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Complex mathematical horizontal parallax easing function

I'm currently diving into parallax effects on the web.
Parallax scrolling is a technique in computer graphics and web design, where background images move by the camera slower than foreground images, creating an illusion of depth in a 2D scene and adding to the immersion.
~ Wikipedia
I want to create a little container (could be an image, or any block level element) and move it across the screen horizontally as the user scrolls.
The effect should be scalable across all viewports. Meaning that the hight and the width of the element that the element is moving across should not matter.
When the user has scrolled half of the height of the screen the "moving element" should be in the exact center. Since the user will have scrolled half of the screen the element will be vertically already. We're only worried about horizontally right now.
I've thought about this question for a while and came up with a pretty good idea of how.
Take the hight and the width of the element you want the "moving element" to move across. For example a screen that is 1000px tall and 600px wide.
Divide the width by the height. For example (600px / 1000px = 3/5 = 0.6)
Take the amount of pixels the user scrolled and multiply it by the number we just created. For example (500px * 0.6 = 300px). As you can see this is the exact center.
Move the element across the screen by the amount of pixels just calculated.
This calculation works fine even for every screen size, however it's linear. Meaning that the element will move at the same speed across the screen all the time. Let me show you what I mean.
Let's draw out a screen size. (Let's say 1000 * 500)
Calculate two points for this graph ->
screen factor: (500 / 1000) = 0.5
1. The first point is going to be easy. Let's say we scrolled exactly 0px -> (0.5 * 0) = 0
The "Moving element" will not have moved at all.
2. For the second element we'll take the center. Just for convenience.
The vertical center is at 500px -> (0.5 * 500) = 250 px (Exactly the horizontal center)
Put the results in a graph and draw a line through the points.
In the graph above you can see that whenever the user scrolls down the "moving element" will follow the line (the values on the x-axis).
My question
I really hope I described all that well enough to understand. Now on to my question.
What I want to create is a moving element that would go faster on the edge of the screen and slow down a bit in the middle. If we were to draw that out in the same way we just did. (Creating a graph where we can take the amount of pixels scrolled and see where the element should be positioned horizontally) it would look like this:
Sorry for the poor quality of the image but this is the part I'm having problems with.
As you can see in this graph the "moving element" wouldn't be moving all that much in the middle of the graph. (I over did it a bit in my drawing but you get the general idea.)
What I need is a mathematical function that takes three parameters (Screen height, width and the amount of pixels scrolled) and returns the horizontal position of the "moving element".
My idea:
My idea was to position the element in the dead center of the page and then to move it left and right (translations using CSS and JavaScript) based on how far there has been scrolled.
The graph would look something like this:
The (Hand drawn) graph above would be true for a screen that's 1000x600px since the "moving element" translates -300px when no scrolling has been done and 300px when 100% has been scrolled.
However I have no idea on how to create a mathematical function that would be true for every screen size.
To be clear I need a function that "always" starts at Y= (-screen-width/2) and X = 0. It should always cross the point (sreen-height; (screen-width//2)) and the function should be in a form of x^3 (To get the right easing)
I really hope I explained myself well enough and I really hope somebody can help me out here.

Answer from math.stackexchange
Besides asking my question here I also went ahead and posted this question on math.stackexchange.com. Stackoverflow's mathematical sister site. Somebody there helped me find an answer to my question.
The function had to be a function that would output the moving-element it's horizontal position in pixels relative to the horizontal center of the page based on the amount of pixels scrolled since the element was first visible. The function would have to be "steeper" on the edges and ease into a short stop in the middle and be usable across every possible screen size. Meaning that the mathematical function would have to be positioned based on two variables, screen height and -width.
The answer they came up with on math.stackexchange:
In this example s-width is the width of the screen in pixels. s-height is the height of the screen in pixels. px-scrolled is the amount of pixels scrolled since the element was first visible.
The output pos is the moving elements horizontal position in pixels relative to the center of the screen.
If you put all this math into JavaScript you get the following:
var pos = ((4*win_width)/(Math.pow(win_height, 3))) * Math.pow(px_since_visible - (win_height/2),3)
There is a working example available on codepen. You can find it here.

