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Representation of a curve by a function of a parameter
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters
Parametric_equation
Mathematical formula expressing equality
integers A transcendental equation is an equation involving a transcendental function of its unknowns A parametric equation is an equation in which the solutions
Equation
Straight figure with zero width and depth
single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes. Parametric equations are also used
Line_(geometry)
Surface specified with parameters
A parametric surface is a surface in the Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} which is defined by a parametric equation with two parameters
Parametric_surface
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Plane curve: conic section
with equation x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1} can be described by several parametric equations: Through
Hyperbola
Circle that passes through the vertices of a triangle
containing the circle, n ^ , {\displaystyle {\widehat {n}},} one parametric equation of the circle starting from the point P0 and proceeding in a positively
Circumcircle
Topics referred to by the same term
Look up parametric in Wiktionary, the free dictionary. Parametric may refer to: Parametric equation, a representation of a curve through equations, as functions
Parametric
Variable used for specification
parametric equation this can be written ( x , y ) = ( cos t , sin t ) . {\displaystyle (x,y)=(\cos t,\sin t).} The parameter t in this equation would
Parameter
Horizontal and vertical axes/coordinate numbers of a 2D coordinate system or graph
describes the point's location along some path, e.g. the parameter of a parametric equation. Used in this way, the abscissa can be thought of as a coordinate-geometry
Abscissa_and_ordinate
Family of geometric shapes
g_{2}(\nu )\\f_{2}(\mu )\end{pmatrix}}} This is similar to the typical parametric equation of a sphere: x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin
Superquadrics
Harmonic oscillator whose parameters oscillate in time
sinusoid, the equation is called a Mathieu equation. Harmonic oscillator Mathieu equation Optical parametric amplifier Optical parametric oscillator Case
Parametric_oscillator
Plane curve
coordinate equation: x 1 a 2 x + y 1 b 2 y = 1. {\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.} A vector parametric equation of the tangent
Ellipse
Engineering design method
method incorporated the main features of a computational parametric model (input parameters, equation, output): The string length, birdshot weight, and anchor
Parametric_design
Describing something mathematical with variables
process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process
Parametrization_(geometry)
Category of regression analysis
constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent
Nonparametric_regression
(that is, when the dependent variables are x and y and are given by parametric equations in t). Let x(t) and y(t) be the coordinates of the points of the
Parametric_derivative
In general, exponentiation fails to be commutative
nontrivial solutions in positive real numbers are expressed as the parametric equation x = v 1 / ( v − 1 ) , y = v v / ( v − 1 ) . {\displaystyle
Equation_xy_=_yx
Triangle whose side lengths and area are integers
have a common factor, that factor must be the sum of two squares. A parametric equation or parametrization of Heronian triangles consists of an expression
Heronian_triangle
Geometric civil engineering calculation technique
from these parametric equations will yield the non-parametric equation of the Mohr circle. This can be achieved by rearranging the equations for σ n {\displaystyle
Mohr's_circle
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
Mathematical concept
curve can also often be represented in Cartesian coordinates by a parametric equation of the form ( x , y ) = ( x ( t ) , y ( t ) ) {\displaystyle (x,y)=(x(t)
Plane_curve
Symbol representing the heart
approximations of the heart shape. Heart curve on TI-89 graphing calculator Parametric equation of heart curve on TI-89 graphing calculator Cordata, Cordatum and
Heart_symbol
Cubic plane curve
the Tschirnhausen cubic has this singularity, it can be given a parametric equation, expressing both of its Cartesian coordinates as polynomial functions
Tschirnhausen_cubic
In mathematics, vector subspace
homogeneous system of linear equations, the subset of Euclidean space described by a system of homogeneous linear parametric equations, the span of a collection
