Parametric Playground

Random points distributed inside a triangle.

2011-02-19T20:31:00.002+00:00

For a particular project for the office where I work, this month I had to develop a random distribution of points inside a triangle using Grasshopper. The aim is then to CNC cut holes in a complex triangular mesh, so I started from a single triangle to understand the geometry and the mathematical possibilities to perform this operation.

If you need to create a random series of points on a surface, it is very simple: you just divide the surface using U and V parameters... and you can randomize U and V to obtain a random result.

The problem is that the distribution is not entirely random, but it follows the U and V pattern.
For a Sphere you will obtain points more concentrated on the poles.
In this particular case, in a triangle, you will obtain the points more concentrated in one of the vertices.

(the image show just the UV subdivision, without any random function)

After a brief research I discovered some possible solutions to solve this issue.

0. USING A GRID OF POINTS
I will skip this option as it is not what I want, but you can always create a grid of points on the same plane of the triangle, and than pick only the points that are inside the borders.
Not random, and not what I am looking for.

1. USING MATH AND RANDOM NUMBERS

One possible approach is to use math and random numbers.

There is a mathematical way to create a random point inside any given triangle. You can see an hardcore reference here. or an easier and more understandable version here. (at least for non-mathematicians)

The basic formula is:

P = aA + bB + cC.

"where a+b+c=1 and a,b,c are each ≥ 0. Knowing a and b permits you to calculate c=1-a-b. So if you can generate two random numbers a and b, each in [0,1], such that their sum ≤ 1, you’ve got a random point in your triangle."

I used this basic math in Grasshopper generating a C# script to generate points.

This is the script for your reference:

C# CODE:

private void RunScript(double a, double b, double Ax, double Ay, double Bx, double By, double Cx, double Cy, double Az, double Bz, double Cz, ref object Point)
{
    double c = 0;
    double px, py, pz;
    if (a + b > 1)
    {
      a = 1 - a;
      b = 1 - b;
    }
    c = 1 - a - b;

    px = (a * Ax) + (b * Bx) + (c * Cx);
    py = (a * Ay) + (b * By) + (c * Cy);
    pz = (a * Az) + (b * Bz) + (c * Cz);
    Point3d point1 = new Point3d(px, py, pz);
    Point = point1;
}

Running this script you can generate any number of points inside a triangle.

This is the result:

The problem now is... how can you control the distribution?!

The answer is, you can't. The points are randomly generated... If you could control them, they were not random. (mmm)

But in my particular case I wanted to generate points with a certain distance between each other... kind of "equally distributed in a random way...".

My solution (I'm not sure if it is the only possible solution...) is then to filter the result, eliminating all the points that are too close between each other with a certain threshold value.
To do this you need a big for-loop that checks all the distances between all the points, ad exclude the once that are in a certain radius. Really complex. (at least for me).

Thanks god there is a great tool Kangaroo that added a command in Grasshopper: removeDuplicatePts.

Exactly what it is needed in this case...!

After a filter with this tool, plus filtering all the points too close to the borders, this is the result:

(second image shows only the selected points, the circles represent the minimum distance between selected points)

(Just as a reference, to filter the points too close to the borders, I've created an offset of the borders, and tested if the points were inside or outside.)

This is a video to see the random generation changing the radius between points, and changing the quantity of points generated:

2. USING KANGAROO

A different approach is to use the incredible tool called Kangaroo. (in the previous approach I've used only one of the math commands, but in this approach I will use the physics simulation)

You can create a definition similar to the one used by Daniel Picker in this video:

adapting it for a triangle or for a triangular mesh.

This is the result I've obtained in my case...: (the quality of the video is a bit poor... sorry...)

On the left side you see the definition by Daniel Picker adapted for this case. When I move the points of the triangle, Kangaroo re-calculate a repulsion with springs and re-distribute the points in the new resulting shape.

The problem that I have encountered is that the points are generated from a starting condition...

This means that probably a better way to do it would be to start with all the points in the center, and only then run Kangaroo.

This is the result with this second attempt:

The result is more accurate, or better to say that the points are distributed in a more even way.

3. USING GALAPAGOS

There is surely a way to the same with Galapagos... but as I reached the desired result with the previous methods, I leave this to someone else... if you create a definition that solve the issue with Galapagos... well let me know, and I can add it here ;)

----

I hope to receive comments to understand if the approaches are correct, and if there are better ways to do the same thing, please let me know.
And I hope this will help someone with the same problem to solve!

