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

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-2
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1answer
36 views

use the Parameterization in u and v to write the term $x^2+y^2$

Given that : $u=xy$ $v=x^2-y^2$ we want to write the term $x^2+y^2 $ using only $u$ and $v$. how can we do this ? update: please reread my question I have edited it. I think it is clear now ...
1
vote
1answer
22 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
0
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0answers
9 views

Parametric equation of a particle moving around a circle at known speed

A runner is running around a circular track of radius $r$ meters at $q$ meters per minute. The track is oriented on a Cartesian coordinate system with center at the origin and such that the runner ...
1
vote
1answer
19 views

What will the graph of this parametric equation look like?

What will the graph of this parametric equation look like? $$x = 2t$$ $$y = t + 5, \quad -2 ≤ t ≤ 3$$ Does "$-2 ≤ t ≤ 3$" represent the domain?
0
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0answers
19 views

Write an equation for the line through $A =(3, 1)$ and $B = (1, 2)$.

The line passes through $B$ and is parallel to $B - A$. So, the equation is $X = B + t(B - A)$. My question is: can we say that the following equations are correct as well? $X = B + t(A - B).$ $X ...
0
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0answers
25 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...
0
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1answer
22 views

Parameterizing the path of a point on a circle rolling on another circle

Problem: A wheel of radius $a$ rolls on the outside of a circle with radius $b$ (see figure). Find the parameterization for the curve a point on the wheel follows. You may choose freely how you ...
0
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2answers
16 views

What does $v = v_0 + t_1(v_1 - v_0) + t_2(v_2 - v_0)$ parameterize where $t_i$s are scalars and $v_i$s are vectors?

On the one hand $v$ looks like it describes a plane. On the other hand, $v_0 + t_1(v_1 — v_0)$ describes a line in $3$-space. Since we need two vectors(?) to describe a line, $t_2(v_2 - v_0)$ is ...
0
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2answers
24 views

Solving a Cartesian and parametric equation at a intersection.

A curve C has parametric equations: $x=4cos(2t)$ and $y=3sin(t)$ $-\frac{\pi}{2} < t < \frac{\pi}{2}$ The normal of a point A$(2,1.5)$ on curve C has the equation $6y-16x+23=0$ The curve and ...
0
votes
1answer
15 views

What does $v = v_0 + t_1v_1 + t_2v_2$ parameterize?

Let $v_1$ and $v_2$ be given vectors. $v = t_1v_1 + t_2v_2$ varies over the plane determined by the two vectors. The plane is parameterized by $t_1$ and $t_2$. Let $v_0$ be another given vector. ...
0
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1answer
14 views

What is the vector equation of the line through the head of $v_0$ and parallel to $v_p$?

$v_0$ and $v_p$ are vectors. Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 ...
2
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0answers
14 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
0
votes
1answer
32 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
0
votes
1answer
17 views

Obtain an equation for a parametrized curve segment

I am trying to express the segment of a curve in terms of $t$. For example, for a straight line between $(1,1)$ and $(2,2)$, I can express it like: $$ (x,y) = (1,1) + t (1,1), \space 0\le t \le 1 $$ ...
0
votes
0answers
9 views

Any special case satisfying $\arctan{\frac{dy(c+s)} {dx(c+s)}}$

There is a mysterious parametric curve: $$ x(s),y(s)$$ defined on three intervals (continuity unkown) around a, b, and c $(a< b\leq c)$, thus the curve consists of three segments : ...
1
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1answer
16 views

Finding the points of intersection between parametric equations and a circle.

A curve has the equation $ x=2t^{2} $ and $ y=3t $ and a circle has the equation $ x^{2} + y^{2}-6x-1 =0 $ What are the coordinates of the intersections between the objects? I tried subbing the x ...
3
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4answers
42 views

Given $\vec r(t)$, what are $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$?

I have come to a problem that simply states that we have a parametric curve $$\vec r(t) = (2\sin t, 3\cos t), \ \ t\in \mathbb R$$ and asks that we find $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$. ...
0
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0answers
31 views

Drawing a Parametric Equation from the intersection of surfaces

I need help with the second part of this problem. Show that any point on $$x^{2}+y^{2} = z^{2}$$ can be written in the form $$(zcos\theta ,zsin\theta ,z)$$ for some $\theta$. Use this to find a ...
2
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1answer
30 views

Pythagorean Quadruple Parametric Equation in 3 variables

I am looking for a pythagorean quadruple generator in 3 variables. I know this one with 4 variables. $$a=2mp+2nq \\ b=2np-2mq \\ c=p^2+q^2-(n^2+m^2) \\ d=p^2+q^2+n^2+m^2 $$ Anyway to do this?
0
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0answers
19 views

Pythagoren triple generation with 4 variables

I am trying to find parametric equations of 4 variables to find Pythagorean triples. I know one with 2 variables, but I would like 3 or 4. For example: a=M^2-n^2 b=2mn c=m^2+n^2 I would like p,q,m, ...
0
votes
1answer
43 views

Gradient of a parametric form

I want to find the gradient of a parametric form. So say you have the form $$(x,y,z) = (f(u,v),g(u,v),h(u,v))$$ and now I want to find and the gradient in parametric form. How do I do that? The ...
2
votes
2answers
42 views

Are there non-parametrizable surfaces?

Are there any surfaces that cannot be parameterized? (I'm in multivariable calc and we were talking about parametrizing surfaces for Stokes' Theorem so I was wondering if there are any surfaces that ...
0
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1answer
43 views

Parametrization of the intersection of a cone and plane.

