = getAngles(segments, len);
for(var i = 0 ; i < segments ; i++){
let c = color('blue');
fill(c);
noStroke();
// draw shape
if(i % 2){
// console.log(i, ' ', angles[i]);
// try rotating around the object's center
push();
// translate(points[i].x, points[i].y)
rotate(PI/angles[i]);
arc(points[i].x, points[i].y, len*3, len*3, 0, 0 + PI);
pop();
}
// draw spiral
strokeWeight(20);
stroke(0,0,100,(20+i/segments));
if(i > 0) line(points[i].x,points[i].y,points[i-1].x,points[i-1].y);
}
}
function getAngles(segment, len){
let angles = [];
let radius = 0;
let angle = 0;
for(var i =0; i < segments; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
angles[i] = angle;
}
return angles;
}
function getTheodorus(segments,len){
var result = [];
var radius = 0;
var angle = 0;
for(var i = 0 ; i < segments ; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
result[i] = new p5.Vector(cos(angle) * radius*len,sin(angle) * radius*len);
}
return result;
}
Note that your drawing shows Archimedean spiral while link refers to Theodorus one.
Archimedean spiral is described by equation in polar coordinates (rho-theta)
r = a + b * Theta
where a is initial angle, b is scale value (describes distance between arms), r is radius.
And angle Theta + Pi/2 describes normal to spiral in point at parameter Theta
If you need an approximation to divide spiral into (almost) equal segments - use Clackson formula (example here)
theta = 2 * Pi * Sqrt(2 * s / b)
for arc length s
I'm applying rotation matrix on Stewart platform joints to get their position referenced to basement.
Code used is here:
for (int i=0; i<6; i++) {
float mx = baseRadius*cos(radians(baseAngles[i]));
float my = baseRadius*sin(radians(baseAngles[i]));
baseJoint[i] = new PVector(mx, my, 0);
}
for (int i=0; i<6; i++) {
float mx = platformRadius*cos(radians(platformAngles[i]));
float my = platformRadius*sin(radians(platformAngles[i]));
platformJoint[i] = new PVector(mx, my, 0);
q[i] = new PVector(0, 0, 0);
l[i] = new PVector(0, 0, 0);
A[i] = new PVector(0, 0, 0);
}
for (int i=0; i<6; i++) {
// rotation
q[i].x = cos(rotation.z)*cos(rotation.y)*platformJoint[i].x +
(-sin(rotation.z)*cos(rotation.x)+cos(rotation.z)*sin(rotation.y)*sin(rotation.x))*platformJoint[i].y +
(sin(rotation.z)*sin(rotation.x)+cos(rotation.z)*sin(rotation.y)*cos(rotation.x))*platformJoint[i].z;
q[i].y = sin(rotation.z)*cos(rotation.y)*platformJoint[i].x +
(cos(rotation.z)*cos(rotation.x)+sin(rotation.z)*sin(rotation.y)*sin(rotation.x))*platformJoint[i].y +
(-cos(rotation.z)*sin(rotation.x)+sin(rotation.z)*sin(rotation.y)*cos(rotation.x))*platformJoint[i].z;
q[i].z = -sin(rotation.y)*platformJoint[i].x +
cos(rotation.y)*sin(rotation.x)*platformJoint[i].y +
cos(rotation.y)*cos(rotation.x)*platformJoint[i].z;
// translation
q[i].add(PVector.add(translation, initialHeight));
l[i] = PVector.sub(q[i], baseJoint[i]);
Each point it's correctly initialized providing their positions by angle. Providing a rotation vector, I've as result a rotated platform based on It's center.
What I'd like to do, is to perform a rotation on arbitrary point (distant x,y,z from platform center) that I provide.
I've thought about 2 methods:
Geometric calculation of each joint respect to a provided point (complex if It's also not on same level). Apply this calculations also on base to have their center perpendicular.
Apply an additional rotation matrix translated by a vector
I'd like a tip about which way is correct (computationally light) or If there's already known way to move rotational center of this points.
I'm trying to find the visible size of a sphere in pixels, after projection to screen space. The sphere is centered at the origin with the camera looking right at it. Thus the projected sphere should be a perfect circle in two dimensions. I am aware of this 1 existing question. However, the formula given there doesn't seem to produce the result I want. It is too small by a few percent. I assume this is because it is not correctly taking perspective into account. After projecting to screen space you do not see half the sphere but significantly less, due to perspective foreshortening (you see just a cap of the sphere instead of the full hemisphere 2).