You can position it with a function f that actually draws that trajectory.
This is the idea I propose:
Create the function trajectory f such that f(0) = 0, and f(1) = 1 (add more constraints in order to reproduce the effect you are looking for, ex: f(0.5) = 0.5)
Within each scroll event, set x as the amount scrolled and position the element using the coordinates (f(x) * (w - s), x * (h - s)), where w is the document width, h is the document height and s is the size of the element
I can see that cubic functions are plotted like the trajectory you want, so I've been testing with different functions and I've got this working example https://codepen.io/anon/pen/YZJxGa
var element = $('.element')
var height = $(document).height()
var scrollable = $('body').height() - $(window).height()
var width = $('body').width();
$(window).scroll(function () {
var winH = $(window).scrollTop()
var x;
// Determine the amount of px scrolled since the element first came into view.
console.log(winH, scrollable)
x = (winH/scrollable) > 1 ? 1 : (winH/scrollable);
var posY = x * (height - 120);
var posX = (-1.000800320128*x
+6.0024009603841*x**2-4.0016006402561*x**3)*(width - 120)
console.log(posY)
if (x > 0) {
console.log(`translate(${posX}px, ${posY}px, 0)`)
element.css({
'transform': `translate3d(${posX}px, ${posY}px, 0)`
})
}
})
You can generate more cubic functions using this tool I've just found http://skisickness.com/2010/04/28/ or solve a couple of systems of linear equations using the fact that you just want to find values for a, b, c and d for f(x) = ax^3 + bx^2 + cx + d

Zooming in and out on a point

I'm trying to make a stage that can be zoomed by mousewheel. I'm using the stage.scale and jquery event. But I have not been able to make the zoom done with the mouse pointer as center. So when my mouse is on a point, I want to zoom in it centrically.
I've make this example, that doesen't work for now :
http://fiddle.jshell.net/rr7pLapz/1/
Here is my zoom function :
$("#cont").bind('mousewheel', function(e) {
var x = e.offsetX;
var y = e.offsetY;
if (e.originalEvent.wheelDelta > 0) {
STAGE.scale({
x:STAGE.scale().x+1,
y:STAGE.scale().y+1
});
STAGE.x(-x + 500/(STAGE.scale().x));
STAGE.y(-y + 350/(STAGE.scale().y));
STAGE.batchDraw();
}
else {
if (STAGE.scale().x > 1) {
STAGE.x(-x + 500/(STAGE.scale().x));
STAGE.y(-y + 350/(STAGE.scale().y));
STAGE.scale({
x:STAGE.scale().x-1,
y:STAGE.scale().y-1
});
STAGE.batchDraw();
}
}
});
Can you help ? Many thanks.

Here is a manual way to do it:
First, store the mouse position in stage coordinates. Then apply the scale, and get the new mouse position in stage coordinates. Then offset the stage by oldPosition - newPosition. This effectively puts the same point in the stage back under the mouse.
Or as Mark said, you can do it via layer offsets. Though that might not be the most convenient way to do it, depending how your project is setup to use kinetic's features. See these links:
Scaling to a fixed point in KineticJS
KineticJS Canvas - Scale group from center point
Also some side notes:
You have some duplication in that code. Why not only have one line under each wheelDelta condition, to determine the newScale, then use that variable in the following lines.
I've found that it makes a smooth zoom to do scale *= 1.2 and scale /= 1.2 rather than the straight adding you're doing there. Just a suggestion.

Parallax effect with zoom and rotating

I am currently experimenting with parallax effect that i am planning to implement to my HTML5-canvas game engine.
The effect itself is fairly easy to achieve, but when you add zooming and rotating, things get a little more complicated, at least for me. My goal is to achieve something like this:Youtube video.
As you can see, you can zoom in and out "to the center", and also rotate around it and get the parallax effect.
In my engine i want to have multiple canvases that are going to be my parallax layers, and i am going to translate them.
I came up with something like this:
var parallax = {
target: {
x: Mouse.x,
y: Mouse.y
},
offset: {
x: -ctx.width / 2,
y: -ctx.height / 2
},
factor: {
x: 1,
y: 1
}
}
var angle = 0;
var zoomX = 1;
var zoomY = 1;
var loop = function(){
ctx.canvas.width = ctx.canvas.width; //Clear the canvas.
ctx.translate(parallax.target.x * parallax.factor.x, parallax.target.y * parallax.factor.y);
ctx.rotate(angle);
ctx.scale(zoomX, zoomY);
ctx.translate((-parallax.target.x - parallax.offset.x) * parallax.factor.x, (-parallax.target.y - parallax.offset.y) * parallax.factor.y);
Draw(); //Function that draws all the objects on the screen.
}
This is a very small and simplified part of my script, but i hope that's enough to get what i am doing. The object "parallax" contains the target position, the offset(the distance from the target), and the factor that is determining how fast the canvas is moving away relatively to the target. ctx is the canvas that is moving in the opposite direction of the target.(In this example i am using only one layer.) I am using the mouse as the "target", but i could also use the player, or some other object with x and y property. The target is also the point around which i rotate and scale the canvas.
This method works completely fine as long as the factor is equal to 1. If it is something else, the whole thing suddenly stops working correctly, and when i try to zoom, it zooms to the top-left corner, not the target. I also noticed that if i zoom out too much, the canvas is not moving in the opposite way of the target, but in the same direction.
So my question is: What is the correct way of implementing parallax with zooming and rotating?
P.S. It is important to me that i am using canvases as the layers.

