Family of closed mathematical curves
A
superellipse
, also known as a
Lame curve
after
Gabriel Lame
, is a closed curve resembling the
ellipse
, retaining the geometric features of
semi-major axis
and
semi-minor axis
, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.
In two dimentional
Cartesian coordinate system
, a superellipse is defined as the set of all points
on the curve that satisfy the equation
where
and
are positive numbers referred to as
semi-diameters
or
semi-axes
of the superellipse, and
is a positive parameter that defines the shape. When
, the superellipse is an ordinary ellipse. For
, the shape is more rectangular with rounded corners, and for
, it is more pointed.
[1]
[2]
[3]
In the
polar coordinate system
, the superellipse equation is (the set of all points
on the curve satisfy the equation):
Specific cases
[
edit
]
This formula defines a
closed curve
contained in the
rectangle
−
a
≤
x
≤ +
a
and
−
b
≤
y
≤ +
b
. The parameters
and
are the semi-diameters or semi-axes of the curve. The overall shape of the curve is determined by the value of the exponent
, as shown in the following table:
|
The superellipse looks like a four-armed star with
concave
(inwards-curved) sides.
For
, in particular, each of the four arcs is a segment of a
parabola
.
An
astroid
is the special case
,
|
|
|
The curve is a
rhombus
with corners (
) and (
).
|
|
|
The curve looks like a rhombus with the same corners but with
convex
(outwards-curved) sides.
The
curvature
increases without
limit
as one approaches its extreme points.
|
|
|
The curve is an ordinary
ellipse
(in particular, a
circle
if
).
|
|
|
The curve looks superficially like a
rectangle
with rounded corners.
The curvature is zero at the points (
) and (
).
|
|
If
, the figure is also called a
hypoellipse
; if
, a
hyperellipse
. When
and
, the superellipse is the boundary of a
ball
of
in the
-norm
. The extreme points of the superellipse are (
) and (
), and its four "corners" are (
,
), where
(sometimes called the "superness"
[4]
).
Mathematical properties
[
edit
]
When
n
is a positive
rational number
(in lowest terms), then each quadrant of the superellipse is a
plane algebraic curve
of order
.
[5]
In particular, when
and
n
is an even integer, then it is a
Fermat curve
of degree
n
. In that case it is non-singular, but in general it will be
singular
. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.
The curve is given by the
parametric equations
(with parameter
having no elementary geometric interpretation)
where each
can be chosen separately so that each value of
gives four points on the curve. Equivalently, letting
range over
where the
sign function
is
Here
is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals
while in the parametric expressions
Area
[
edit
]
The
area
inside the superellipse can be expressed in terms of the
gamma function
as
or in terms of the
beta function
as
- [6]
Perimeter
[
edit
]
The
perimeter
of a superellipse, like that of an
ellipse
, does not admit
closed-form solution
purely using
elementary functions
. Exact solutions for the perimeter of a superellipse exist using
infinite summations
;
[7]
these could be truncated to obtain approximate solutions.
Numerical integration
is another option to obtain perimeter estimates at arbitrary precision.
A closed-form approximation obtained via
symbolic regression
is also an option that balances parsimony and accuracy. Consider a superellipse centered on the origin of a 2D plane. Now, imagine that the superellipse (with shape parameter
) is stretched such that the first quadrant (e.g.,
,
) is an arc from
to
, with
. Then, the arc length of the superellipse within that single quadrant is approximated as the following function of
and
:
[8]
h + (((((n-0.88487077) * h + 0.2588574 / h) ^ exp(n / -0.90069205)) + h) + 0.09919785) ^ (-1.4812293 / n)
This single-quadrant arc length approximation is accurate to within ±0.2% for across all values of
, and can be used to efficiently estimate the total perimeter of a superellipse.
Pedal curve
[
edit
]
The
pedal curve
is relatively straightforward to compute. Specifically, the pedal of
is given in
polar coordinates
by
[9]
Generalizations
[
edit
]
| This section
needs expansion
. You can help by
adding to it
.
(
June 2008
)
|
The generalization of these shapes can involve several approaches.The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.
Higher Dimensions
[
edit
]
The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.
[10]
- A
superellipsoid
extends the superellipse into three dimensions, creating shapes that vary between ellipsoids and rectangular solids with rounded edges. The superellipsoid is defined as the set of all points
that satisfy the equation:
where
and
are positive numbers referred to as the semi-axes of the superellipsoid, and
is a positive parameter that defines the shape.
