What is Parametric equations: Definition and 208 Discussions

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object.For example, the equations







{\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}
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 such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors:




{\displaystyle (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 manifolds and algebraic varieties of higher dimension, 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, where the trajectory 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.

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  1. chwala

    Solve this problem that involves parametric equations

    My take; Part (a); ##\dfrac{dy}{dx}=\dfrac{1}{t}## therefore, ##y-2at=\dfrac{1}{t}(x-at^2)## ##ty-2at^2=x-at^2## ##ty=x+at^2## implying that ##T## has co-ordinates ##(-at^2,0)##. ##SP=\sqrt{(a-at^2)^2+(0-2at)^2}## ##SP=\sqrt{4a^2t^2-2a^2t^2+a^2t^4+a^2}## ##SP=\sqrt{a^2t^4+2a^2t^2+a^2}##...
  2. chwala

    Solve the given problem involving parametric equations

    My take; ##y=\dfrac{c^2}{x}## ##y+x\dfrac{dy}{dx}=0## ##\dfrac{dy}{dx}=\dfrac{-y}{x}## ##y-\dfrac{c}{t}=-\dfrac{y}{x}(x-ct)## ##yt-c=-\dfrac{yt}{x}(x-ct)## ##xyt-cx=-yt(x-ct)## ##c^2t-cx=-cx+yct^2## ##c^2t-cx=-cx+ytct## ##c^2t-cx=-cx+c^2t## ##⇒-cx=-cx## ##⇒cx=cx## Therefore it...
  3. chwala

    Prove that PA=2BP in the problem involving parametric equations

    My take; ##\dfrac{dy}{dx}=\dfrac{-1}{t^2}⋅\dfrac{1}{2t}=\dfrac{-1}{2t^3}## The equation of the tangent line AB is given by; ##y-\dfrac{1}{t}=\dfrac{-1}{2t^3}(x-t^2)## ##ty=\dfrac{-1}{2t^2}(x-t^2)+1## At point A, ##(x,y)=(3t^2,0)## At point B, ##(x,y)=(0,1.5t)##...
  4. chwala

    Find the Cartesian equation given the parametric equations

    hmmmmm nice one...boggled me a bit; was trying to figure out which trig identity and then alas it clicked :wink: My take; ##x=(\cos t)^3 ## and ##y=(\sin t)^3## ##\sqrt[3] x=\cos t## and ##\sqrt[3] y=\sin t## we know that ##\cos^2 t + \sin^2t=1## therefore we shall have...
  5. R

    Correct Parametrization for Calculating Area of a Tube?

    Hi, I'm trying to find the area of this tube using ##\int \int ||\vec{N}|| ds d\theta##. However, I get 0 as result which is wrong. So at this point, I'm wondering if I made a mistake during the parametrization of the tube. This is how I parametrized the tube. ##S(s, \theta) = (cos(s), sin(s)...
  6. B

    MHB Parametric Eqs: Find Line & Plane, Find Triangle Area

    Let P (1, 2, 3), Q (2, 3, 1), and R (3, 1, 2). (a) Derive the parametric equations for the line that passes through P and Q without resorting to the known formula. (b) Derive the equation of the plane that passes through the points P, Q, and R without resorting to the known formula. (c) Find the...
  7. I

    How Does the Point of Tangency Move in Circular Motion?

    Solution: The point of tangency of the string moves around the circle at ##2\pi## radians per second. First, we compute the position of the point of tangency of the string with the bobbin. Because this is simply a revolution around a circle of radius 10, the parameterization of the point of...
  8. karush

    MHB 243 parametric equations and motion direction

    11.1 Parametric equations and a parameter interval for the motion of a particle in the xy-plane given. Identify the paritcals path by finding a Cartestian equation for it $x=2\cos t, \quad 2 \sin t, \quad \pi\le t \le 2\pi$ (a) Identify the particles path by finding a Cartesian Equation the...
  9. Specter

    Find the scalar, vector, and parametric equations of a plane

    Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5) Homework EquationsThe Attempt at a Solution Finding the scalar equation: Ax+By+Cz+D=0 3x-4y+6z+D=0 3(9)-4(2)+6(-5)+D=0 -11+D=0 D=11...
  10. J

    B How do you create a + and π sign using multivariable (x,y,z)

    I am taking a high school multivariable calculus class and we have an end-of-semester project where we trace out some letters etc., except that they all have to be connected, continuous and differentiable everywhere. My group's chosen to do Euler's formula, but so far we are having problems...
  11. Specter

    Writing vector and parametric equations for a line that....