How can I derive an exact 2D bounding circle?
Indeed, with a perspective projection you need to compute the height of the sphere "horizon" from the eye / center of the camera (this "horizon" is determined by rays from the eye tangent to the sphere).
Notations:
d: distance between the eye and the center of the sphere
r: radius of the sphere
l: distance between the eye and a point on the sphere "horizon", l = sqrt(d^2 - r^2)
h: height / radius of the sphere "horizon"
theta: (half-)angle of the "horizon" cone from the eye
phi: complementary angle of theta
h / l = cos(phi)
but:
r / d = cos(phi)
so, in the end:
h = l * r / d = sqrt(d^2 - r^2) * r / d
Then once you have h, simply apply the standard formula (the one from the question you linked) to get the projected radius pr in the normalized viewport:
pr = cot(fovy / 2) * h / z
with z the distance from the eye to the plane of the sphere "horizon":
z = l * cos(theta) = sqrt(d^2 - r^2) * h / r
so:
pr = cot(fovy / 2) * r / sqrt(d^2 - r^2)
And finally, multiply pr by height / 2 to get the actual screen radius in pixels.
What follows is a small demo done with three.js. The sphere distance, radius and the vertical field of view of the camera can be changed by using respectively the n / f, m / p and s / w pairs of keys. A yellow line segment rendered in screen-space shows the result of the computation of the radius of the sphere in screen-space. This computation is done in the function computeProjectedRadius().
projected-sphere.js:
"use strict";
function computeProjectedRadius(fovy, d, r) {
var fov;
fov = fovy / 2 * Math.PI / 180.0;
//return 1.0 / Math.tan(fov) * r / d; // Wrong
return 1.0 / Math.tan(fov) * r / Math.sqrt(d * d - r * r); // Right
}
function Demo() {
this.width = 0;
this.height = 0;
this.scene = null;
this.mesh = null;
this.camera = null;
this.screenLine = null;
this.screenScene = null;
this.screenCamera = null;
this.renderer = null;
this.fovy = 60.0;
this.d = 10.0;
this.r = 1.0;
this.pr = computeProjectedRadius(this.fovy, this.d, this.r);
}
Demo.prototype.init = function() {
var aspect;
var light;
var container;
this.width = window.innerWidth;
this.height = window.innerHeight;
// World scene
aspect = this.width / this.height;
this.camera = new THREE.PerspectiveCamera(this.fovy, aspect, 0.1, 100.0);
this.scene = new THREE.Scene();
this.scene.add(THREE.AmbientLight(0x1F1F1F));
light = new THREE.DirectionalLight(0xFFFFFF);
light.position.set(1.0, 1.0, 1.0).normalize();
this.scene.add(light);
// Screen scene
this.screenCamera = new THREE.OrthographicCamera(-aspect, aspect,
-1.0, 1.0,
0.1, 100.0);
this.screenScene = new THREE.Scene();
this.updateScenes();
this.renderer = new THREE.WebGLRenderer({
antialias: true
});
this.renderer.setSize(this.width, this.height);
this.renderer.domElement.style.position = "relative";
this.renderer.autoClear = false;
container = document.createElement('div');
container.appendChild(this.renderer.domElement);
document.body.appendChild(container);
}
Demo.prototype.render = function() {
this.renderer.clear();
this.renderer.setViewport(0, 0, this.width, this.height);
this.renderer.render(this.scene, this.camera);
this.renderer.render(this.screenScene, this.screenCamera);
}
Demo.prototype.updateScenes = function() {
var geometry;
this.camera.fov = this.fovy;
this.camera.updateProjectionMatrix();
if (this.mesh) {
this.scene.remove(this.mesh);
}
this.mesh = new THREE.Mesh(
new THREE.SphereGeometry(this.r, 16, 16),
new THREE.MeshLambertMaterial({
color: 0xFF0000
})
);
this.mesh.position.z = -this.d;
this.scene.add(this.mesh);
this.pr = computeProjectedRadius(this.fovy, this.d, this.r);
if (this.screenLine) {
this.screenScene.remove(this.screenLine);
}
geometry = new THREE.Geometry();
geometry.vertices.push(new THREE.Vector3(0.0, 0.0, -1.0));
geometry.vertices.push(new THREE.Vector3(0.0, -this.pr, -1.0));
this.screenLine = new THREE.Line(
geometry,
new THREE.LineBasicMaterial({
color: 0xFFFF00
})
);
this.screenScene = new THREE.Scene();
this.screenScene.add(this.screenLine);
}
Demo.prototype.onKeyDown = function(event) {
console.log(event.keyCode)
switch (event.keyCode) {
case 78: // 'n'
this.d /= 1.1;
this.updateScenes();
break;
case 70: // 'f'
this.d *= 1.1;
this.updateScenes();
