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Circle with radius of one
mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius
Unit_circle
Concept in geometry
from the area of a unit circle. Consider the unit circle circumscribed by a square of side length 2. The transformation sends the circle to an ellipse by
Area_of_a_circle
Simple curve of Euclidean geometry
respectively. The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes
Circle
Two-dimensional packing problem
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If
Circle_packing_in_a_circle
Technique in analytic number theory
so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle. The circle method is specifically how to compute
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Extension of the domain of an analytic function (mathematics)
since the set formed by all such roots is dense on the boundary of the unit circle, there is no analytic continuation of L c ( z ) {\displaystyle {\mathcal
Analytic_continuation
Relation between sine and cosine
definition of defining x = cos θ and y sin θ for the unit circle and thus x = c cos θ and y = c sin θ for a circle of radius c and reflecting our triangle in the
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
Integer side lengths of a right triangle
points on the unit circle (Trautman 1998). In fact, a point in the Cartesian plane with coordinates (x, y) belongs to the unit circle if x2 + y2 = 1
Pythagorean_triple
Concept within complex analysis
be viewed as closed vector subspaces of the complex Lp spaces on the unit circle T = { z ∈ C : | z | = 1 } {\displaystyle \mathbb {T} =\{z\in \mathbb
Hardy_space
Two-dimensional packing problem
Circle packing in a square is a packing problem in recreational mathematics where the aim is to pack n unit circles into the smallest possible square
Circle_packing_in_a_square
Fundamental trigonometric functions
any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series
Sine_and_cosine
orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane
Orthogonal polynomials on the unit circle
Orthogonal_polynomials_on_the_unit_circle
SI derived unit of angle
the center of a plane circle by an arc that is equal in length to the radius. The unit is defined in the SI as the coherent unit for plane angle, as well
Radian
Functions of an angle
real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions
Trigonometric_functions
Area of geometry, about angles and lengths
Acid". Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. In this setting
Trigonometry
Polynomial whose coefficients are all 1 or −1
polynomials are named after J. E. Littlewood, who studied their values on the unit circle and posed several influential extremal problems about them in the 1960s
Littlewood_polynomial
Sphere with radius one, usually centered on the origin of the space
space; the unit circle is a special case, the unit 1 {\displaystyle 1} -sphere in the plane. An (open) unit ball is the region inside of a unit sphere
Unit_sphere
Set of points at distance less than one from a given point
} Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk
Unit_disk
Topics referred to by the same term
performed Unit angle, a full turn equal to an angle of 1 Unit circle, a circle with a radius of length 1 Unit cube, a cube with sides of length 1 Unit fraction
Unit
Complex numbers with unit norm and both real and imaginary parts rational numbers
In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1
Group of rational points on the unit circle
Group_of_rational_points_on_the_unit_circle
Linear transform from the time domain to the frequency domain
evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right
Z-transform
Lie group of complex numbers of unit modulus; topologically a circle
complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers T = { z ∈ C : | z | = 1 } . {\displaystyle
Circle_group
Geometric figure
\textstyle r={\sqrt {x^{2}-y^{2}}}} . Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1
Unit_hyperbola
Nonlinear equation which arises on linear optimal control problems
stable if and only if all of its eigenvalues are strictly inside the unit circle of the complex plane. A solution to the algebraic Riccati equation can
Algebraic_Riccati_equation
Hashing technique
idea is to use a hash function that maps both the BLOB and servers to a unit circle, usually 2 π {\displaystyle 2\pi } radians. For example, ζ = Φ %
Consistent_hashing
Particular mapping that projects a sphere onto a plane
through the origin intersects the unit sphere in a great circle, called the trace of the plane. This circle maps to a circle under stereographic projection
Stereographic_projection
Representation of a curve by a function of a parameter
form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of t
Parametric_equation
Length in a vector space
function. The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented
Norm_(mathematics)
American mathematician (1933–2017)
Function Theory on the Unit Circle were made available by the math department at VPI. 1994. Sub-Hardy Hilbert Spaces in the Unit Disk. This book developed
Donald_Sarason
Perimeter of a circle or ellipse
circumferēns 'carrying around, circling') is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up
Circumference
Type of operator in Fourier analysis
norm and Lp space. In the setting of periodic functions defined on the unit circle, the Fourier transform of a function is simply the sequence of its Fourier
Multiplier_(Fourier_analysis)
On converting relations to functions of several real variables
curve, one has y = f ( x ) {\displaystyle y=f(x)} . An example is the unit circle, whose points ( x , y ) {\displaystyle (x,y)} satisfy x 2 + y 2 − 1 =
Implicit_function_theorem
Branch of mathematics
defined relative to a unit circle, squigonometry focuses on analogous relationships and functions within the context of a unit squircle. The term squigonometry
Squigonometry
Notion in computational learning
four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set
Shattered_set
Number, approximately 3.14
example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x 2 + y 2 = 1 {\textstyle
