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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a parametric equation defines a group of quantities as functions of one or more
independent variables Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. Parametric equations are commonly used to express the
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
of the points that make up a geometric object such as a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
or
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. For example, the equations :\begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the
unit circle In 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 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, where ''t'' is the parameter: A point (''x'', ''y'') is on the unit circle
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
there is a value of ''t'' such that these two equations generate that point. Sometimes the parametric equations for the individual
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
output variables are combined into a single parametric equation in vectors: :(x, y)=(\cos t, \sin t). Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations. In addition to curves and surfaces, parametric equations can describe
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of higher
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is ''one'' and ''one'' parameter is used, for surfaces dimension ''two'' and ''two'' parameters, etc.). Parametric equations are commonly used in
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
, where the
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled ''t''; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.


Applications


Kinematics

In
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
, objects' paths through space are commonly described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter (usually time). Used in this way, the set of parametric equations for the object's coordinates collectively constitute a
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
for position. Such parametric curves can then be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as :\mathbf(t) = (x(t), y(t), z(t)) then its
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
can be found as :\mathbf(t) = \mathbf'(t) = (x'(t), y'(t), z'(t)) and its
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
as :\mathbf(t) = \mathbf''(t) = (x''(t), y''(t), z''(t)).


Computer-aided design

Another important use of parametric equations is in the field of
computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
(CAD). For example, consider the following three representations, all of which are commonly used to describe planar curves. Each representation has advantages and drawbacks for CAD applications. The explicit representation may be very complicated, or even may not exist. Moreover, it does not behave well under geometric transformations, and in particular under
rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. On the other hand, as a parametric equation and an implicit equation may easily be deduced from an explicit representation, when a simple explicit representation exists, it has the advantages of both other representations. Implicit representations may make it difficult to generate points on the curve, and even to decide whether there are real points. On the other hand, they are well suited for deciding whether a given point is on a curve, or whether it is inside or outside of a closed curve. Such decisions may be difficult with a parametric representation, but parametric representations are best suited for generating points on a curve, and for plotting it.


Integer geometry

Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's parametrization of
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s such that the lengths of their sides and their hypotenuse are
coprime integers In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. As ''a'' and ''b'' are not both even (otherwise and would not be coprime), one may exchange them to have even, and the parameterization is then :a = 2mn, \ \ b = m^2 - n^2, \ \ c = m^2 + n^2, where the parameters and are positive coprime integers that are not both odd. By multiplying and by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.


Implicitization

Converting a set of parametric equations to a single
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit functi ...
involves eliminating the variable t from the simultaneous equations x=f(t),\ y=g(t). This process is called implicitization. If one of these equations can be solved for ''t'', the expression obtained can be substituted into the other equation to obtain an equation involving ''x'' and ''y'' only: Solving y=g(t) to obtain t=g^(y) and using this in x=f(t) gives the explicit equation x=f(g^(y)), while more complicated cases will give an implicit equation of the form h(x,y)=0. If the parametrization is given by
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s :x=\frac,\qquad y=\frac, where are set-wise
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
polynomials, a
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over t ...
computation allows one to implicitize. More precisely, the implicit equation is the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over t ...
with respect to of and In higher dimensions (either more than two coordinates or more than one parameter), the implicitization of rational parametric equations may by done with
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbn ...
computation; see Gröbner basis § Implicitization in higher dimension. To take the example of the circle of radius ''a'', the parametric equations :\begin x &= a \cos(t) \\ y &= a \sin(t) \end can be implicitized in terms of ''x'' and ''y'' by way of the
Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations b ...
: As :\begin \frac &= \cos(t) \\ \frac &= \sin(t) \\ \end and : \cos(t)^2 + \sin(t)^2 = 1, we get : \left(\frac\right)^2 + \left(\frac\right)^2 = 1, and thus :x^2+y^2=a^2, which is the standard equation of a circle centered at the origin.


Examples in two dimensions


Parabola

The simplest equation for a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, :y = x^2\, can be (trivially) parameterized by using a free parameter ''t'', and setting :x = t, y = t^2 \quad \mathrm -\infty < t < \infty.\,


Explicit equations

More generally, any curve given by an explicit equation :y = f(x)\, can be (trivially) parameterized by using a free parameter ''t'', and setting :x = t, y = f(t) \quad \mathrm -\infty < t < \infty.\,


Circle

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation : x^2 + y^2 = 1.\, This equation can be parameterized as follows: :(x,y)=(\cos(t),\; \sin(t))\quad\mathrm\ 0\leq t < 2\pi.\, With the Cartesian equation it is easier to check whether a point lies on the circle or not. With the parametric version it is easier to obtain points on a plot. In some contexts, parametric equations involving only
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s (that is fractions of two
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s) are preferred, if they exist. In the case of the circle, such a ''rational parameterization'' is :\begin x &= \frac \\ y &= \frac \end. With this pair of parametric equations, the point is not represented by a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
value of , but by the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of and when tends to
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
.


