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Dividing A Polygon In Any Given Number Of Equal Areas

This post will detail a closed form solution to split polygons into any number of equal areas, while ensuring minimum length of line based cuts. The solution works for both convex and concave polygons, as long as they don’t have non-manifold vertices, and can be traversed with a single loop. Credit for this concept goes to Dr. Rossignac, who also introduced me to the power of medial axis.

Figure 1: Examples of Convex (Left) and Concave (Right) Polygon division. Red lines show the cuts splitting polygons into equal areas

Base Case: 4-Sided Polygon
Let’s start with a 4-sided polygon, and say we need to split this into N sub-polygons. We use the divide and conquer methodology:

  • Calculate the area of the polygon, say A_{poly}. Therefore the target area of each of the N sub-polygons would need to be A_{poly} \over N
  • Split the polygon into two sub-polygons – smaller one with area of A_{poly} \over N, and bigger one with area of {(N-1)\over N} A_{poly}
  • Continue using the same approach on the bigger polygon till the bigger polygon reduces into polygon with desired area.

The problem of splitting polygon N-ways has now been reduced to splitting polygon into two sub-polygon with specific area. Since the cut is a line, it would start from one edge and end at another. There are six possible edge pairs in a 4-sided polygon, that could lead to a potential cut. Three of these cuts – C1, C2, C3 are shown in the images below as a reference. We can select the minimum cut from the all potential cuts obtained from each edge pair.

Figure 2: Three potential cuts for splitting polygon ABCD

Now, the problem is reduced to making a cut on a given edge pair, such that the cut splits the polygon into two sub-polygons of specific area. Continue reading

Prototyping Acrylic Based Microfluidics Using Thermal Bonding

A key aspect of translational research in the medical devices domain is migrating to thermoplastic polymers like PMMA (commonly referred to as acrylic). The fabrication of microfluidic channels involves two surfaces. One surface has the open channels and the features necessary for microfluidics. The second surface provides the closing interface almost like a lid. The challenge is to create a hermetically sealed interface between these two surfaces without using any adhesives. There are several bonding technologies like laser welding and ultra-sonic welding that are perfect for such applications, but they are expensive and time consuming. In this post, I will describe a low cost prototyping technique that delivers optically clear, hermetically sealed bonding for thermoplastic microfluidics.

Basic Concept
The thermoplastic surfaces to be bonded are sandwiched between two aluminum plates. The bottom plate is heated, and precise pressure is applied from the top. The performance of the bonding will depend upon

  • Temperature of the hot plate
  • Pressure applied
  • Bonding Time
  • Surface roughness of the bonding plates (important for optically clear finish)

Any issues like uneven bonding, bubbles, feature deformation can be overcome by adjusting the above parameters. More on these common mistakes, once I have covered the basic process. Continue reading

Minimum Viable Product On A Horse Carriage

Dyson, the leader in high-end vacuum cleaners, recently announced their new robotic vacuum cleaner, the 360 eye. It has every bell and whistle you can imagine in a vacuum cleaner – it has more cleaning power than any of its competitors. It can visualize the entire house via a 360 degree camera and collection of infra-red sensors and plan the most efficient cleaning route. It has army tank style treads, so it can simply step over uneven surfaces like carpets and rugs. If all that’s not enough, it comes with a smart phone app, so you can schedule a cleaning session remotely thirty minutes before you arrive in your house with an unexpected guest.

This impressive set of features took sixteen years, and $47 million for Sir James Dyson to improve a product that was already the best in the market. It got me wondering about the first guy who thought of making a vacuum cleaner. There were products as early as 1860. This is an era well before electricity was available as a public utility, and well before plastics arrived, but hey – the carpets and the house are not going to clean themselves. Let’s say – you live in the nineteenth century, and you are crazy enough to dream about a motorized product to clean the house – what would the first version of your product look like, and how would you convince everyone that they can’t live without it?

I love what Hubert Cecil Booth did in 1901. He created a vacuum pump powered by a gasoline engine. The engine was heavy, so he used a horse carriage to carry the engine from one neighborhood to another. All cleaning was done by suction through long tubes with nozzles on the ends.

He used transparent hoses so customers could watch the dirt leaving their house. His staff was dressed in clean professional uniforms. The entire cleaning service was offered as an enjoyable lavish experience while making his customers as the envy of their neighborhood. Booth encouraged his customers to host vacuum tea parties, great for enhancing the whole experience, and perfect for him to attract new customers.

Booth’s vacuum service is an excellent example of a minimum viable product. He didn’t lock himself in a cave to make the most sleek, elegant looking product. He modified existing technologies to test the product hypothesis with minimal resources. He got his product to early customers as soon as possible. His first customer was a local restaurant owner. Booth agreed to clean the dining room for free, but the immense publicity from that event made sure he never looked back.

Surfing & Entrepreneurship

My concept of having a good time on the beach is to find a good spot, lie around all day reading a book, and soak in the nice ocean view. I might even take an occasional dip in the water or at least get my feet wet, but I can’t be in the water for long. I found it too boring, that is till our recent trip to Santa Catalina. I got myself a six-foot surfing board, and didn’t know that would make such a difference. I found surfing to be fun and addictive. My wife had to literally drag me out of the water every day. For a beginner like me – the learning curve was huge, but the joy of standing up on the waves, even if it was for fraction of a second, kept me going. Another interesting aspect about this experience was the number of parallels I found myself drawing between surfing and entrepreneurship. Here are a few:

Inspiration is better served up close and personal
Its a thing of beauty to watch a crafty surfer dance on the waves. While you can certainly watch youtube videos and get inspired, there’s something different about watching a master practice in person. Perhaps it has to do with getting a full experience and not just the highlights. You get to see their imperfect attempts and the efforts they continue to put. Its the same reason I find myself more inspired by the entrepreneurial ventures of my friends. I see them fighting their battles every day, and can relate.

You will fall down
It is a sure thing in surfing. Falling down is an integral part of the surfing experience. It is not looked down upon as a failure. What matters is you get up and go back for another wave. You try again, fail again, and try to fail better the next time till you succeed.
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Book Notes: The Design of Everyday Things

This is an excellent book for those who aspire to do good work in the field of design, engineering, architecture, consumer products, or anything creative. After reading the book, you will have an enhanced appreciation for everyday things around you – things as trivial as a toaster, refrigerator, and pen. You will learn what separates a good product from a bad one. Dan Norman, the author is a renowned cognitive scientist and usability expert. He worked as a user experience architect at Apple, and popularized the term “user-centered-design”.

Notes from the book:

Far too many items in the world are designed, constructed, and foisted upon us with no understanding for how they will be used. Three key things stand out of the book:

  1. Not the user’s fault: If people are having trouble with your product, its not their fault – it’s the fault of design
  2. Design Principles: Don’t criticize something unless you can offer a solution.
    • Feedback: The user must be able to see the effect of his action. If he presses something, let him know through the design interface.
    • Constraints: The surest way to make something easy to use, with few errors, is to make it impossible to do otherwise. e.g. Battery of memory cards go only one way in properly designed products
    • Affordances: A good designer makes sure that appropriate actions are perceptible and inappropriate ones invisible.
  3. The power of observation: The path to be a good designer, starts by learning to watch, and learning to observe. Yogi Berra said “You can observe a lot by watching”. Problem is you have to know how to watch.

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