Questions tagged [parametric]

For questions about parametric equations, their application, equivalence to other equation types and definition.

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Implicit equation of all points that a circle that traces along a 2d parametric curve.

I want to find an implicit equation that contains points that fall within a circle that has an origin that follows a 2d parametric curve, which would look like you painted a circle along that curve. I ...
Allan J.'s user avatar
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2 answers
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How can you find multiple answers for vectors a and v in parametric vector form given by the Cartesian equations?

For example you have the cartesian equation: (x-2)/-2 = y/3 = (z-1)/3 One possible choice for a is the vector <2, 0, 1>. How can you find a different ...
Markus H's user avatar
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parametric reflection of one curve along another

In this other question the user asks for a parametric curve and "imposing" one curve onanother. You can find a demonstration here. I have been meaning to use the tangent to answer a similar ...
vallev's user avatar
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How to parametrically express a parametric plane curve of a 3D plot as an axis in different 2D plot?

Preface Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$): $$x=x\left(t\right)$$ $$y=y\left(t\right)$$ In addition, there is a scalar function $f\left(x,y\right)$ defined ...
Dave's user avatar
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Solve for t: x = t cos(a) - sin(a) f(t) [closed]

I was trying to generate a method of rotating a function f(x) by angle a $y = f(x)$ is the Cartesian equation, which can be represented in parametric form in terms of $t$ as $$x = t \;\; \text{ and } \...
Zuctiv Azenci's user avatar
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What is $b$ in $r=b\theta$ of Archimedean spiral?

This Wikipedia entry says Equivalently, in polar coordinates $(r, θ)$ it (Archimedean spiral) can be described by the equation $r = b\theta$, with real number $b$. Changing the parameter $b$ controls ...
User's user avatar
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Parametric equation of inward pointing half sin waves

Parametric equation of inward pointing half sin waves I can create a circle in red and I can create a sin wave that goes around a circle in green. Parametric equation: ...
Rick T's user avatar
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orienting a point in polar coordinates along a particular unit vector

I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle. I wish to parameterize this circle and ...
Stan Shunpike's user avatar
3 votes
1 answer
69 views

Shortest distance between vertex of a circular cone and a quarter of its conical helix

I was given with the question below: ...
kingking's user avatar
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Parametric prolate epicycloid modelling and integration

I was trying to model an epicycloid for my math assignment but none of the parametric equations I found ended up helping me model it on desmos. One of the more prominent equations I found on the ...
TheShadowSider101's user avatar
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Is there any hope to find solution of the integral $\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx $?

Is there any hope to find a closed-form solution to this integral $$\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx=? \qquad\textstyle{with}\qquad a,b>0 \qquad\textstyle{and}\qquad a,b\in\...
math2021's user avatar
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Non-linear parametric equation zeros analytical formula

As in the title, I have to find the zeros of a non-linear parametric equation with several parameters as functions of the parameters themselves. I tried using "Solve" in Mathematica and &...
Giuseppe's user avatar
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1 answer
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parametric equation for hyperbola like $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

How can I get this parametric equation $x=\frac a2\left(t+\frac1t\right)$ and $y=\frac b2\left(t-\frac1t\right)$ for hyperbola like $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ using this $xy=1$ ($x=t$, $y=...
Mohamad Masaada's user avatar
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The length of an arc through parametric equations

Prove that the length l of an arc given by the parametric equations $x = \theta$ and $y = (\sec\theta)^2 $ from $\theta = 0$ to $\theta = \frac{\pi}{4}$ is given by $l = \ln(1 + \sqrt{2})$. I have ...
Robert Mdee's user avatar
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1 answer
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Show that $F(z)=\int_{-\infty}^{\infty} e^{-t^2+4 z t} d t$ is holomorphic in $\mathbb{C}$

I am trying to show that $$F(z)=\int_{-\infty}^{\infty} e^{-t^2+4 z t} d t$$ is a holomorphic function in $\mathbb{C}$. The idea is to use the following theorem for parametric integrals: Suppose $I$ ...
ImHackingXD's user avatar
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Stitching together parametric tubes?