Linear_subspace
Curve traced by a string as it is unwrapped from another curve
It is special as it contains no cusp. By serial expansion, it has parametric equation { x ( s ) = 18 5 s 5 − 126 5 s 9 + O ( s 13 ) y ( s ) = − 2 s 3 +
Involute
Mathematical idealization of the surface of a body
{\begin{aligned}x&=\cos(u)\cos(v)\\y&=\sin(u)\cos(v)\\z&=\sin(v)\,.\end{aligned}}} Parametric equations of surfaces are often irregular at some points. For example, all
Surface_(mathematics)
Point on a curve where motion must move backwards
singular point of a curve. For a plane curve defined by an analytic, parametric equation x = f ( t ) y = g ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t)
Cusp_(singularity)
Type of spline curve
defined by a polynomial parametric equation for which the speed (the derivative of arc length) also has a polynomial parametric equation. This allows the arc
Pythagorean_hodograph_curve
Simplest non-trivial closed knot with three crossings
trefoil knot can be defined as the curve obtained from the following parametric equations: x = sin t + 2 sin 2 t y = cos t − 2 cos 2 t z = − sin
Trefoil_knot
Type of plane curve
(\varphi )={\tfrac {8}{3}}a\sin {\tfrac {\varphi }{2}}\,.} Hence the parametric equations of the evolute are X ( φ ) = 4 a sin 2 φ 2 cos φ − 8 3 a sin
Cardioid
Fastest curve descent without friction
which is the differential equation of an inverted cycloid generated by a circle of diameter D=2r, whose parametric equation is: x = r ( φ − sin φ )
Brachistochrone_curve
Curve generated by the projections of a fixed point on the tangents of another curve
{\pi }{2}}.} These equations may be used to produce an equation in p and α which, when translated to r and θ gives a polar equation for the pedal curve
Pedal_curve
Quadric surface that looks like a deformed sphere
(conjugate vectors), such that the ellipse can be represented by the parametric equation x = f 0 + f 1 cos t + f 2 sin t {\displaystyle \mathbf {x} =\mathbf
Ellipsoid
Branch of statistics to estimate models based on measured data
Matched filter Maximum entropy spectral estimation Nuisance parameter Parametric equation Pareto principle Rule of three (statistics) State estimator Statistical
Estimation_theory
Curve formed by a hanging chain
{a^{2}+p^{2}}}}+{\frac {T_{0}p}{Ea}}\,.\end{alignedat}}} Integrating gives the parametric equations x = a arsinh ( p a ) + T 0 E p + α , y = a 2 + p 2 + T 0 2 E a
Catenary
Transcendental plane curve
{\displaystyle 0\leq t\leq 12\pi } , the curve is given by the following parametric equations: x = sin t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) y = cos
Butterfly curve (transcendental)
Butterfly_curve_(transcendental)
Geometric figure
u=\arccos(x/r)} into the parametric equation for z(u) given in § Parametric characterization yields the following equation for z in terms of x: z ( x
Mylar_balloon_(geometry)
Two Advanced Placement courses and exams
AP Calculus AB topics plus integration by parts, infinite series, parametric equations, vector calculus, and polar coordinate functions, among other topics
AP_Calculus
Curve where spinning and moving lines cross
that the lines trace out. Then the quadratrix can be described by parametric equations that give the coordinates of each point on the curve as a function
Quadratrix_of_Hippias
Distance along a curve
continuously differentiable, then it is simply a special case of a parametric equation where x = t {\displaystyle x=t} and y = f ( t ) . {\displaystyle
Arc_length
CAD software
or where the resulting geometry may be complex or based upon equations. Creo Parametric provides a complete set of design, analysis and manufacturing
Creo_Parametric
geometry software List of mathematical artists Mathematical software Parametric surface Procedural modeling suites Ray tracing "Fractal Art: An Introduction
List of mathematical art software
List_of_mathematical_art_software
Cubic plane curve
{2}{3}}.} The parametric equation x = t 2 , y = a t 3 {\displaystyle x=t^{2},\quad y=at^{3}} can also be deduced from the implicit equation by putting t
Semicubical_parabola
Plane curve: conic section
side of the equation uses the Hesse normal form of a line to calculate the distance | P l | {\displaystyle |Pl|} ). For a parametric equation of a parabola
Parabola
Mathematical curve outputted from a specific pair of parametric equations
figure or Bowditch curve /ˈbaʊdɪtʃ/, is the graph of a system of parametric equations x = A sin ( a t + δ ) , y = B sin ( b t ) , {\displaystyle x=A\sin(at+\delta
Lissajous_curve
Serpent-like curve