    Homework Statement [/B] Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1). Homework EquationsThe Attempt at a Solution First I find the direction vector for PQ. PQ=Q-P = (2,7,1)-(-3,5,2) =[2-(-3),7-5,1-2] =5,2,-1 PQ= (5,2,-1) Now I...
  12. E

    I Eliminating the Parameter from Helix Equation

    Let's say you have a helix defined parametrically as r(t) =<sin(t), cos(t), t> Is it possible to eliminate t and write an equation for this helix just in terms of x, y, and z?
  13. opus

    B Parametric Equations- Ball travel

    Suppose a baseball is hit 3 feet above the ground, and that it leaves the bat at a speed of 100 miles an hour at an angle of 20° from the horizontal. I've got the parametric equations in terms of x and in terms of y, and I have values plotted and a graph sketched. My question is in regards to...
  14. T

    Finding a Piecewise Smooth Parametric Curve for the Astroid

    Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...
  15. Physics345

    Find the scalar, vector, and parametric equations of a plane

    Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n→=(3,−4,6) and passes through the point P(9, 2, –5). Homework Equations Ax+By+Cz+D=0 (x,y,z)=(x0,y0,z0)+s(a1,a2,a3)+t(b1,b2,b3) x=x0+sa1+tb1 y=y0+sa2+tb2 z=z0+sa3+tb3 The Attempt at a...
  16. karush

    MHB 12.5.4 Find parametric equations .

    $\textsf{Find parametric equations .}$ $\textsf{of the line through the point }$ $$P(-3, -4, -2)$$ $\textsf{and perpendicular to the vectors }$ $$u = -6i + 2j + 8k$$and $$v = -7i + 5 j - 2k$$ $\textit{Answer:$\displaystyle x = -44t - 3 , y = -68t - 4, z = -16t - 2 $} $ ok how is this done with...
  17. K

    Second derivative in parametric equations

    Homework Statement Only the second part Homework Equations Second derivative: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$ The Attempt at a Solution $$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$ Do i differentiate the differential dt? $$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...
  18. Mr Davis 97

    I Second derivative of a curve defined by parametric equations

    Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...
  19. doktorwho

    Determining the velocity function

    Homework Statement Given the ## r(t) = ae^{kt}## , ##θ(t)=kt## find the velocity function that is dependent on ##r##. ##v(r)=?## Homework Equations 3. The Attempt at a Solution [/B] My attempt: 1)##r(t) = ae^{kt}## 2)##{\dot r(t)} = ake^{kt}## From the first equation: ##\ln...
  20. W

    Parametric equation of a circle intersecting 3 points

    Homework Statement Given the points P0 = (0,a), P1 = (b,0), P2 = (0,0), write the parametric equation of a circle that intersects the 3 points. Assume that b > a and both are positive. Homework Equations X = h + rcos(t) Y = k + rsin (t) r = √((x-h)2 + (y-k)2 Cos (t) = (x-h)/r Sin (t) =...
  21. Erenjaeger

    Solving Parametric Equations for the Equation of a Plane

    Homework Statement parametric equations are x=s+2t y=2s+3t z=3s+4t trying to solve for equation of the plane and then take the coefficients of the equation to get the vector normal to the planeHomework EquationsThe Attempt at a Solution apparently t=2x-y and s=2y-3x and therefore the equation...
  22. H

    I Does derivative formula work for all parametric equations

    The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by ##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}## The proof of the above formula requires that ##y## be a function of ##x##, as seen in...
  23. Isaac0427

    I What are some cool parametric equations for a butterfly curve?

