Squircle
Encyclopedia
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
with properties between those of a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
and those of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. It is a special case of superellipse
Superellipse
A superellipse is a geometric figure defined in the Cartesian coordinate system as the set of all points withwhere n, a and b are positive numbers....
. The word "squircle" is a portmanteau of the words, "square" and "circle".
Equation
In a Cartesian coordinate systemCartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
, the squircle centered on the point (a, b) with axes parallel to the coordinate axes is described by the equation:
where r is the minor radius of the squircle (cf.
Cf.
cf., an abbreviation for the Latin word confer , literally meaning "bring together", is used to refer to other material or ideas which may provide similar or different information or arguments. It is mainly used in scholarly contexts, such as in academic or legal texts...
equation of a circle).
Generalisation
The squircle is a specific case (found by setting n = 4) of the class of shapes known as "supercircles", which have the equation
Unfortunately, the taxonomy
Taxonomy
Taxonomy is the science of identifying and naming species, and arranging them into a classification. The field of taxonomy, sometimes referred to as "biological taxonomy", revolves around the description and use of taxonomic units, known as taxa...
is not consistent - some authors refer to the class as "supercircles" and the specific case as a squircle, while others adopt the opposite naming convention. Supercircles in turn are a subgroup of the even more general "superellipse
Superellipse
A superellipse is a geometric figure defined in the Cartesian coordinate system as the set of all points withwhere n, a and b are positive numbers....
s", which have the equation
where ra and rb are the semi-major
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
and semi-minor
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...
axes. Superellipses were extensively studied and popularised by the Danish
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Piet Hein
Piet Hein (Denmark)
Piet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...
.
Similar shapes
Another similar shape is defined as the boolean intersection
Constructive solid geometry
Constructive solid geometry is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects...
of a square and a concentric circle whose diameter is greater than the length of the side of the square, but less than the length of the diagonal of the square. Such truncated circle shapes lack the tangent continuity possessed by both superellipses and rounded squares.
Uses
Squircles are useful in opticsOptics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
. If light is passed through a two-dimensional square aperture, the central spot in the diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
pattern can be closely modelled by a squircle or supercircle. If a rectangular aperture is used, the spot can be approximated by a superellipse
Superellipse
A superellipse is a geometric figure defined in the Cartesian coordinate system as the set of all points withwhere n, a and b are positive numbers....
.
Squircles have also been used to construct dinner plate
Plate (dishware)
A plate is a broad, concave, but mainly flat vessel on which food can be served. A plate can also be used for ceremonial or decorative purposes.-Materials:...
s. A squircular plate has a larger area (and can thus hold more food) than a circular one with the same radius, but still occupies the same amount of space in a rectangular or square cupboard
Cupboard
A cupboard or press is a type of cabinet, often made of wood, used indoors to store household objects such as food, crockery, textiles and liquor, and protect them from dust,vermin and dirt....
. The same is true of a square plate, but there are various problems (such as wiping up sauce) associated with the corners of square plates.
Sumvision manufactures an SD card
Secure Digital
Secure Digital is a non-volatile memory card format developed by the SD Card Association for use in portable devices. The SD technology is used by more than 400 brands across dozens of product categories and more than 8,000 models, and is considered the de-facto industry standard.Secure Digital...
based MP3 player named Squircle for the budget market. It is not, however, a genuine squircle, although vaguely similar in shape.
Nokia
Nokia
Nokia Corporation is a Finnish multinational communications corporation that is headquartered in Keilaniemi, Espoo, a city neighbouring Finland's capital Helsinki...
is closely associated with the squircle, having used it as a touchpad button in many phones.
See also
- Egg of ColumbusEgg of ColumbusAn egg of Columbus or Columbus's egg refers to a brilliant idea or discovery that seems simple or easy after the fact. The expression refers to a popular story of how Christopher Columbus, having been told that discovering the Americas was no great accomplishment, challenged his critics to make an...
- SupereggSupereggIn geometry, a superegg is a solid of revolution obtained by rotating an elongated super-ellipse with exponent greater than 2 around its longest axis. It is a special case of super-ellipsoid....
- EllipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
- Astroid
- Ellipsoid
- OvalOvalAn oval is any curve resembling an egg or an ellipse, such as a Cassini oval. The term does not have a precise mathematical definition except in one area oval , but it may also refer to:* A sporting arena of oval shape** a cricket field...