Center for Architecture Exhibition in New York

2010-12-18T22:15:00.001+00:00

I passed by the Center for Architecture of New York yesterday evening, and if you are around there, it is worth getting inside for a little tour!

I know this exhibition started over a month ago, so this post may look old, but I thought it was interesting to share it anyway

Even if this blog should be focused mainly on Parametric Architecture, I wanted to post this as I found it quite interesting: the current exhibition focuses on innovation in Architecture, integrating ecological features in the façades of the buildings.
The need of innovation in architecture is evident on many sides: if you consider a solar year, buildings consumes the 40% of the energy produced in the year. (the remaining percentage is shared between transport and production of goods...) And this is considering only the consumption for the maintainance, not the actual building phase.

The exhibition concentrate the attentions on the possible innovation in façade systems, that could create energy, insulate better the buildings, and keep them cooler in the hotter periods.
What I found interesting is that in the exhibition there are large scale physical models of moulded façade elements, with construction schemes, and deep technical explanations.

The following video shows one of the main exhibited elements:

Liquid Wall Prototype for Innovate:Integrate Exhibition from arbuckle industries on Vimeo.

The exhibition will be there until Jan 15, free.

Smart Geometry 2011 in Copenhagen announced!

2010-11-30T03:45:00.002+00:00

A double news!

1. First news is that the annual SmartGeometry event will take place in Copenhagen between 28th March and 2nd April 2011.

The clusters (the groups of research) are still not confirmed yet, but will come soon. If you want you can apply to propose a subject before the 13 of December.

All infos at: http://www.smartgeometry.org/

"Building the invisible" - Smart Geometry 2011

But there is a second news...!
In 2012 RPI will host the event!

Being in NY, I can only wait one year and a half to attend the next one, but fore everyone in Europe, it is an un-missable event. I attended in Barcelona in 2010, and it has been an amazing experience that I would recommend to any architect / designer into computational design to be updated to the latest researches in the field.

Advances in Architectural Geometry - Book

2010-11-12T22:27:00.001+00:00

Geometry is obviously a fundamental subject that lies at the core of Architecture. This September took place an important conference in Vienna (Advances in Architectural Geometry 2010), that ended with the publication of this book, that summarize in great details the last researches in the field of computational design.

I got a copy and I browsed quickly inside... It will take a while to read it, so I cannot give you a review here, but at the bottom there is the list of what you can find inside...

(I know, as a list is pretty boring...)

The interesting thing is that every project / research, is explained in full details, with all the references to bibliography used during the researches.

Quite interesting is the article about the research done at SmartGeometry 2010, Deep Surfaces.

Unfortunately I was attending a different Cluster, but thanks to this publication you can take a look at what they did. Really cool.

Basically...if you are really deep into the subject, you can't miss this book... and worst case, it will make a great impression if you leave it on the desk of the office...!

------

- Cristiano Ceccato, Builder-Geometer.

- Barbara Cutler, Joshua Nasman, Interpreting Physical Sketches as Architectural Models.

- Martin Tamke, Jacob Rüber, Hauke Jungjohann et al. ,Lamella Flock.

- Michael Eigensatz, Mario Deuss, Alexander Schiftner et al., Case Studies in Cost-Optimized Paneling of Architectural Freeform Surfaces.

- Johannes Wallner, Alexander Schiftner, Martin Kilian et al., Tiling Freeform Shapes With Straight Panels: Algorithmic Methods.

- Tomohiro Tachi , Freeform Rigid-Foldable Structure Using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh.

- Klaus Bollinger, Manfred Grohmann, Oliver Tessmann , The Sphere Project – Negotiate Geometrical Representations. From Design to Production.

- Ilija Bentscheff, Christoph Gengnagel ,Towards Teaching Generative Design in Architecture. - William Bergeron-Mirsky, Jason Lim, John Gulliford et al., Architectural Acoustics for Practioners.

- Ralph Bärtschi, Michael Knauss, Tobias Bonwetsch et al., Wiggled Brick Bond.

- Maurizio Brocato, Lucia Mondardini , Geometric Methods and Computational Mechanics for the design of Stone Domes Based on Abeille’s Bond.

-Benjamin S. Koren , Louvre Abu Dhabi 1/33 – Fabrication of a Large-Scale Physical Light-Test Model.