Say I have two parametric tubes of the form $C_i(u,v) = (x(u, v), y(u, v), z(u,v))$. The most common case would be sweeping surfaces along a path, but the sweep can have a scaling factor. These ...
Makogan's user avatar
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4 votes
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Jerack curve - $\int_{-1}^{1}\frac{\sqrt{4w^{4}-20w^{3}+24w^{2}-20w+13}}{(w-2)^{2}\sqrt{1-w^{2}}}dw$

I wanted to calculate the length of the Jerabek curve which has equation: $$(x^2+y^2)(x-1)^2=4(x^2+y^2-x)^2$$ My work I used the coordinates to transform it into parametric form and arrived at: $$\...
Efesto's user avatar
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Solving Envelope Equation [closed]

I have a family of curves: $$F(x, y, t) = \sin(2t)x-\cos(2t)y-sin(t)$$ I'm trying to solve the equation for the envelope, that is this systems of equations: $$ \text{} \left\{ \begin{align} F(x, y, \...
sherwoodbirdin's user avatar
2 votes
2 answers
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Order of Differential Equation using number of arbitrary constants

Consider $$y=C_1 \sin ^{-1} x+C_2 \cos ^{-1} x+C_3 \tan ^{-1} x+C_4 \cot ^{-1} x$$ [ where $C _1, C _2, C _3$ and $C _4$ are arbitrary constants ] It is a family of curves. When we convert it to ...
Maths_Rocks's user avatar
9 votes
3 answers
188 views

Cartesian parametrization of the Greek Meandre

As the title suggests, I've been trying to find a parametric equation to describe the Greek meander pattern that's seen in a lot of historical architecture. I've created a series of X and Y ...
Ventry's user avatar
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Sketch a parametric function

We have a parametric function here and we want to obtain a rough sketch of it. $$x(t)=t-\sin t\quad y(t)=1-\cos t$$ I found that $\frac{dx}{dt}=y$. But I don’t know how that will help me sketch the ...
YANGyu's user avatar
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Line integral over multiple lines by parameterizing the curves

I have solved this problem by setting a relationship between x and y for the 2 lines and got 11. However, when I try solving by the method of parameterization, I come up with 36. I tried setting $C_1 =...
Saveer Jain's user avatar
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1 answer
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How to find a parametric function that fits (5 variables)

I have collected data for a parametric function and id like to explain a bit of the methodology first. If I launch a ball, the things I need to take into account are elevation change between me and ...
Gabe Spound's user avatar
3 votes
3 answers
202 views

Father duck swim along $y=-x^2$,baby duck swim along $y=x^2$, Father duck’s movements parameterized by: $𝒓(𝑡)=(𝑡−5,−(𝑡−5)^2)$, equation for baby?

A father duck and a baby duck are swimming along opposing parabolic paths, with the father along $𝑦 = −𝑥^2$ and the baby along 𝑦 = $𝑥^2$. The father duck’s movements can be parameterized by: $𝒓_𝟏...
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2 votes
2 answers
264 views

Finding the plane curve resulting from a reflection onto another plane curve

Question: What is the best way to describe the set of reflected points of one plane curve along the perpendicular lines at points of another plane curve? I am trying to find a way to find the ...
vallev's user avatar
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Normalize speed of parametric function

I have a parametric function for an ellipse: $$f\left(t\right)=\left(a\cos\left(t\right),b\sin\left(t\right)\right)$$ As the function goes linearly through t from 0 to 2pi, the point speeds up near ...
Kitty Craft0's user avatar
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In Statistics, how do we call a parameters constrained to be equal to another one?

I want to know how we can refer to the parameters of a model that was constrained to be equal to another parameter of this model. For example: Consider a parametric model (wiki definition) $\mathcal{P}...
Renato Fernandes's user avatar
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1 answer
65 views

Families of orthogonal curves to parabola.