    Hi all! I have recently taught myself parametrics, and I stumbled upon the butterfly curve. So, I was wondering about some cool equations I can plug into a parametric graphing calculator.
  24. N

    Parametric Equations of Tangent Line

    Homework Statement z = 2x^2 + 5y^2 +2 C is cut by the plane x = 2 Find parametric eqns of the line tangent to C @ P(2, 1, 15) Homework Equations z = 5y^2 + 10 dz/dx = 10y dz/dx (1) = 10 The Attempt at a Solution z = 10y + 15 y = t + 1 if the slope is 10/1 then delta z = 10 and delta y = 1...
  25. nomadreid

    Tangents to parametric equations

    The site http://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx talks of "the" unit tangent vector of r→(t) = f(t)*i→(t)+g(t)*j→(t)+h(t)*k→(t) as finding "the" tangent vector r'→(t) = f'(t)*i→(t)+g'(t)*j→(t)+h'(t)*k→(t) and normalizing it, and further with finding "the" tangent...
  26. P

    Second derivative with parametric equations

    http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx On this page the author makes it very clear that: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided ##\frac{dx}{dt} \neq 0##. In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...
  27. BruceW~

    Trajectory vs. time from parametric equations of motion

    Homework Statement Sketch the trajectory over the time interval 0 ≤ t ≤ 10 of the particle whose parametric equations of motion are given by X= t−3sint . And y = 4 − 3cost find the value of x,y,t Remember that you should be in Radian mode! Answer Do you isolate the t first and plug it in...
  28. D

    Parametric equations questions

    Hi everybody, i just joined. Here's my first post I have a few questions. When looking at curve C and by the eye test, it doesn't pass the VLT, even though x=f (t ) and y=g (t) are functions,...then overall with curve C not passing the VLT, is it still a function since f (t) and g (t) are? I...
  29. Nile Anderson

    Mathematical Modelling, Vectors and Parametric Equations

    Homework Statement Sorry to disappoint the math fanatics but no this is not a question that integrates all three topics at once but individual ones. I still need assistance though with the following more so in the reasoning behind them as I feel my logic is flawed...
  30. B

    Integrating parametric equations

    Homework Statement Why does \int_a^b \, y \; dx become \int_\alpha^\beta \, g(t) f^\prime(t) \; dt if x = f(t) and y = g(t) and alpha <= t <= beta? Homework Equations Substitution rule?The Attempt at a Solution I'm not sure how y = y(x) in the integrand turns into g(t). Isn't y a...
  31. R

    Parametric equations of various shapes

    How we can know the parametric equation for any curve? Is there some trick? Like for parabola ## y^2 =4a x ## It has general coordinates## (at^2 , 2at) ## It will satisfy the equation but how in first place we know it? Also we can have ##(a/t^2, -2a/t) ##, how?
  32. R

    Parametric Equations and slope

    Homework Statement Find all the points on the following curves that have the given slope: x=4cost y=4sint slope=1/2 Homework EquationsThe Attempt at a Solution Im not to sure what to do with this question.. I found dy/dx to be -cot(t) but I am not sure if that is even needed for this...
  33. F

    Points on lines with parametric equations (linear algebra)

    Homework Statement "Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line: x = 1 +t, y = 2+ t, z =2t . a. Do the lines intersect? If so, find the point of intersection. b. Find the point P on the graph of L1 that is closest to the graph of L2...
  34. B

    MHB Parametric equations and augmented coefficient matrices

    hi, I'm currently really struggling with an assignment that I've been tasked with. https://ss1002.files.wordpress.com/2015/01/assignment.pdf It's mostly theoretical proof questions, which I find difficult. Actual questions I'm fine with. I have done the first question without issue, as...
  35. TheDemx27

    Graphing Functions in n Dimensions, Parametric Equations

    So I was watching this video on Khan Academy, and it talks about graphing functions that have values in multiple dimensions. It shows how to represent a linear function in 3 dimensions with a set of vectors, L ={p1 + t(p1-p2)|t∈R} where p1 and p2 are vectors that lie on the line you want to...
  36. K