To prepare for the next animation frame, you must undo any previous transforms in the reverse order they were executed:
context.translate(x,y);
context.scale(sx,sy);
context.rotate(r);
// draw stuff
context.rotate(-r);
context.scale(-sx,-sy);
context.translate(-x,-y);
Alternatively, you can use context.save / context.restore to undo the previous transforms.
Adjust your parallax values for the current frame,
Save the un-transformed context state using context.save(),
Do your transforms (translate, scale, rotate, etc),
Draw you objects as if they were in non-transformed space with [0,0] at your translate point,
Restore your context to it's untransformed state using context.restore()/
Either way will correctly give you a default-oriented canvas to use for your next animation frame.
The exact parallax effects you apply are up to your own design, but using these methods will make the canvas return to a normal default state for you to design with.

Masking effect using HTML5 canvas

Using canvas and mousemove I create a masking effect.
Example: FIDDLE
The masking effect works but on fast scrolling performance isn't ideal.
How can I improve the framerate?

I fixed it simply by using the window mouse coordinates, instead of the listener on the canvas (assuming you want the picture to just be black).
http://jsfiddle.net/gfZ5C/166/
I also changed to requestAnimationFrame, you'll notice a complete FPS difference in the movement of the circle, instead of it moving according to the listener.
window.onmousemove = function (e) {
circle.posX = e.pageX-can.offsetLeft;
circle.posY = e.pageY-can.offsetTop;
circle.hide = (circle.posX >= 550 || circle.posY >= 550) ? true : false;
}
if (!circle.hide) ctx.arc(circle.posX, circle.posY, 70, 0, Math.PI * 2, true);
I left it at 550, so that you could see it actually disappear, you can change it to a big number if you want the circle to just drag right off the canvas.
Also there is no lag on the circle following the mouse now.
Update:
Fixed the fiddle so that the circle adjusts for window offset too.
The offset is relevant to its parent element, so depending on how far deep your canvas is, you will have to subtract each elements offset that it is inside.
Update:
Switched to handle 'onmouse' events, this will work much better on a real site, because mousing over the iFrame doesn't work well, but it still works. Will work 100% as intended on a real site.

Pixel perfect 2D mouse picking with Canvas

I'm writing a 2D game in html5 using Canvas which requires mouse click and hover events to be detected. There are 3 problems with this: detections must be pixel-perfect, objects are not rectangular (houses, weird-shaped UI buttons...), and it is required to be fast and responsive. (Obviously brute force is not an option)
So what I want to ask is how do I find out which object the mouse is on, and what are the possible optimizations.
P.S: I did some investigation and found a guy who used QuadTree here.

I have a (dated) tutorial that explains the concept of a ghost canvas which is decent for pixel-perfect hit detection. The tutorial is here. Ignore the warning about a newer tutorial, the newer one does not use the ghost canvas concept.
The idea is to draw the image in question to an in-memory canvas and then use getImageData to get the single pixel of the mouse click. Then you see if that single pixel is fully transparent or not.
If its not fully transparent, well, you've got your target.
If it is fully transparent, draw the next object to the in-memory canvas and repeat.
You only have to clear the in-memory canvas at the end.
getImageData is slow but it is your only option if you want pixel-perfect hit detection and aren't pre-computing anything.
Alternatively you could precompute a path or else an array of pixels with an offset. This would be a lot of work but might be faster. For instance if you have a 40x20 image with some transparency you'd compute an array[40][20] that would have true or false corresponding to transparent or not. Then you'd test that against the mouse position, with some offset, if the image is drawn at (25, 55) you'd want to subtract that from the mouse position and then test if the new position is true when you look at array[posx][posy].
That's my answer to your question. My Suggestion? Forget pixel-perfect detection if this is a game.
Seriously.
Instead make paths (not in canvas, in plain javascript code) that represent the objects but are not pixel perfect, for instance a house might be a square with a triangle on the top that is a very close approximation of the image but is used in its stead when it comes to hit testing. It is comparatively extremely fast to compute if a point is inside a path than it is to do pixel-perfect detection. Look up point in polygon winding number rule detection. That's your best bet, honestly.