[6]
- A
hyperellipsoid
is the
-dimensional analogue of an
ellipsoid
(and by extension, a superellipsoid). It is defined as the set of all points
that satisfy the equation:
where
are positive numbers referred to as the semi-axes of the hyperellipsoid, and
is a positive parameter that defines the shape.
[11]
Different Exponents
[
edit
]
Using different exponents for each term in the equation, allowing more flexibility in shape formation.
[12]
For two-dimentional case the equation is
where
either equals to or differs from
. If
, it is the Lame's superellipses. If
, the curve possesses more flexibility of behavior, and is better possible fit to describe some experimental information.
[11]
For the three-dimensional case, three different positive powers
,
and
can be used in the equation
. If
, a super-ellipsoid is obtained. If any two or all three powers differ from each other, a solid is obtained that may possess more flexibility in representing real structural data than the super ellipsoid. A three-dimensional super-ellipsoid with
,
and the semi-diameters
,
represents the structure of the National Centre for the Performing Arts in China.
[11]
In the general
?dimensional case, the equation is
, where In general,
may differ from each other. It is the superellipsoid only if
.
[11]
Related shapes
[
edit
]
Superquadrics
are a family of shapes that include superellipsoids as a special case. They are used in computer graphics and geometric modeling to create complex, smooth shapes with easily adjustable parameters.
[13]
While not a direct generalization of superellipses,
hyperspheres
also share the concept of extending geometric shapes into higher dimensions. These related shapes demonstrate the versatility and broad applicability of the fundamental principles underlying superellipses.
Anisotropic Scaling
[
edit
]
Anisotropic
scaling involves scaling the shape differently along different axes, providing additional control over the geometry. This approach can be applied to superellipses, superellipsoids, and their higher-dimensional analogues to produce a wider variety of forms and better fit specific requirements in applications such as computer graphics, structural design, and data visualization. For instance, anisotropic scaling allows the creation of shapes that can model real-world objects more accurately by adjusting the proportions along each axis independently.
[14]
History
[
edit
]
The general Cartesian notation of the form comes from the French mathematician
Gabriel Lame
(1795–1870), who generalized the equation for the ellipse.
Hermann Zapf
's
typeface
Melior
, published in 1952, uses superellipses for letters such as
o
. Thirty years later
Donald Knuth
would build the ability to choose between true ellipses and superellipses (both approximated by
cubic splines
) into his
Computer Modern
type family.
The superellipse was named by the
Danish
poet and scientist
Piet Hein
(1905?1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in
Stockholm
,
Sweden
announced a design challenge for a
roundabout
in their city square
Sergels Torg
. Piet Hein's winning proposal was based on a superellipse with
n
= 2.5 and
a
/
b
= 6/5.
[15]
As he explained it:
Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.
Sergels Torg was completed in 1967. Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.
[16]
By rotating a superellipse around the longest axis, he created the
superegg
, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a
novelty toy
.
In 1968, when negotiators in
Paris
for the
Vietnam War
could not agree on the shape of the negotiating table, Balinski,
Kieron Underwood
and Holt suggested a superelliptical table in a letter to the
New York Times
.
[15]
The superellipse was used for the shape of the 1968
Azteca Olympic Stadium
, in
Mexico City
.
The second floor of the original
World Trade Center
in New York City consisted of a large, superellipse-shaped overhanging balcony.
Waldo R. Tobler
developed a
map projection
, the
Tobler hyperelliptical projection
, published in 1973,
[17]
in which the
meridians
are arcs of superellipses.
The logo for news company
The Local
consists of a tilted superellipse matching the proportions of Sergels Torg. Three connected superellipses are used in the logo of the
Pittsburgh Steelers
.
In computing, mobile operating system
iOS
uses a superellipse curve for app icons, replacing the
rounded corners
style used up to version 6.
[18]
See also
[
edit
]
References
[
edit
]
- ^
Shi, Pei-Jian; Huang, Jian-Guo; Hui, Cang; Grissino-Mayer, Henri D.; Tardif, Jacques C.; Zhai, Li-Hong; Wang, Fu-Sheng; Li, Bai-Lian (15 October 2015).
"Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape"
.
Frontiers in Plant Science
.
6
: 856.
doi
:
10.3389/fpls.2015.00856
.
ISSN
1664-462X
.
