##### 2:48 AM, Thursday November 4th 2021

Just one quick note about your second point - the shape of the chest itself doesn't actually factor into the ellipse we use for figuring out how the lid opens. The reason it's an ellipse is actually completely independent of the box/chest itself.

Rather, it has to do with the fact that we're rotating something around an axis. What is being rotated is the edge of the lid, along the axis defined by the hinges. Those are the only parts that matter - the rest of the chest could be as long or as tall as you'd like, but all we care about is the fact that we're rotating a distance (the edge of the lid) is what creates a circle in 3D space, with the length of that edge as its radius. Here's what I mean (though excuse the sloppy drawing and accidental swastika). No matter how far the lid opens, the tip of that edge will always fall on the given circle.

##### 4:40 AM, Thursday November 4th 2021 edited at 10:38 AM, Nov 4th 2021

I see, there's one thing still bugging me, how do you illustrate the length of the lid when it's being rotated along the axis if the box is longer than its height? Is there a tool for this?

edited at 10:38 AM, Nov 4th 2021
##### 6:31 PM, Friday November 5th 2021

I'm honestly still a little confused by what you're asking, because I don't really see why the relationship between the length/height would be relevant.

There are three aspects of the lid that we need to determine, once it has been rotated in space:

• Its depth - so the distance from the front of the lid to the back of the lid. This is the edge we're rotating in the first place, so we maintain that distance by drawing an ellipse that represents a circle in 3D space, with the axis defined by the hinges serving as the ellipse's minor axis.

• Its height. I eyeballed this in the video demo for the challenge, but to be more precise, you'd just repeat the step for the depth, but for the top-side edge (where for the depth we used the bottom-side edge). If your lid isn't a simple box (and therefore might not have a top-side edge), you'd just end up using a box to figure out the correct rotation, then build whatever fancy lid inside of it.

• Its length - so the distance from one side to the other. This is achieved by simply repeating the same process on the other side. You'll see in the demo video that while I eyeballed the height (instead of adding two ellipses on either side), I did place an ellipse on both sides.

So, in essence, all three dimensions of the rotated lid can be determined by applying the same ellipse/circle based approach, and none of this has anything to do with the chest's height. The only dimension of the chest that actually matters here is the depth, since the depth is what becomes the radius of each of these circles in 3D space.

##### 3:32 PM, Wednesday November 10th 2021

Ah, alright. I'm just wondering if there's a more precise way to draw a square around the minor axis that can work for chests of any dimension (chests with long/short depth), so I can draw a more precise ellipse. Since the depth is converging to the "horizontal" vanishing point, how do you measure the height part of the square, which is vertical?

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