The common solution in traditional game development is to build a click mask. You can re-render everything onto a separate off-screen canvas in a solid color (the rendering should be very quick). When you want to figure out what was clicked on, you simply sample the color at the x/y co-ordinate on the off-screen canvas. You end up building a color-->obj hash, akin to:
var map = {
'#000000' : obj1
, '#000001' : obj2
, ...
};
You can also optimize the rendering to the secondary canvas to only happen when the user clicks on something. And using various techniques, you can further optimize it to only draw the part of the canvas that the user has clicked on (for example, you can split you canvas into an NxN grid, e.g. a grid of 20x20 pixel squares, and flag all of the objects in that square -- you'd then only need to re-draw a small number of objects)

HTML5 Canvas is just a drawing plane, where you can set different transforms before calling each drawing API function. Objects cannot be created and there is no display list. So you have to build these features yourself or you can use different libraries available for this.
http://www.kineticjs.com/
http://easeljs.com/
A few months before I got interested in this and even wrote a library for this purpose. You can see it here : http://exsprite.com. Ended up facing a lot of performance issues, but because of lack of time I couldn't optimize it. It was really interesting, so waiting for some time to make it perfect.

I believe the comments should suffice. This is how I determine user intention in my 2d isometric scroller, currently located at http://untitled.servegame.com
var lastUp = 0;
function mouseUp(){
mousedown = false; //one of my program globals.
var timeNow = new Date().getTime();
if(mouseX == xmouse && mouseY == ymouse && timeNow > lastUp + 100){//if it was a centralized click. (mouseX = click down point, xmouse = mouse's most recent x) and is at least 1/10th of a second after the previous click.
lastUp = new Date().getTime();
var elem = document.elementFromPoint(mouseX, mouseY); //get the element under the mouse.
var url = extractUrl($(elem).css('background-image')); // function I found here: http://webdevel.blogspot.com/2009/07/jquery-quick-tip-extract-css-background.html
imgW = $("#hiddenCanvas").width(); //EVERY art file is 88px wide. thus my canvas element is set to 88px wide.
imgH = $(elem).css('height').split('p')[0]; //But they vary in height. (currently up to 200);
hiddenCanvas.clearRect(0, 0, imgW, imgH); //so only clear what is necessary.
var img = new Image();
img.src = url;
img.onload = function(){
//draw this elements image to the canvas at 0,0
hiddenCanvas.drawImage(img,0,0);
///This computes where the mouse is clicking the element.
var left = $(elem).css('left').split('p')[0]; //get this element's css absolute left.
var top = $(elem).css('top').split('p')[0];
offX = left - offsetLeft; //left minus the game rendering element's absolute left. gives us the element's position relative of document 0,0
offY = top - offsetTop;
offX = mouseX - offX; //apply the difference of the click point's x and y
offY = mouseY - offY;
var imgPixel = hiddenCanvas.getImageData(offX, offY, 1, 1); //Grab that pixel. Start at it's relative X and it's relative Y and only grab one pixel.
var opacity = imgPixel.data[3]; //get the opacity value of this pixel.
if(opacity == 0){//if that pixel is fully transparent
$(elem).hide();
var temp = document.elementFromPoint(mouseX, mouseY); //set the element right under this one
$(elem).show();
elem = temp;
}
//draw a circle on our hiddenCanvas so when it's not hidden we can see it working!
hiddenCanvas.beginPath();
hiddenCanvas.arc(offX, offY, 10, 0, Math.PI*2, true);
hiddenCanvas.closePath();
hiddenCanvas.fill();
$(elem).css("top", "+=1"); //apply something to the final element.
}
}
}
In conjunction with this:
<canvas id="hiddenCanvas" width="88" height="200"></canvas>
Set the CSS positioning absolute and x = -(width) to hide;