a b x x 2 + a 2 {\displaystyle y={\frac {abx}{x^{2}+a^{2}}}} Its parametric equation for 0 < t < π {\displaystyle 0<t<\pi } is x = a cot ( t ) {\displaystyle
Serpentine_curve
Rules related to the mathematical principles of origami
there is a unique fold that passes through both of them. In parametric form, the equation for the line that passes through the two points is : F ( s )
Huzita–Hatori_axioms
Affine subspace of a Euclidean space
flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter: x = 2 + 3 t , y = − 1
Flat_(geometry)
Nonlinear transduction mechanism
A parametric array, in the field of acoustics, is a nonlinear transduction mechanism that generates narrow, nearly side lobe-free beams of low frequency
Parametric_array
Study of geometry using a coordinate system
lines can not be described by a single linear equation, so they are frequently described by parametric equations: x = x 0 + a t {\displaystyle x=x_{0}+at}
Analytic_geometry
Integer side lengths of a right triangle
primitive Pythagorean triples. The unit circle may also be defined by a parametric equation x = 1 − t 2 1 + t 2 , y = 2 t 1 + t 2 . {\displaystyle x={\frac {1-t^{2}}{1+t^{2}}}
Pythagorean_triple
Measure of local oscillation behavior
is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite
Total_variation
Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius
x^{2/3}+y^{2/3}=a^{2/3}.} This means that an astroid is also a superellipse. Parametric equations are x = a cos 3 t = a 4 ( 3 cos ( t ) + cos ( 3 t ) ) , y =
Astroid
Set of points equidistant from a center
the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity. A parametric equation for
Sphere
Cubic plane curve
{2at^{2}}{1+t^{2}}}\\&y=tx={\frac {2at^{3}}{1+t^{2}}}\end{aligned}}} are parametric equations for the cissoid. Converting the polar form to Cartesian coordinates
Cissoid_of_Diocles
Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space
of the sphere in terms of longitude (θ) and latitude (φ), gives parametric equations for the Roman surface as follows: x = r 2 cos θ cos φ sin φ
Roman_surface
Type of ordinary differential equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle
Clairaut's_equation
Geometric intersection of a line and plane in 3D space
the point on the line equal to the point on the plane, giving the parametric equation: l a + l a b t = p 0 + p 01 u + p 02 v . {\displaystyle \mathbf {l}
Line–plane_intersection
{1}{2}}} . The parametric equations for a fish curve correspond to those of the associated ellipse. For an ellipse with the parametric equations x = a cos
Fish_curve
Association of one output to each input
Functional decomposition Functional predicate Functional programming Parametric equation Set function Simple function This definition of "graph" refers to
Function_(mathematics)
Topics referred to by the same term
specifically to: Parametrization (geometry), the process of finding parametric equations of a curve, surface, etc. Parametrization by arc length, a natural
Parametrization
Mathematical function
codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real variable
Function_of_a_real_variable
Spiral with constant distance from itself
dt&=y\end{aligned}}} The above equations can be integrated by applying integration by parts, leading to the following parametric equations: x = ( v t + c ) cos
Archimedean_spiral
Curve traced by the crossing of two lines revolving about poles
{\displaystyle p} is a parameter. Then converting the polar equation above to parametric equations produces x = a sin [ m p + θ 0 ] cos n p sin [ ( m
Sectrix_of_Maclaurin
Special case of the Euler-Lagrange equations
,} which can be solved with the result put in the form of parametric equations x = A ( ϕ − sin ϕ ) {\displaystyle x=A(\phi -\sin \phi )} y = A
Beltrami_identity
Partial differential equation
mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability
Fokker–Planck_equation
Rendering method
tracing and finally unbiased path tracing, which provides the rendering equation framework that has allowed computer-generated imagery to be faithful to
Ray_tracing_(graphics)
Theorem in calculus relating line and double integrals
rewritten as the union of four curves: C1, C2, C3, C4. With C1, use the parametric equations: x = x, y = g1(x), a ≤ x ≤ b. Then ∫ C 1 L ( x , y ) d x = ∫ a b
Green's_theorem
Equations modelling predator–prey cycles
Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used
Lotka–Volterra_equations
Simple curve of Euclidean geometry
x_{0}-r} to x 0 + r {\displaystyle x_{0}+r} . The equation can be written in parametric form using the trigonometric functions sine and cosine as