break;
case 77: // 'm'
this.r /= 1.1;
this.updateScenes();
break;
case 80: // 'p'
this.r *= 1.1;
this.updateScenes();
break;
case 83: // 's'
this.fovy /= 1.1;
this.updateScenes();
break;
case 87: // 'w'
this.fovy *= 1.1;
this.updateScenes();
break;
}
}
Demo.prototype.onResize = function(event) {
var aspect;
this.width = window.innerWidth;
this.height = window.innerHeight;
this.renderer.setSize(this.width, this.height);
aspect = this.width / this.height;
this.camera.aspect = aspect;
this.camera.updateProjectionMatrix();
this.screenCamera.left = -aspect;
this.screenCamera.right = aspect;
this.screenCamera.updateProjectionMatrix();
}
function onLoad() {
var demo;
demo = new Demo();
demo.init();
function animationLoop() {
demo.render();
window.requestAnimationFrame(animationLoop);
}
function onResizeHandler(event) {
demo.onResize(event);
}
function onKeyDownHandler(event) {
demo.onKeyDown(event);
}
window.addEventListener('resize', onResizeHandler, false);
window.addEventListener('keydown', onKeyDownHandler, false);
window.requestAnimationFrame(animationLoop);
}
index.html:
<!DOCTYPE html>
<html>
<head>
<title>Projected sphere</title>
<style>
body {
background-color: #000000;
}
</style>
<script src="http://cdnjs.cloudflare.com/ajax/libs/three.js/r61/three.min.js"></script>
<script src="projected-sphere.js"></script>
</head>
<body onLoad="onLoad()">
<div id="container"></div>
</body>
</html>
Let the sphere have radius r and be seen at a distance d from the observer. The projection plane is at distance f from the observer.
The sphere is seen under the half angle asin(r/d), so the apparent radius is f.tan(asin(r/d)), which can be written as f . r / sqrt(d^2 - r^2). [The wrong formula being f . r / d.]
The illustrated accepted answer above is excellent, but I needed a solution without knowing the field of view, just a matrix to transform between world and screen space, so I had to adapt the solution.
Reusing some variable names from the other answer, calculate the start point of the spherical cap (the point where line h meets line d):
capOffset = cos(asin(l / d)) * r
capCenter = sphereCenter + ( sphereNormal * capOffset )
where capCenter and sphereCenter are points in world space, and sphereNormal is a normalized vector pointing along d, from the sphere center towards the camera.
Transform the point to screen space:
capCenter2 = matrix.transform(capCenter)
Add 1 (or any amount) to the x pixel coordinate:
capCenter2.x += 1
Transform it back to world space:
capCenter2 = matrix.inverse().transform(capCenter2)
Measure the distance between the original and new points in world space, and divide into the amount you added to get a scale factor:
scaleFactor = 1 / capCenter.distance(capCenter2)
Multiply that scale factor by the cap radius h to get the visible screen radius in pixels:
screenRadius = h * scaleFactor
I want to create a 3rd person camera similiar to example. The camera should stick behind the object and rotate if the rotation difference between camera and object is too high (maybe above ten percent).
This is my actual camera code:
var targetPosition = this.getTargetPosition();
var targetRotation = this.getTargetRotation();
var tmpQuaternion = new THREE.Quaternion();
tmpQuaternion.setFromAxisAngle(new THREE.Vector3(0, 1, 0), 180 * (Math['PI'] / 180));
this.camera.quaternion = targetRotation;
this.camera.position = targetPosition;
this.camera.quaternion.multiplySelf(tmpQuaternion);
this.camera.quaternion.normalize();
this.camera.updateMatrix();
this.camera.translateZ(200);
this.camera.translateY(50);
But there are several problems right now. The camera quaternion should not set directly to the target rotation. But I dont know how to calculate the difference between camera quaternion and target quaternion and use maybe this if the distance is too high:
var qm = new THREE.Quaternion();
THREE.Quaternion.slerp(targetRotation, this.camera.quaternion, qm, time);
this.camera.quaternion = qm;
The second problem is the position itself. Currently I set camera position to the object position and translate it back to view behind, but the translation should be already in target position and the camera position should be translated to the target position.