Pi
Second order recursive digital linear filter
inside the unit circle for it to be stable. In general, this is true for all discrete filters i.e. all poles must be inside the unit circle in the Z-domain
Digital_biquad_filter
Roots of the Chebyshev polynomials of the first kind
set of equispaced points on the unit circle onto the real interval [ − 1 , 1 ] {\displaystyle [-1,1]} , the circle's diameter. There are two kinds of
Chebyshev_nodes
Mathematical relation consisting of a multi-variable function equal to zero
variables (often a polynomial). For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0. {\displaystyle x^{2}+y^{2}-1=0.} An implicit function
Implicit_function
Model of hyperbolic geometry
the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle
Poincaré_disk_model
Number with a real and an imaginary part
the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation
Complex_number
In control theory, when an LTI system and its inverse are causal and stable
the s-plane representation (in discrete time, respectively, inside the unit circle of the z plane). Since inverting a system function leads to poles turning
Minimum_phase
Geometric representation of the complex numbers
convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z =
Complex_plane
Extends the Jordan curve theorem to characterize the inner and outer regions
^{2}\to \mathbb {R} ^{2}} such that f ( C ) {\displaystyle f(C)} is the unit circle in the plane. Elementary proofs can be found in Newman (1939), Cairns
Schoenflies_problem
Change of variable for integrals involving trigonometric functions
{x}{2}}} . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation
Tangent half-angle substitution
Tangent_half-angle_substitution
Constant equal to twice pi
irrational. When radians are used as the unit of angular measure there are τ radians in one full turn of a circle, and the radian angle is aligned with the
Tau_(mathematics)
Graphical calculator used in electrical engineering
{\displaystyle \operatorname {Re} (z)\geq 0} ). Regions outside the unit circle ( Re ( z ) < 0 {\displaystyle \operatorname {Re} (z)<0} ) correspond
Smith_chart
Relates the tangent of half of an angle to trigonometric functions of the entire angle
{\tfrac {1}{2}}(a+b)}}={\frac {\sin a+\sin b}{\cos a+\cos b}}.} In the unit circle, application of the above shows that t = tan 1 2 φ {\textstyle t=\tan
Tangent_half-angle_formula
Algorithm
uniformly and randomly to multiple points on a unit circle called tokens. Objects are also mapped to the unit circle and placed in the site owning the token
Rendezvous_hashing
Distance along a curve
length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y
Arc_length
Theory of interwoven space and time by Albert Einstein
functions. Fig. 7-1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry
Special_relativity
Sum in algebraic number theory
additive group R+ into the unit circle, and χ is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r, where it takes the
Gauss_sum
Polynomial sequence
, which is equivalent to Var ( Z ) unit circle = 1 {\displaystyle \operatorname {Var} (Z)_{\text{unit circle}}=1} . The functions are a basis defined
Zernike_polynomials
reproducing kernel Hilbert space. In general, elements of L2 on the unit circle are given by ∑ n = − ∞ ∞ a n e i n φ {\displaystyle \sum _{n=-\infty
H_square
Root-finding algorithm
distribution of the roots of a complex polynomial with respect to the unit circle in the complex plane. It is based on two auxiliary polynomials, introduced
Lehmer–Schur_algorithm
Method of evaluating certain integrals along paths in the complex plane
taken to be the unit circle traversed counterclockwise (or any positively oriented Jordan curve about 0). In the case of the unit circle there is a direct
Contour_integration
Triangle in hyperbolic geometry
{\displaystyle M(z)={\overline {z}}} maps the unit circle into itself and maps the interior of the unit circle into itself and maps hyperbolic lines into
Hyperbolic_triangle
Plane curve
ellipse uses affine transformations: Any ellipse is an affine image of the unit circle with equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} . Parametric
Ellipse
Mathematical functions
to the way circular angle measure is the arc length of an arc of the unit circle in the Euclidean plane or twice the area of the corresponding circular
Inverse_hyperbolic_functions
Complex matrix whose conjugate transpose equals its inverse
diagonal and unitary. The eigenvalues of U {\displaystyle U} lie on the unit circle. That is, if the complex number λ is an eigenvalue of U then |λ| = 1
Unitary_matrix
Study of angle-preserving transformations
inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves
Inversive_geometry
Mapping which preserves all topological properties of a given space
line is not homeomorphic to the unit circle as a subspace of R 2 {\displaystyle \mathbb {R} ^{2}} , since the unit circle is compact as a subspace of Euclidean
Homeomorphism
On reflection in a spherical mirror
the unit circle, or fail to give a valid reflection path, but the valid solutions can all be found among the roots. The root on the unit circle minimizing
Alhazen's_problem
Right triangle with a feature making calculations on the triangle easier
angles. The side lengths of these triangles can be deduced based on the unit circle, or with the use of other geometric methods; and these approaches may
Special_right_triangle
Signal processing operation
[ s ] = 0 {\displaystyle \mathrm {Re} [s]=0} , in the s-plane to the unit circle, | z | = 1 {\displaystyle |z|=1} , in the z-plane. Other bilinear transforms
Bilinear_transform
Complex exponential in terms of sine and cosine
interpreted as saying that the function eiφ is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real
Euler's_formula
Angular measurement, thousandth of a radian
measurement (e.g. artillery replaced "units of base" with metres) the Red Army expanded the 600 unit circle into a 6000 mil circle. Hence the Russian mil has a
Milliradian
Principal square root of minus 1
cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication. Written as a special case