Circle
Critical point where a periodic solution arises
\right)+{\mathcal {O}}(\mu ^{2})\end{aligned}}} This provides a parametric equation for the limit cycle. This is plotted in the illustration on the right
Hopf_bifurcation
Plane algebraic curve
y^{2}=\left({\sqrt {8x^{2}+a^{2}}}-a\right){\frac {a}{2}}-x^{2}} As a parametric equation: x = a cos t 1 + sin 2 t , y = a sin t cos t 1 + sin 2
Lemniscate_of_Bernoulli
Statement based on repeated empirical observations that describes some natural phenomenon
configuration space, i.e. the curve q(t), parameterized by time (see also Parametric equation). The action is a functional rather than a function, since it depends
Scientific_law
Mathematical equation
d x = tan φ . {\displaystyle {\frac {dy}{dx}}=\tan \varphi .} Parametric equations for the curve can be obtained by integrating: x = ∫ cos φ d s
Whewell_equation
Term in mathematics
integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Integral curves are
Integral_curve
Geometric shape
and from tissues. Kuchel, Philip W.; Fackerell, Edward D. (1999). "Parametric-equation representation of biconcave erythrocytes". Bulletin of Mathematical
Biconcave_disc
Space curve that winds around a line
slope a/b (or pitch 2πb) expressed in Cartesian coordinates as the parametric equation t ↦ ( a cos t , a sin t , b t ) , t ∈ [ 0 , T ] {\displaystyle
Helix
Coordinates comprising a distance and an angle
at any given point, the curve is first expressed as a system of parametric equations. x = r ( φ ) cos φ y = r ( φ ) sin φ {\displaystyle
Polar_coordinate_system
Curve for which the time to roll to the end is equal for all starting points
2 {\textstyle r={\frac {g}{4\omega ^{2}}}\,} , we see that these parametric equations for x {\displaystyle x} and y {\displaystyle y} are those of a point
Tautochrone_curve
Formulation of classical mechanics using momenta
Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. Hamilton's equations usually
Hamiltonian_mechanics
Tool for measuring area
respect to change in angle varies quadratically with the radius. For a parametric equation in polar coordinates, where both r and θ vary as a function of time
Planimeter
Type of mathematical curve
{\displaystyle c(x,tx)-q(x,y)=x^{2}(xc(1,t)-q(1,t)),} giving the parametric equation x = q ( 1 , t ) c ( 1 , t ) , y = t q ( 1 , t ) c ( 1 , t ) . {\displaystyle
Cubic_plane_curve
Determining where a point is in relation to a coplanar polygon
can be solved easily by use of a barycentric coordinate system, parametric equation or dot product. The dot product method extends naturally to any convex
Point_in_polygon
Geometric surface
pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: x = a cos u sin v y = a sin u sin v z = a ( cos v + ln
Dini's_surface
Physical system that responds to a restoring force proportional to displacement
damping β {\displaystyle \beta } . Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the
Harmonic_oscillator
Spiral that surrounds equal area per turn
by y cancels the a√φ parts of the parametric equations, leaving the simpler equation x/y = cot φ. From this equation, substituting φ by φ = r2/a2 (a
Fermat's_spiral
Line-clipping algorithm in computer graphics
which can be used only on a rectangular clipping area. Here the parametric equation of a line in the view plane is p ( t ) = t p 1 + ( 1 − t ) p 0 {\displaystyle
Cyrus–Beck_algorithm
Statistical price volatility chart
chart to the magnitude and frequency of price changes, similar to parametric equations in signal processing or control systems. Bollinger Bands consist
Bollinger_Bands
Course designed to prepare students for calculus
trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences and series are other common topics of
Precalculus
solve the above equation in an implicit form. Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (2011). "Construction of Exact Parametric or Closed Form
Abel equation of the first kind
Abel_equation_of_the_first_kind
Line-clipping algorithm
first published in early 1984. The Liang–Barsky algorithm uses the parametric equation of a line and inequalities describing the range of the clipping window
Liang–Barsky_algorithm
Cubic plane curve
arctangent function. The witch of Agnesi can also be described by parametric equations whose parameter θ is the angle between OM and OA, measured clockwise:
Witch_of_Agnesi
Locus of the zeros of a polynomial of degree two
simplifying the expression of the last coordinate, one obtains the parametric equation { x 1 = 2 t 1 1 + t 1 2 + ⋯ + t n − 1 2 ⋮ x n − 1 = 2 t n − 1 1 +