In a typical ODE course, we learn that families of orthogonal curves to parabola $y=Ax^2$ are given by families of ellipses, given by $x^2/2+y^2=c^2$. However, there is this thing called parabolic ...
Sean's user avatar
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Two ways of finding families of curve orthogonal to circle

Given the family of curves $x=r\cos(\theta)$ and $y=r\sin(\theta)$ with $\theta \in [0,2\pi)$, I wish to find another family of curves orthogonal to it. One method is as follows: $$x^2+y^2=r^2 \...
Sean's user avatar
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1 vote
1 answer
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Can't spot my error in calculating 3D Parametric Arc Length

We're asked to find a function s(t), for the arc length of a curve centered at point t=0, as a function of t. The function is as follows... $\gamma (t) =e^t i + \sqrt{2} tj-e^{-t}k$ My work is as ...
Glen Gaige's user avatar
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1 answer
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Questions about Analytical Geometry: Relationship Between Lines and Planes in $3D$ Space

I'm currently working on an exercise in analytical geometry in three-dimensional space, and I have a set of questions regarding the problem. The exercise involves two lines, $L1: {x = 2 − t, y = 3t − ...
Ayesca's user avatar
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1 vote
2 answers
100 views

Find $a\in\mathbb{R}$ such that the following three vectors from a basis of vector space $\mathbb{R}^3$

$\text{Let } v_1=\begin{pmatrix}a \\ 1 \\ 1\end{pmatrix}, \ v_2 = \begin{pmatrix}1 \\ a \\ 1\end{pmatrix},\ v_3 = \begin{pmatrix}1 \\ 1 \\ a\end{pmatrix}.\ \text{Determine } a\in\mathbb{R} \text{ such ...
J__n's user avatar
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0 answers
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Question about the definition of parametric derivative

Consider the parametric equations $x=x(t),y=y(t)$, then the parametric derivative is $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}, \text{ provided }\frac{dx}{dt}\not=0.$$ My question is: Is the ...
dessert's user avatar
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1 answer
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How to modify the coefficients of this parametric equation to mimick a heart/bean shape please?

My current equation is: (x(t), y(t)) = (1.53cos(t), 0.99sin(t)+0.42cos(2t)) From my understanding, this is an explicitly-defined parametric equation. (See this screenshot for a visualized version of ...
C_Y's user avatar
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2 votes
2 answers
45 views

Single equation to graph a line in $3$-d

I know that to represent a line in $3$-d mathematically we need a pair of equations which represent two surfaces. Then their intersection becomes the desired line. I wonder if there is any way to ...
Mystic mystic's user avatar
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1 answer
85 views

How to convert this shape/equation into a Parametric Equation please?

From my understanding, but please correct me if I am wrong, this equation is implicit (See the attached screenshot). However, the goal is to create an identical (or similar) shape that is defined ...
C_Y's user avatar
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How to map a plane onto a Octahedron without polar distortions?

https://www.math3d.org/ZfulehTVK this is a link to where all my visualizations are. While researching seamless procedural texturing I noticed people creating a 4d hypersphere to loop noise back on ...
local idiot's user avatar
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0 answers
31 views

Streamline adjacency from first principals

Some Context I'm looking for an approach to print the stress trajectory of a part via composite 3D printing. If you imagine the stress trajectory as streamlines through a field of stress vectors, this ...
Lyndon Alcock's user avatar
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1 answer
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Parametric equations for a point traversing the circumference of a laterally accelerating circle (not rolling) with a max speed for the point.