Imaginary_unit
Feature of some stochastic processes
trend. If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first
Unit_root
Unit of plane angle where a full circle equals 1
(symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One
Turn_(angle)
1 minus the cosine of an angle
in the original context for their definition, a unit circle: For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of
Versine
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
Smyth, C. J. (1971). "On the product of the conjugates outside the unit circle of an algebraic integer". Bulletin of the London Mathematical Society
Lehmer's_conjecture
Theorem in statistical mechanics
of an external field, then all zeros are purely imaginary (or on the unit circle after a change of variable). The first version was proved for the Ising
Lee–Yang_theorem
On distances between points on a circle
chromatic circle, the points of which represent classes of equivalent tones. Mathematically, this circle can be described as the unit circle in the complex
Three-gap_theorem
Topological space that locally resembles Euclidean space
small piece of a line. Considering, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow
Manifold
Mathematical function, denoted exp(x) or e^x
is wrapped around the unit circle at a constant angular rate values with negative real parts are mapped inside the unit circle values with positive real
Exponential_function
Matrix that commutes with its conjugate transpose
are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it is not the case that
Normal_matrix
Representation of a type of random process
(z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}} must lie outside the unit circle, i.e., each (complex) root z i {\displaystyle z_{i}} must satisfy | z
Autoregressive_model
On triangles inscribed in a circle with a diameter as an edge
Let O = (0, 0), A = (−1, 0), and C = (1, 0). Then B is a point on the unit circle (cos θ, sin θ). We will show that △ABC forms a right angle by proving
Thales's_theorem
Figure formed by two rays meeting at a common point
"measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen
Angle
Spiral that surrounds equal area per turn
unit circle has in polar coordinates the simple description (r, φ) ↦ (1/r, φ). The image of Fermat's spiral r = a√φ under the inversion at the unit
Fermat's_spiral
Signal processing filter
an unstable system that is outside of the unit circle can be canceled and reflected inside the unit circle. Bridged T delay equaliser Lattice phase equaliser
All-pass_filter
Process in geometric function theory
removed, along their boundary circles. This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk and its complement into
Conformal_welding
Analytic function in mathematics
the open unit disk. Nevertheless, f has dense singularities on the unit circle, and cannot be analytically continued outside of the open unit disk, as
Lacunary_function
Exploring properties of the integers with complex analysis
applicability. For example, the circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it
Analytic_number_theory
Object movement along a circular path
}{T}}=2\pi f={\frac {d\theta }{dt}}} and the units are radians/second. The speed of the object traveling the circle is: v = 2 π r T = ω r {\displaystyle v={\frac
Circular_motion
Mathematical transform that expresses a function of time as a function of frequency
and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely
Fourier_transform
Curve used in computer graphics and related fields
from an outer control point on a unit circle. More generally, an n-piece cubic Bézier curve can approximate a circle, when each inner control point is
Bézier_curve
Diagram showing the singularities of a given control system's transfer function
{\displaystyle z=Ae^{j\phi }} Real frequency components are along its unit circle In general, a rational transfer function for a continuous-time LTI system
Pole–zero_plot
Force directed to the center of rotation
form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ(t) as r(t). When the
Centripetal_force
Theorem in complex analysis
sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the
Carathéodory's theorem (conformal mapping)
Carathéodory's_theorem_(conformal_mapping)
Indication of rate and sense of rotation
\sin(t))} has a positive frequency of +1 radian per unit of time and rotates counterclockwise around a unit circle, while the vector ( cos ( − t ) , sin (
Negative_frequency
System of moving vectors in differential geometry
unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.
Parallel_transport
Lowest possible mass of the celestial object
(which also determine orbital inclinations). In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate
Minimum_mass
Method for estimating new data outside known data points
part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular, the compactification point
Extrapolation
Mathematical concept
used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle. The space of functions that are the limits on T of functions
Poisson_kernel
Theorem in linear algebra
ρ(A) by the remark above. It might not be the only eigenvalue on the unit circle: and the associated eigenspace can be multi-dimensional. If A is row-stochastic
Perron–Frobenius_theorem
Mathematical function
defined on the unit circle with radius r = 1 {\displaystyle r=1} and angle φ = {\displaystyle \varphi =} arc length of the unit circle measured from the
Jacobi_elliptic_functions
Theorem in complex analysis
\gamma (t)=e^{it}\quad t\in \left[0,2\pi \right],} which traces out the unit circle. Here the following integral: ∫ γ 1 z d z = 2 π i ≠ 0 , {\displaystyle
Cauchy's_integral_theorem
Point at infinity in hyperbolic geometry
Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk
Ideal_point
In geometry, set whose intersection with every line is a single line segment
^{2}:x^{2}+y^{2}\leq 1\}} are its extreme points, namely the points on the unit circle S 1 = { ( x , y ) ∈ R 2 : x 2 + y 2 = 1 } {\displaystyle S^{1}=\{(x,y)\in
Convex_set
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