- Gershon Elber , Ortho-Pictures: 3D Objects From Independent 2D Data Sets.

- Lorenz Lachauer, Toni Kotnik, Geometry of Structural Form.

- Sean Ahlquist, Achim Menges, Realizing Formal and Funcional Complexity for Structurally Dynamic Systems in Rapid Computational Means.

- Alexander Schiftner, Jonathan Balzer, Statics-Sensitive Layout of Planar Quadrilateral Meshes.

Galapagos tests - How random are the solutions?

2010-11-06T14:41:00.004+00:00

From the previous post about creating a very simple curve with Galapagos, I've been asked why the solution is always different and random.

As I didn't really know exactly how to answer, I started to do some tests myself to understand it better.

I've used the same very simple file, just to not complicate things too much...
Always generating a line from 4 points, I moved the two central points using Galapagos, to obtain the curve closer to an attraction point. (you can get the GH file here).

The 4 points start as indicated (A, B), and in the image below you can see what happens after 10 tests (50 generations each):

As I was expecting, all the results are different... (but valid, as they are very close to the attractor point)... cool!

But if you do the test starting from this configuration (see below...) you will obtain all the solutions identical! This is an image with 10 solutions all identical, after running 50 generations 10 times:

I was quite surprised.

So why before I was obtaining random results, and now not?

The "randomness" seems to be related to the starting value of the sliders (starting position of the points).

If you start with the points very close to an acceptable solution, the final results are very similar between each other... (in the previous case, the starting points were coincident to a solution, so after generations, there was no change, so all the solutions are identical)
If you start with the points very far from an acceptable solution, the final results are very different from each other.

How random the results are of course depends also on the "freedom" of the genes... So if I increase the sliders range, I would obtain more random results...

If you re-do the test, starting with points distant from the solution, this is the result after 10 tests (again 50 generations each):

I'm looking forward for a manual about Galapagos, to have a better understanding of the generative process!
For the moment a great tool to learn more is the lecture that David Rutten did at AA in London:

Lecture of David Rutten - evolutionary principles.

Genetic geometry - Optimize a curve with Galapagos in Grasshopper

2010-11-02T02:29:00.004+00:00

In this days I'm doing very-very simple tests on genetic geometry, learning the complexity of the tool Galapagos that you can find in Grasshopper.

The video below shows a simple application of the genetic tool applied on a simple problem (in this example):
What is the closest curve to an attractor point (ATT) that I can generate moving 2 points (A and B)?

Galapagos allows to create genetic populations that get closer to a fitness target that you set.
In this case the target was to have the distance between the final curve and the attractor = 0.

Even if the solution of the problem seems easy, there are potentially infinite different solutions. It only depends on the genetic algorithm used. In this video, after few generations, the solution is displayed.
If you run again the program, you may get different results, but all of them will solve the problem.

The overall idea is to use the same genetic tools of the nature to obtain a better result generation over generation. If the problem has no solutions, the generation may be random and never lead to a final result.
The potential of this tool is enormous.

Some interesting stuff about it can be read here:

- Algorithmic Nature (pdf download, 220 pages)

And some other books you can order if you are still hungry:

- Algorithmic Architecture

- Autogenic Structures

- Of Growth and Form

In the meantime if you want, feel free to play with my definition, that you can get here.
(You need the latest grasshopper version to run it)
But as I said at the beginning is just a quick tests.
Hope it may help to get you started in discovering Galapagos more.

Parametric Playground...welcome!

2010-11-01T11:42:00.001+00:00

Hi! First of all, welcome to Parametric Playground!

The idea of this blog is to share with everyone my researches on parametric design, with the aim to enlarge the already big and growing community of designers that uses parametric softwares such as Grasshopper.

Check on the right side panel some other cool rocking stuff by other bloggers.

All of the things I've learnt so far (not too much!) are thanks to people that shared their knowledge.

Internet is populated by a growing number of blogs about parametric design and hopefully publishing here my researches will help other researches, and I have the hope that all the comments to my posts will help me in developing the right results!

The name playground comes from the fact that my researches are done just playing around during my free time, so don't expect something too scientific... but still my hope is to be as clear as possible, explaining the basics of every step of the research I'm doing to help everyone interested to follow them.

At the moment I'm playing with Grasshopper and Galapagos for genetic design.

I will post soon the first results...

Anyway, I'll try to be not boring as a church sermon!

Keep in touch