    The geometric shape of parametric equations

    Hello everyone, I have another question mark buzzing inside my head. After the elimination steps of a matrix, I'm having some problems about imagining in 3D. For example, x=t , y=2t, z=3t what it shows us? Or, x=t+2, y=t,,z=t ? Or another examples you can think of. ( Complicated ones of...
  37. RJLiberator

    Parametric Equations describing curves

    Homework Statement Homework EquationsThe Attempt at a Solution For part A) my answer was:[/B] \int_a^b \sqrt{(dx/dt)^2+(dy/dt)^2}dt The work I used for part A was based off this sites explanation: http://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx For part B) I simple took...
  38. E

    Rearranging parametric equations

    Homework Statement I am given a line that passes through the points (x0, y0, z0) and (x`,y`,z`) and a plane in 3D space being defined by these three nonlinear points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). Im directed to use cramers rule to find the intersection of the line and plane...
  39. grassstrip1

    Planes and parametric equations

    Hi everyone! I'm having some issues with this problem for linear algebra. I understand parametric equations fairly but I'm confused about the unit vector notation 1) Consider the plane r(s,t)=2i + (t-s) j + (1+3s-5t) k find the z component of the point (2,-1, z0) For what values of s and t is...
  40. M

    Derivation of parametric Equations?

    Homework Statement Hi, so confused abou this question that I probably haven't even posted it in the correct section.Here's the question. A wheel of radius ,r, is situated at the top of a ramp having an angle θ = π/6 rad. At t= 0 the wheel is at rest with its centre at coordinates (0,r) and...
  41. G

    Tangent to Parametric Equations

    Homework Statement Consider the curve with parametric equations: x = t - cos t, y = sin t. Determine exactly the equation of the tangent to the curve at the point where t=-0.5pi. Homework EquationsThe Attempt at a Solution The equation of a line is y - y1 = m ( x - x1 ) I substituted t = -pi/2...
  42. T

    How to interpret parametric equations

    Homework Statement Eliminate the parameter to find a description of the following circles or circular arc's in terms of x and y and find the center and radius and indicate the positive orientation x=cos(t) , y = 3sin(t) ; 0< t < pi/2 (should be less than or equal to) Homework Equations Not...
  43. I

    MHB Graphing 3D Parametric Equations on Nspire: Did I Do It Right?

    i graphed the parametric equations x=t y=t^2 and z=2 on my nspire in 3d and it came out looking like this (Blush) i have made them pretty so you can see the graph properly anyways what my question is, is that ( :confused: "is is" ) is the graph supposed to look like that? on a 3d plane? did...
  44. Yae Miteo

    Find parametric equations for the tangent line

    Homework Statement Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Homework Equations x = 1+2 \sqrt{t}, \quad y = t^3 - t, \quad z = t^3 + t, \quad (3, 0, 2) The Attempt at a Solution I began by...
  45. M

    How to graph parametric equations?

    Hi, so after doing some calculus for projectile motion with air resistance, I obtained two equations of vx(t) and vy(t) that describes the vertical and horizontal motion of the projectile. Please tell me if I'm wrong, but I believe since both vx and vy are functions of t, can't they be...
  46. I

    MHB Sketch Parametric Curve and Find Cartesian Equation

    sketch the parametric curve and eliminate the parameter to find the cartesian equation of the curve $x=\cos\left({\theta}\right)$, $y=\sec\left({\theta}\right)$, $0\le \theta < \pi/2$ $y=\frac{1}{\cos\left({\theta}\right)}=\frac{1}{x}$ i sketched the curve. how do i do the second part?
  47. sheldonrocks97

    Parametric equations for the portion of the parabola y=x^2?

    Homework Statement Find the parametric equations for the portion of the parabola y=x^2 from (-1,1) to (3,9) Homework Equations None that I know of. The Attempt at a Solution Using knowledge of parametric equations I am not sure how to start. My teacher never went over this...