[Plugin] Superellipse 1.2

pilou

Cool!

Next Will be Super Cycloïd?

Regular Polygon

@unknownuser said:

Next Will be Super Cycloïd?

Yeah, that sounds like a plugin I would work on. They tend to have that sort of mathematical flavor.

SU Reviewer

Welcome! Another great and simple plugin.

It works really well, though I have a note on the Exponent value. There is no known scale for the exponent input that I could tell. As of now its pretty much a shot in the dark as to how square the superellipse will come out. Perhaps having a scale of 1-100 (100 being a perfect square) the plugin would be more manageable?

Good job!

remus

Why replace one arbitrary scale with another? they'll both serve the same purpose, and as long as the user understands 'big n = more square' you cant go too far wrong.

Regular Polygon

@su reviewer said:

As of now its pretty much a shot in the dark as to how square the superellipse will come out. Perhaps having a scale of 1-100 (100 being a perfect square) the plugin would be more manageable?

Otherwise great plugin!

Thanks for your review!

I can see your point. It is not real intuitive to me either how square the superellipse will be for a given exponent value.

We could replace the exponent with a scale that runs from 1-100. If 1 represents a completely round shape, then we would associate it with an ellipse. At the other end, 100 represents a completely square shape, so we associate it with a rectangle. But it is not so obvious to me what the numbers in the middle represent. Which superellispe should correspond to a scale factor of say 50?

boofredlay

Oh man, I could have used this about 4 days ago. This is one of those quick scripts I imagine using quite often.

Thank you very much.

Ben Ritter

Thank you Regular Polygon.

thomthom

Is "Regular Polygon" a reference to Flat Land?

Dave R

Thank you for this plugin. It is very nice.

I think the exponent makes sense. After all, it's a logical input considering the formula you're solving. Length and width are a and b so why not define n? I would prefer that to some arbitrary 1-100 scale.

$http://upload.wikimedia.org/math/1/7/1/171490937fc13b93677c00b8c80ffb76.png$

brookefox

@dave r said:

I think the exponent makes sense. After all, it's a logical input considering the formula you're solving. Length and width are a and b so why not define n? I would prefer that to some arbitrary 1-100 scale.

$http://upload.wikimedia.org/math/1/7/1/171490937fc13b93677c00b8c80ffb76.png$

Silly me thought you figured that formula out for yourself, Dave.

Since I have no sense at this point where the exponent variable will take me, though I guess that comes with practice, the 1-100 option has some appeal. I should hold my tongue until I practice, but I won't.

Thanks for the contribution, rp, and for adding the loci to the ellipse.

remus

A quick question about using values of n<1 if i may: how do you handle it, because surely you'll end up trying to take roots of negative numbers which'll give you imaginary numbers, unless you just take the imaginary part and use that?

Regular Polygon

@remus said:

A quick question about using values of n<1 if i may: how do you handle it, because surely you'll end up trying to take roots of negative numbers.

Here is the code that computes the coordinates.


def sgn(num)
  return  0  if num.zero?
  return +1  if ( num > 0 )
  return -1
end


# Compute the points for the vertices of the superellipse
def points
  e = 2.0 / @exp
  pts = []
  delta = 2 * Math;;PI / @edges
  for i in 0...@edges do
    phi = i * delta
    x = @a * sgn(Math.cos(phi)) * Math.cos(phi).abs**e
    y = @b * sgn(Math.sin(phi)) * Math.sin(phi).abs**e
    pts.push([x, y, 0])
  end
  pts.push(pts[0])  # close the loop
  pts
end

The only trick is to take the absolute value before you raise a number to a fractional power. After that, multiply it by +1 or -1, depending on the sign the original number had.

remus

That makes sense, cheers.

Regular Polygon

@thomthom said:

Is "Regular Polygon" a reference to Flat Land?

No, not really.

"Regular Polygon" is kind of a play on words. It could mean an ordinary, everyday, run-of-the-mill polygon (i.e. the basic element of 3D graphics). But, in the mathematical sense, it means a symmetrical polygon whose angles and edges are all equal.

Regular Polygon

I have just released a new version 1.1 of the Superellipse plugin. This version allows you to specify the "squareness" of a superellipse using a scale. The exact association between the scale and the squareness is spelled out in gory detail in the post on my blog. But a picture is probably worth a 1000 words.

scale 8.PNG

Basically, 0 corresponds to an ellipse, 100 corresponds to a rectangle, and a scale factor of 50 corresponds to a superellipse that stretches 50% of the way in between. I think this makes assigning a squareness value fairly intuitive.

You can run both versions (1.0 and 1.1) at the same time to compare them. To download the plugin, or read more about it, please visit my blog at http://regularpolygon.blogspot.com/

Thanks.

olishea

thanks for sharing

simon le bon

thanks for sharing
we will see, but it would prove itself often useful in time
++simon

Regular Polygon

Hey, thanks everyone for the big welcome, and your show of support, for this first -- albeit simple -- plugin.

pyroluna

Really nice... but it reminds me... I used to use POV-Ray some years ago, in which it was possible to create a 3d-superellipsoid.
Could you make that??? would be so awesome, I used to use it a lot!
Perhaps in SU it would need a variable for level of detail, though...

@unknownuser said:

The superellipsoid object creates a shape known as a superquadric ellipsoid object. It is an extension of the quadric ellipsoid. It can be used to create boxes and cylinders with round edges and other interesting shapes. Mathematically it is given by the equation:

The values of e and n, called the east-west and north-south exponent, determine the shape of the superquadric ellipsoid. Both have to be greater than zero. The sphere is given by e = 1 and n = 1.

The syntax of the superquadric ellipsoid is:

SUPERELLIPSOID:
superellipsoid
{
<Value_E, Value_N>
[OBJECT_MODIFIERS...]
}

The 2-D vector specifies the e and n values in the equation above. The object sits at the origin and occupies a space about the size of a box{<-1,-1,-1>,<1,1,1>}.

Two useful objects are the rounded box and the rounded cylinder. These are declared in the following way.

#declare Rounded_Box = superellipsoid { <Round, Round> }
#declare Rounded_Cylinder = superellipsoid { <1, Round> }

The roundedness value Round determines the roundedness of the edges and has to be greater than zero and smaller than one. The smaller you choose the values, the smaller and sharper the edges will get.

Very small values of e and n might cause problems with the root solver (the Sturmian root solver cannot be used).

Regular Polygon

@unknownuser said:

Really nice... but it reminds me... I used to use POV-Ray some years ago, in which it was possible to create a 3d-superellipsoid.
Could you make that??? would be so awesome, I used to use it a lot!
Perhaps in SU it would need a variable for level of detail, though...

Thanks, glad you like it. Funny you should ask because, in fact, I do have a superellipsoid plugin. It lets the user enter the number of lines of latitude and longitude to control the level of detail.

It was interesting to see how POV-Ray is dealing with the exponents in the formula. They let the user specify a Roundedness value, whereas I came up with a Squareness value. I'll have to try out both approaches to see which I like better.

Anyway, I should get the superellipsoid plugin polished up, and posted on my blog, sometime next week. Stay tuned!