Quadric
Plane curve formed by rolling a circle on the outside of another
point is at a distance d from the center of the exterior circle. The parametric equations for an epitrochoid are: x ( θ ) = ( R + r ) cos θ − d cos ( R
Epitrochoid
Graphical representation of [[Black box]] structure
block (always with composite aggregation). Constraint blocks define parametric equations that constrain values. Ports and Interfaces: Blocks have ports, depicted
Block_definition_diagram
Family of closed mathematical curves
algebraic curve in different orientations. The curve is given by the parametric equations (with parameter t having no elementary geometric interpretation)
Superellipse
Mathematical knot with crossing number 7
The 74 knot is a Lissajous knot, representable for example by the parametric equation x = cos ( 2 t + 0.22 ) y = cos ( 3 t + 1.10 ) z = cos 7 t {\displaystyle
74_knot
Spiral curve of the form r = a*sin(θ)/θ
{\displaystyle (x^{2}+y^{2})\arctan {\frac {y}{x}}=ay,} or the parametric equations x = a sin t cos t t , y = a sin 2 t t . {\displaystyle x={\frac
Cochleoid
Branch of mathematics
example, the circle of equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} is a rational curve, as it has the parametric equation x = 2 t 1 + t 2 {\displaystyle
Algebraic_geometry
PARAMETRIC EQUATION
PARAMETRIC EQUATION
PARAMETRIC EQUATION
PARAMETRIC EQUATION
Boy/Male
American, Celebrity, Danish, Finnish, French, German, Greek, Hindu, Indian, Italian, Swedish
Italy; Son of Mist; Lion; Man from Naples; Lion of the Woodland; Lion of the New City
Boy/Male
Hindu
Lord Shiva, Shapely, Diverse, Changed
Male
Egyptian
, child of the moon + support + the sun.
Girl/Female
Hindu
Goddess Durga
Male
Arthurian
, king of Cornwall.
Girl/Female
Tamil
Music
Boy/Male
Muslim
Beautiful
Surname or Lastname
German
German : from a variant of the Germanic personal name Gambert, or some other personal name formed with Old High German gam(an) ‘joy’, ‘play’.English : variant spelling of Gamble.
Boy/Male
French
Dark one; the Moor.
Boy/Male
Hindu, Indian, Modern
Lovely
PARAMETRIC EQUATION
PARAMETRIC EQUATION
PARAMETRIC EQUATION
PARAMETRIC EQUATION
PARAMETRIC EQUATION
n.
The change, as of an equation or quantity, into another form without altering the value.
a.
Alt. of Paracentrical
a.
Alt. of Barometrical
n.
The act of solving, or the state of being solved; the disentanglement of any intricate problem or difficult question; explanation; clearing up; -- used especially in mathematics, either of the process of solving an equation or problem, or the result of the process.
a.
Alt. of Pyrometrical
n.
The bringing of any term of an equation from one side over to the other without destroying the equation.
v. t.
To bring, as any term of an equation, from one side over to the other, without destroying the equation; thus, if a + b = c, and we make a = c - b, then b is said to be transposed.
a.
Alt. of Perimetrical
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
a.
Recurring once a month; monthly; gone through in a month; as, the menstrual revolution of the moon; pertaining to monthly changes; as, the menstrual equation of the sun's place.
adv.
By means of a barometer, or according to barometric observations.
a.
Pertaining to the barometer; made or indicated by a barometer; as, barometric changes; barometrical observations.
a.
Indicating equal barometric pressure.
n.
An instrument for measuring heights by observation of barometric pressure; esp., one for determining heights by ascertaining the boiling point of water. It consists of a vessel for water, with a lamp for heating it, and an inclosed thermometer for showing the temperature of ebullition.
n.
The curve whose ordinates are proportional to the sines of the abscissas, the equation of the curve being y = a sin x. It is also called the curve of sines.
a.
Of or pertaining to the perimeter, or to perimetry; as, a perimetric chart of the eye.
v. t.
Making a large angle with the plane of the horizon; ascending or descending rapidly with respect to a horizontal line or a level; precipitous; as, a steep hill or mountain; a steep roof; a steep ascent; a steep declivity; a steep barometric gradient.
n.
An identical equation.
n.
A quantity which may increase or decrease; a quantity which admits of an infinite number of values in the same expression; a variable quantity; as, in the equation x2 - y2 = R2, x and y are variables.
n.
A movement of the atmosphere opposite in character, as regards direction of the wind and distribution of barometric pressure, to that of a cyclone.