PMC
4606055
.
PMID
26528316
.
- ^
Barr (1981).
"Superquadrics and Angle-Preserving Transformations"
.
IEEE Computer Graphics and Applications
.
1
(1): 11?23.
doi
:
10.1109/MCG.1981.1673799
.
ISSN
1558-1756
.
S2CID
9389947
.
- ^
Liu, Weixiao; Wu, Yuwei; Ruan, Sipu; Chirikjian, Gregory S. (2022).
"Robust and Accurate Superquadric Recovery: A Probabilistic Approach"
.
2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
. pp. 2666?2675.
arXiv
:
2111.14517
.
doi
:
10.1109/CVPR52688.2022.00270
.
ISBN
978-1-6654-6946-3
.
S2CID
244715106
.
- ^
Donald Knuth:
The METAFONTbook
, p. 126
- ^
"Astroid"
(PDF)
.
Xah Code
. Retrieved
14 March
2023
.
- ^
a
b
"Ellipsoids in Higher Dimensions"
.
analyticphysics.com
. Retrieved
19 June
2024
.
- ^
"Superellipse (Lame curve)"
(PDF)
. Archived from
the original
(PDF)
on 31 March 2022
. Retrieved
9 November
2023
.
- ^
Sharpe, Peter.
"AeroSandbox"
. GitHub
. Retrieved
9 November
2023
.
- ^
J. Edwards (1892).
Differential Calculus
. London: MacMillan and Co. pp.
164
.
- ^
Boult, Terrance E.; Gross, Ari D. (19 February 1988). Casasent, David P.; Hall, Ernest L. (eds.).
"Recovery Of Superquadrics From 3-D Information"
.
SPIE Proceedings
. Intelligent Robots and Computer Vision VI.
0848
. SPIE: 358.
Bibcode
:
1988SPIE..848..358B
.
doi
:
10.1117/12.942759
.
- ^
a
b
c
d
Ni, B. Y.; Elishakoff, I.; Jiang, C.; Fu, C. M.; Han, X. (1 November 2016).
"Generalization of the super ellipsoid concept and its application in mechanics"
.
Applied Mathematical Modelling
.
40
(21): 9427?9444.
doi
:
10.1016/j.apm.2016.06.011
.
ISSN
0307-904X
.
- ^
Cheng, Xinyu; Li, Chengbo; Peng, Yixue; Zhao, Chuang (17 April 2021).
"Discrete element simulation of super-ellipse systems"
.
Granular Matter
.
23
(2): 50.
doi
:
10.1007/s10035-021-01107-4
.
ISSN
1434-7636
.
- ^
"SuperQuadrics - Applications"
.
www.cs.mcgill.ca
. Retrieved
18 June
2024
.
- ^
Land, Richard; Foley, James D.; Dam, Andries Van (1984).
"Fundamentals of Interactive Computer Graphics"
.
Leonardo
.
17
(1): 59.
doi
:
10.2307/1574879
.
ISSN
0024-094X
.
JSTOR
1574879
.
- ^
a
b
Gardner, Martin
(1977),
"Piet Hein's Superellipse"
,
Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American
, New York:
Vintage Press
, pp.
240?254
,
ISBN
978-0-394-72349-5
- ^
The Superellipse
, in
The Guide to Life, The Universe and Everything
by
BBC
(27 June 2003)
- ^
Tobler, Waldo (1973), "The hyperelliptical and other new pseudocylindrical equal area map projections",
Journal of Geophysical Research
,
78
(11): 1753?1759,
Bibcode
:
1973JGR....78.1753T
,
CiteSeerX
10.1.1.495.6424
,
doi
:
10.1029/JB078i011p01753
.
- ^
Mynttinen, Ivo.
"The iOS Design Guidelines"
.
External links
[
edit
]
- Sokolov, D.D. (2001) [1994],
"Lame curve"
,
Encyclopedia of Mathematics
,
EMS Press
- "Lame Curve"
at MathCurve.
- Weisstein, Eric W.
"Superellipse"
.
MathWorld
.
- O'Connor, John J.;
Robertson, Edmund F.
,
"Lame Curves"
,
MacTutor History of Mathematics Archive
,
University of St Andrews
- "Super Ellipse"
on 2dcurves.com
- Superellipse Calculator & Template Generator
- Superellipse fitting toolbox in MATLAB
- C code for fitting superellipses