css - formular to calculate width/height (relative to parent) of container with translateZ inside of parent container with perspective

Question

What is the formular to calculate the widths/heights of child elements with translateZ inside of parent container with set perspective (keyword: "parallax") relative to its parents width/height?

I'd like to create a site with parallax effect on both axis. I was able to figure out everything i need for my mockup except one thing. How to calculate the childrens widths/heights when its above 100%. Because of parents perspective and childrens translateZ the childrens widths/heights visually don't align with parents width/height anymore.

The formular to scale the child elements is: 1 + (translateZ * -1) / perspective. But i was not able to find a formular for width/height. BTW: When childrens widths/heights <= 100% everything works fine.
But see the result on the image below when width >= 100% (containers have top offset to make things visible).

To be correct the approach in my particular case is to let all child elements have visually the same widths/heights.

---

in SASS (preferred): PEN or SassMeister
in CSS: PEN

---

links from the specs that could help:
https://www.w3.org/TR/css-transforms-1/#recomposing-to-a-3d-matrix
https://www.w3.org/TR/css-transforms-1/#mathematical-description

---

"Googled" a lot but didn't find anything pointing me to the right direction. Thanks in advance...

html, body {
  height: 100%;
  overflow: hidden;
  width: 100%;
}

#projection {
  perspective: 1px;
  perspective-origin: 0 0;
  height: 100%;
  overflow: auto;
  width: 100%;
}

.pro {
  transform: scale(1) translate(0px, 0px) translateZ(0px);
  height: 100%;
  position: absolute;
  transform-origin: 0 0;
  transform-style: preserve-3d;
  width: 100%;
}

.pro--1 {
  transform: scale(4) translate(0px, 0px) translateZ(-3px);
  width: 110%;
}

.pro--2 {
  transform: scale(3) translate(0px, 50%) translateZ(-2px);
  width: 110%;
}

.pro--3 {
  transform: scale(2) translate(0px, 100%) translateZ(-1px);
  width: 110%;
}

.pro {
  background: #333;
  box-shadow: inset 0 0 0 5px orange;
  color: orange;
  font-size: 4em;
  line-height: 1em;
  text-align: center;
}

.pro--2 {
  background: rgba(75, 75, 75, 0.5);
  box-shadow: inset 0 0 0 5px green;
  color: green;
  line-height: 4em;
}

.pro--3 {
  background: rgba(75, 75, 75, 0.5);
  box-shadow: inset 0 0 0 5px white;
  color: white;
  line-height: 7em;
}

<div id="projection">
  <div class="pro pro--1">pro--1</div>
  <div class="pro pro--2">pro--2</div>
  <div class="pro pro--3">pro--3</div>
</div>

Answer 1

You've already solved your problem. Your code does exactly what you need it to do, it's just a CSS layout issue now.

https://codepen.io/anon/pen/xLWGzp?editors=0100

Because of the perspective changes, if you hang everything off the x-axis center everything will begin to line up properly:

(I'm just adding in the code changes here, I've left everything else the same)

#projection
    perspective-origin: center top

.pro
    transform-origin: center top

Now everything's lining up better, but it's still a bit off - you can change the $width variable to anything other than 100% to see the problem (60% is a good one)

So the problem now is just due to the positioning of the elements, when you set position: absolute they're default positioned to the left, change the width and add scale and transform and you get this equal-width/not-equal-position, so center them by adding:

#projection
    position: relative

.pro
    left: 50%
    margin-left: $width * -.5

(info here as to why that works to center: https://css-tricks.com/quick-css-trick-how-to-center-an-object-exactly-in-the-center/)

So now jiggle $width around to double-check, I tested it from 20% up to 150% and it works fine.