Ellipse

An
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
in canonical position (center at origin, major axis along the ''X''-axis) with semi-axes ''a'' and ''b'' can be represented parametrically as :\begin x &= a\,\cos t \\ y &= b\,\sin t. \end An ellipse in general position can be expressed as :\begin x &= X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi \\ y &= Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi \end as the parameter ''t'' varies from 0 to 2'. Here (X_c, Y_c) is the center of the ellipse, and \varphi is the angle between the X-axis and the major axis of the ellipse. Both parameterizations may be made
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
by using the
tangent half-angle formula In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are th ...
and setting \tan\frac = u.


Lissajous curve

A
Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...
is similar to an ellipse, but the ''x'' and ''y''
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
s are not in phase. In canonical position, a Lissajous curve is given by :\begin x &= a\,\cos(k_xt) \\ y &= b\,\sin(k_yt) \end where k_x and k_y are constants describing the number of lobes of the figure.


Hyperbola

An east-west opening
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
can be represented parametrically by :\begin x &= a\sec t + h \\ y &= b\tan t + k \end\quad or, rationally \quad\begin x &= a\frac + h \\ y &= b\frac + k \end A north-south opening hyperbola can be represented parametrically as :\begin x = b\tan t + h \\ y = a\sec t + k \\ \end\quad or, rationally \quad\begin x = b\frac + h \\ y = a\frac + k \\ \end In all these formulae (''h'', ''k'') are the center coordinates of the hyperbola, ''a'' is the length of the semi-major axis, and ''b'' is the length of the semi-minor axis.


Hypotrochoid

A
hypotrochoid In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The parametric equations f ...
is a curve traced by a point attached to a circle of radius ''r'' rolling around the inside of a fixed circle of radius ''R'', where the point is at a distance ''d'' from the center of the interior circle. Image:2-circles hypotrochoid.gif,
A hypotrochoid for which ''r'' = ''d''
Image:HypotrochoidOutThreeFifths.gif,
A hypotrochoid for which ''R'' = 5, ''r'' = 3, ''d'' = 5
The parametric equations for the hypotrochoids are: :\begin x (\theta) &= (R - r)\cos\theta + d\cos\left(\theta\right) \\ y (\theta) &= (R - r)\sin\theta - d\sin\left(\theta\right) \end


Some sophisticated functions

Other examples are shown: :\begin x &= - b\cos(t)\ + b \cos \left \left(\frac - 1\right)\right\\ y &= - b\sin(t)\ - b \sin \left \left(\frac - 1\right)\right\ k = \frac \end :\begin x &= \cos(a t) - \cos(b t)^j \\ y &= \sin(c t) - \sin(d t)^k \end Image:Param 03.jpg,
''j'' = 3, ''k'' = 3
Image:Param33 1.jpg,
''j'' = 3, ''k'' = 3
Image:Param34 1.jpg,
''j'' = 3, ''k'' = 4
Image:Param34 2.jpg,
''j'' = 3, ''k'' = 4
Image:Param34 3.jpg,
''j'' = 3, ''k'' = 4
:\begin x &= i \cos(a t) - \cos(b t) \sin(c t) \\ y &= j \sin(d t) - \sin(e t) \end Image:Param st 01.jpg,
''i'' = 1, ''j'' = 2


Examples in three dimensions


Helix

Parametric equations are convenient for describing
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s in higher-dimensional spaces. For example: :\begin x &= a \cos(t) \\ y &= a \sin(t) \\ z &= bt\, \end describes a three-dimensional curve, the
helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
, with a radius of ''a'' and rising by 2π''b'' units per turn. The equations are identical in the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...
to those for a circle. Such expressions as the one above are commonly written as :\mathbf(t) = (x(t), y(t), z(t)) = (a \cos(t), a \sin(t), b t), where r is a three-dimensional vector.


Parametric surfaces

A
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
with major radius ''R'' and minor radius ''r'' may be defined parametrically as :\begin x &= \cos(t)\left(R + r \cos(u)\right), \\ y &= \sin(t)\left(R + r \cos(u)\right), \\ z &= r \sin(u). \end where the two parameters ''t'' and ''u'' both vary between 0 and 2π. File:Torus.png, ''R'' = 2, ''r'' = 1/2 As ''u'' varies from 0 to 2π the point on the surface moves about a short circle passing through the hole in the torus. As ''t'' varies from 0 to 2π the point on the surface moves about a long circle around the hole in the torus.


Examples with vectors

The parametric equation of the line through the point \left( x_0, y_0, z_0 \right) and parallel to the vector a \hat\mathbf + b \hat\mathbf + c \hat\mathbf is :\begin x & = x_0 +a t \\ y & = y_0 +b t \\ z & = z_0 +c t \end


See also

*
Curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
*
Parametric estimating Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value ...
*
Position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
*
Vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
*
Parametrization by arc length Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the sy ...
*
Parametric derivative In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the depe ...


Notes


External links

*
Web application to draw parametric curves on the plane
{{DEFAULTSORT:Parametric Equation Multivariable calculus Equations Geometry processing