Month: April 2019

History of the “Squircle”

Posted on by John

Architect Peter Panholzer coined the term “squircle” in the summer of 1966 while working for Gerald Robinson. Robinson had seen a Scientific American article on the superellipse shape popularized by Piet Hein and suggested Panholzer use the shape in a project.

Piet Hein used the term superellipse for a compromise between an ellipse and a rectangle, and the term “supercircle” for the special case of axes of equal length. While Piet Hein popularized the superellipse shape, the discovery of the shape goes back to Gabriel Lamé in 1818.

Squircle with p = 3.162034

You can find more on the superellipse and squircle by following these links, but essentially the idea is to take the equation for an ellipse or circle and replace the exponent 2 with a larger exponent. The larger the exponent is, the closer the superellipse is to being a rectangle, and the closer the supercircle/squircle is to being a square.

Panholzer contacted me in response to my article on squircles. He gives several pieces of evidence to support his claim to have been the first to use the term. One is a letter from his employer at the time, Gerald Robinson. He also cites these links. [However, see Andrew Dalke’s comment below.]

Optimal exponent

As mentioned above, squircles and more generally superellipses, involve an exponent p. The case p = 2 gives a circle. As p goes to infinity, the squircle converges to a square. As p goes to 0, you get a star-shape as shown here. As noted in that same post, Apple uses p = 4 in some designs. The Sergels Torg fountain in Stockholm is a superellipse with p = 2.5. Gerald Robinson designed a parking garage using a superellipse with p = e = 2.71828.

Panholzer experimented with various exponents [1] and decided that the optimal value of p would be the one for which the squircle has an area half way between the circle and corresponding square. This would create visual interest, leaving the viewer undecided whether the shape is closer to a circle or square.

The area of the portion of the unit circle contained in the first quadrant is π/4, and so we want to find the exponent p such that the area of the squircle in the first quadrant is (1 + π/4)/2. This means we need to solve

\int_0^1 (1 - x^p)^{1/p}\, dx = \frac{\Gamma\left(\frac{p+1}{p}\right)^2}{\Gamma\left(\frac{p+2}{p} \right )} = \frac{1}{2} + \frac{\pi}{8}

We can solve this numerically [2] to find p = 3.1620. It would be a nice coincidence if the solution were π, but it’s not quite.

Sometime around 1966 Panholzer had a conference table made in the shape of a squircle with this exponent.

Computing

I asked Panholzer how he created his squircles, and whether he had access to a computer in 1966. He did use a computer to find the optimal value of p; his brother in law, Hans Thurow, had access to a computer at McPhar Geophysics in Toronto. But he drew the plots by hand.

There was no plotter around at that time, but I used transparent vellum over graph paper and my architectural drawing skills with “French curves” to draw 15 squircles from p=2.6 (obviously “circlish”) to p=4.0 (obviously “squarish”).

More squircle posts

[1] The 15 plots mentioned in the quote at the end came first. A survey found that people preferred the curve corresponding to p around 3.1. Later the solution to the equation for the area to be half way between that of a circle and a square produced a similar value.

[2] Here are a couple lines of Mathematica code to find p.

    f[p_] := Gamma[1 + 1/p]^2/Gamma[1 + 2/p]
    FindRoot[f[p] - (1 + Pi/4)/2, {p, 4}]

The 4 in the final argument to FindRoot is just a suggested starting point for the search.

Covered entities: TMPRA extends HIPAA

Posted on by John

The US HIPAA law only protects the privacy of health data held by “covered entities,” which essentially means health care providers and insurance companies. If you give your heart monitoring data or DNA to your doctor, it comes under HIPAA. If you give it to Fitbit or 23andMe, it does not. Government entities are not covered by HIPAA either, a fact that Latanya Sweeney exploited to demonstrate how service dates be used to identify individuals.

Texas passed the Texas Medical Records Privacy Act (a.k.a. HB 300 or TMPRA) to close this gap. Texas has a much broader definition of covered entity. In a nutshell, Texas law defines a covered entity to include anyone “assembling, collecting, analyzing, using, evaluating, storing, or transmitting protected health information.” The full definition, available here, says

“Covered entity” means any person who:

(A) for commercial, financial, or professional gain, monetary fees, or dues, or on a cooperative, nonprofit, or pro bono basis, engages, in whole or in part, and with real or constructive knowledge, in the practice of assembling, collecting, analyzing, using, evaluating, storing, or transmitting protected health information. The term includes a business associate, health care payer, governmental unit, information or computer management entity, school, health researcher, health care facility, clinic, health care provider, or person who maintains an Internet site;

(B) comes into possession of protected health information;

(C) obtains or stores protected health information under this chapter; or

(D) is an employee, agent, or contractor of a person described by Paragraph (A), (B), or (C) insofar as the employee, agent, or contractor creates, receives, obtains, maintains, uses, or transmits protected health information.

Update: Texas has now passed the Texas Data Privacy and Security Act (TDPSA).

Posts on other privacy regulations