#hyperbolic paraboloid

[scavengedluxury]

Szántód ferry terminal, 1968. From the Budapest municipal photography company archive.   

[grlbts]

The shape of Pringles is called a hyperbolic paraboloid

In calculus, a hyperbolic paraboloid is a doubly-curved surface that resembles the shape of a saddle. This shape makes it easier to stack the chips and minimises the possibility of breaking during transport. Due to its saddle shape, there is no predictable way to break it up, which apparently increases that crunchy feeling.

[muisnotforyou]

Today I learned how fun it is to draw a hyperbolic paraboloid.

Pictured above is one example of it, the graph of the function f : IR² → IR defined by the equation f(x,y) := xy, with some points of the input plane marked and height of the graph indicated in four points. It is drawn only for a square around the origin (that makes it easy to draw), but if you imagine extending it on the edges to make a rounder shape, you might be able to see the (probably) most famous rendition of the hyperbolic paraboloid shape, the Pringles chip:

So why is the hyperbolic paraboloid so fun to draw? Well, a defining property (and according to Wikipedia, one of the oldest definitions) of the shape is that it may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines.

Skew lines are lines that don't cross but are also not parallel.

In the above example, this property is easily explained like this: if you fix either x or y into place (treat it like a constant) in the equation z = xy, you get the points z of the cross-section of the graph of f and the plane corresponding to the equation y = c or x = c, depending on which variable you fixed to the constant c. This is now a linear equation, which represents a straight line.

So, whenever you have two points that you know are in the graph, which also lie in the same plane parallel to either the xz-plane or the yz-plane, we now know that the straight line that contains both of them is also completely contained in the graph. This provides us with a fairly simple way to draw (part of) this beautiful shape:

Draw a square around the origin in the xy-plane and for it's corners find the corresponding z values in the graph. In this case, I chose a square with sidelength 2, but the actual values don't matter that much.

Having found four points (above and below the corners of the square), connect them with straight lines parallel to the sides of the square. These are in the graph.

Choose two of those lines which are opposite each other, segment them evenly with the same segment lengths, and connect corresponding points with straight lines.

Technically, in the last step you have to draw infinitely many lines to get the real shape. But then again, technically to get the actual real shape, in the first step you would have to draw an infinitely large square (or connect both of the pairs of opposite lines with infinitely long straight lines, that works too). But the great thing is, your brain will automatically fill in the rest of the shape for you if you have enough segments.

More generally, the definition given above means you can really take any two skew lines, find the plane they are both parallel to, and then connect them with straight lines that lie in planes that cross that plane with a 90 degree angle. I think it's really cool how you can draw such a complicated looking shape with so few and easy instructions, and only using straight lines. In fact, if I understood the Wikipedia arcticle correctly, this property is probably one of the reasons Pringles are even made in this shape, because it makes manufacturing fairly simple.

[karkod-art]

Day 18: Saddle

Behold, the poorly drawn hyperbolic paraboloid.

It will be hard to beat this thing as the worst inktober drawing, look at the typo oof. But let's wait until partial exams begin next week, it could (and will likely) get worse.

[knotty-et-al]

"Home is where love is"

Knotty's art from ~a week ago. [2023/08/17]

[stickydramathegirl4me]

[bluecote]

two saints school, southwark, london. chamberlin, powell & bon 1958-60.

[savageform]

Markham Moor hyperbolic paraboloid