Variables: e -> point on a circle r -> radius of said circle (constant) x(t), y(t) -> parametric equations describing the position of point e v -> velocity of the center of the circle c -&...
Brendan O'Sullivan's user avatar
3 votes
4 answers
145 views

How to solve quadratic equation with parameter for solutions in some range

Given an equation: $2x^2 - (p+3)x + p + 1 = 0$ and I have to find two real solutions and these solutions both have to be in range $(-1,3)$. So assumptions are: $a \neq 0 \land \Delta > 0 \land x_1 \...
Szyszka947's user avatar
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1 answer
125 views

Help explore parametric system in SageMath

i could use some help in presenting a good solution for this problem : Problem Assume the system of equations in $x, y, z, t$ $ax+y+z+t=1$ $x+ay+z+t=b$ $x+y+az+t=b^2$ $x+y+z+at=b^3$ where $a, b$ are ...
PanMath's user avatar
4 votes
1 answer
130 views

Curvature of streamlines as a Scalar field from a given vector field?

Some context I'm looking for an approach to print the stress trajectory of a part via composite 3D printing, however our composite 3D printer only allows for a turn radius of 3mm ($C_r = 3$). I ...
Lyndon Alcock's user avatar
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0 answers
98 views

If $x$ is real, does that mean every equation containing $x$ has real roots?

Question If the normal to the parabola $y^2 = 4ax $ at the point $(at^2,2at)$ cuts the parabola again at $(aT^2,aT)$ , then prove that $T^2 \ge 8$. Answer Normal at $(at^2,2at)$ is $y+tx=2at+at^3$, ...
PandaScientist's user avatar
8 votes
3 answers
747 views

How to parametrize a "SpongeBob flower"

I am trying to figure out a way of generating a 5 lobed shaped like the flower sin the sky of spongebob: I.e. a parametric curve that is at least twice differentiable, closed and that transitions ...
Makogan's user avatar
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3 votes
2 answers
153 views

How to evaluate the limit: $\lim_{t \to \infty} \int_0^1 \frac{e^{it^2(1+y^2)}}{1+y^2} \, dy$

I was evaluating the Fresnel integral: $$ \int_{-\infty}^{\infty} e^{ix^2}dx $$ After some calculations, I successfully evaluated it correctly. However, there is one problem that I don't know how to ...
Nebula's user avatar
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0 answers
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Examples of parametric equations which is impossible to eliminate the parameter from?

In the book "Calculus" by Stewart and others, it is written as follows: "It is not always possible to eliminate the parameter from parametric equations. There are many parametric curves ...
qkqh's user avatar
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1 answer
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(Linear Independence) Show that for two vectors $v,w \in \mathbb{R}^n $, the conditions (i), (ii), (iii) are equivalent

Show that for two vectors $v,w \in \mathbb{R}^n $, the conditions below are equivalent: (i) $v \neq 0$, and there exists no $\rho \in \mathbb{R}$ with $w = \rho \cdot v$ (ii) $w \neq 0$, and there ...
wengen's user avatar
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Find a parametrization for the plane $E_1$ and a describing linear equation for the parametric representation of the plane $E_2$

a) Find a parametrization for the plane $$E_1 = \\{ (x_1, x_2, x_3) \in \mathbb{R}^3 | 3x_1 - 2x_2 + x_3 = -1 \\} $$ b) Give a describing linear equation for the parametric representation $$E_2 = (1, ...
wengen's user avatar
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1 answer
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Parameterization of Curves and Surfaces

I would like to know if the following settings are correct: a) Parameterization of Curve: Given the curve $C = \{(x, y, z) \in \mathbb{R}^3 \,|\, x^2 + y^2 = e^3\}$ I want to find a parameterization ...
Ayesca's user avatar
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2 answers
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Let $P, Q, R \in \mathbb{R}^3$ be three points not on a line. Prove there is exactly one plane $E \subset \mathbb{R}^3$ containing $P,Q,R$, namely

Let $P, Q, R \in \mathbb{R}^3$ be three points not on a line. Prove that there is exactly one plane $E \subset \mathbb{R}^3$ containing $P,Q,R$, namely $$E = P + \mathbb{R} \cdot (P - Q) + \mathbb{R} \...
wengen's user avatar
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