Update 1: I made an example html: http://ssachtleben.github.com/CameraProblem/
Update 2: I made some progress now. Seems like I get quaternion difference with this function:
getAxisAngle = function(quaternion1, quaternion2) {
var tmpQuaternion = new THREE.Quaternion();
tmpQuaternion.setFromAxisAngle(new THREE.Vector3(0, 1, 0), 180 * (Math['PI'] / 180));
var tmpRotation1 = quaternion1.clone();
tmpRotation1.multiplySelf(tmpQuaternion);
tmpRotation1.normalize();
var tmpRotation2 = quaternion2.clone();
if (tmpRotation2.w > 1) {
tmpRotation2.normalize();
}
var angle1 = 2 * Math['acos'](tmpRotation1.w);
var angle2 = 2 * Math['acos'](tmpRotation2.w);
var diff = angle1 > angle2 ? angle1 - angle2 : angle2 - angle1;
return diff;
};
But know I need to freeze the axis if the angle difference is too high. How can I do this?
Any help would be appreciated.
Ok finally the camera is fixed and works as excepted:
var targetPosition = this.getTargetPosition();
var targetRotation = this.getTargetRotation();
var tmpQuaternion = new THREE.Quaternion();
tmpQuaternion.setFromAxisAngle(new THREE.Vector3(0, 1, 0), 180 * (Math['PI'] / 180));
targetRotation.multiplySelf(tmpQuaternion);
targetRotation.quaternion.normalize();
var qm = new THREE.Quaternion();
THREE.Quaternion.slerp(this.camera.quaternion, targetRotation, qm, 0.07);
this.camera.quaternion = qm;
this.camera.quaternion.normalize();
I'm trying to rotate a point in my Canvas from a given point (center). In my MouseDown handler, I save the point where user click (oldPos), and in my MouseMove handler, I'm doing this:
private function onMouseMove(event:MouseEvent):void
{
// Where the user pointer right now
var endPoint:Point = new Point(event.localX,event.localY);
// Calculate angle in radians from the user pointer
var angle:Number = getLineAngleFromHorizontal(oldPos,endPoint);
var rad:Number = Math.PI * (angle / 180);
// Point which I want to rotate
pTop = new Point(oldPos.x,oldPos.y - 30);
var distance:Number = Point.distance(oldPos,pTop);
// Calculate the translation point from previously distance and angle
var translatePoint:Point = Point.polar(distance, rad);
// New point coordinates (in theory)
pTop.x += translatePoint.x;
pTop.y += translatePoint.y;
// Then, draw the line...
}
Where getLineAngleFromHorizontal is a function that returns the angle formed by a center and a give point:
private function getLineAngleFromHorizontal(p1:Point,p2:Point):Number
{
var RotVecOrigen:Point = new Point((p2.x-p1.x),(p2.y-p1.y));
var ModRot:Number = Math.sqrt((RotVecOrigen.x*RotVecOrigen.x)+(RotVecOrigen.y*RotVecOrigen.y));
var ret:Number;
if(((RotVecOrigen.x < 0) && (RotVecOrigen.y <= 0))||((RotVecOrigen.x >= 0) && (RotVecOrigen.y < 0)))
{
ret = Math.round((180.0*(Math.acos(RotVecOrigen.x/ModRot))/Math.PI));
}else{
ret = Math.round((180.0*(-Math.acos(RotVecOrigen.x/ModRot))/Math.PI));
}
return ret;
}
To see an example, watch the image below:
But I don't know why isn't work. I mean, pTop point isn't move where I want, and I think that my calcs are correct.
Can anybody help me? (maybe someone with Math knowledge)
I'm not entirely sure what you want to accomplish. Do you want your new point to be at an 330 degree offset from your center point?
If you want to move your point 330 degrees, use this:
function directionalDistance($start:Point, $direction:Number, $distance:Number, $zeroDegreesUp:Boolean = false):Point{
if($zeroDegreesUp) $direction = ( $direction + 270)%360;
var x:Number = Math.cos($direction * Math.PI / 180) * $distance;
var y:Number = Math.sin($direction * Math.PI / 180) * $distance;
return new Point($start.x +x, $start.y + y);
}
//
var newPoint:Point = directionalDistance(new Point(event.localX,event.localY), 330, 50, true);