EDITED with new progress updates. As the title states, I'm trying to parametrize the intersection of a cone and a plane. The equations are: $z^2 = 2x^2+2y^2$ and $2x+y+3z=4\implies ...
0
votes
1answer
38 views

Show that $y^2 \frac{d^2y}{dx^2}+1=0$ for a set of parametric equations.

A function of $x$ is defined parametrically by $x=t-\sin(t)$ and $y=1-\cos(t).$ How do I answer this question, then? Show that $$y^2 \dfrac{d^2y}{dx^2}+1=0.$$
1
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1answer
56 views

Find all $a$ for which the equation $8x^6+(a-|x|)^3+2x^2-|x|+a=0$ has more than $3$ different roots.

Find all $a$ for which the equation $8x^6+(a-|x|)^3+2x^2-|x|+a=0$ has more than $3$ different roots. I found couple of important things: First a little rearrangement: $8|x|^6+(a-|x|)^3+2|x|^2-|x|+a=0$ ...
1
vote
1answer
51 views

What's the difference betwen parameterizations and variable substitution for solving integrals?

Asumming I have the following integral to solve in the complex plane: $$\int \frac{dz}{z+1} $$ while $|z|=5$ which means a contour of radius 5 around zero. Is it possible to solve this integral using: ...
1
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2answers
68 views

Describe a twisted parabolic trough

I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that? Imagine this is the trough and the $z$ direction ...
4
votes
1answer
55 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
1
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0answers
47 views

Polynomial parametrization of a quadric with two given points

Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function $$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$ where and A, b and c are unknowns. ...
2
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0answers
50 views

Find the length of the parametric curve (Difficult)

Find the length of the parametric curve $$x = t$$ $$y = f(t)$$ $$f(t) = \int_0^t {s \over (s^2-1)} \ \mathrm{d}s$$ $$0\leq t \leq 1/2$$ First I create the $x'$and $y'$ Then put it into the ...
0
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1answer
42 views

Question about continuity of a polynomial curve (Spline)

I'm getting a little bit confused trying to write my own algorithm for calculating a Spline. Let's start saying that for my application I need that the curve, interpolating between more points, must ...
4
votes
1answer
43 views

All $a$ that equation has at least one root. $a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$

Find all $a$ such that the equation has at least one root. $$a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$$ What have I done: substitution $t=x+1$ and some rearrangements ...
0
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1answer
22 views

Given a parametrization, find the function

Consider a parametrization of $x$ and $y$ by $t$ (all real variables), say $y=f(t)$, $x=g(t)$. Given a function $f$ and a function $h$, we would like to find the function $g$ such that $y=h(x)$. ...
1
vote
1answer
63 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
2
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0answers
24 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object A starts at the origin (0,0) and moves straight up the y axis with a speed v. Object B starts at point (1,0), always moves towards object A and has a speed of 2v. ...
0
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2answers
24 views

Show that the surface $x^2+y^2=x$ using $\theta \space and \space z$ can be parametrised by $(\cos^2(\theta), \cos(\theta) \sin(\theta), z)$

I really have no idea how to do this: $x^2-x+y^2=0$ looks like it can be a circle given by: $(x-\frac{1}{2})^2+y^2=\frac{3}{4}$ mostly $x=r\cos(\theta) \space and \space y=r\sin(\theta)$ work as ...
0
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0answers
23 views

Parametrization of a 3D surface

While solving the following problem: The $x$ and $y$ coordinates of a point on the $Paraboloid$ $2z = x^2/a + y^2/b$ are expressed in the form $x = atanθ cosγ $, $y = btanθ sinγ $ where $θ$ is the ...
0
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2answers
37 views

Moebius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Moebius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
1
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0answers
31 views

Interpretation of parametrization

Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function. Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$ ...
2
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2answers
19 views

Give the explicit form of the following parametrized surface

Let $\boldsymbol{X}:\boldsymbol{R}^2\to \boldsymbol{R}^3$ be the paramtrized surface given by$$\boldsymbol{X}(s,t)=(s^2-t^2,s+t,s^2+3t)$$ I'm trying to describe the parametrized surface by an equation ...
2
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2answers
44 views

Is parametric form of a given function unique? [closed]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
2
votes
1answer
23 views

Is $\gamma(t) = (|t|,t)$ plot $y = x$?

I have this parametric curve : $\gamma(t) = (|t|,t)$ with $\gamma(t) : \mathbb{R} \to \mathbb{R}^2$ And I have to say if the plot is the line of equation $y = x$. Here's my answer: $x(t) = |t|$ ...
0
votes
1answer
48 views

What does $\mathbb{R}^2$ domain mean?

I have a parametric curve defined by : $\gamma\colon\left[ 0,+\infty \right] \to \mathbb{R}^2$ defined: $\gamma(t) = (\ln(t), 3\cdot\ln(6t))$ Now I have to say if the plot of this curve is a line of ...
0
votes
3answers
40 views

Determine if 3 parametric curves have same plot

I have to determine if those three parametric curves have the same plot: $\gamma_1(t) = (\cos(t), \sin (t))$ for $t \in \mathbb{R}$ $\gamma_2(t) = (\cos(t), \sin (t))$ for $t \in [0,2\pi]$ ...
0
votes
0answers
7 views

Is there a general way to find out if a curve defined by a set of parametric equations is transcendental?

Suppose we have a curve defined by a set of parametric equations $x_n = f_n(t)$ where $f_n$ may be transcendental. Is there a known general way to find out if the curve defined by those equations is ...
0
votes
2answers
24 views

derivative of closed parametric curve

Suppose, a parametric function $\beta:[0,1]\mapsto\mathbb{R}^2$ is a closed curve, that is $\beta(0)=\beta(1)$. For example $\beta(t)=(\sin 2\pi t,\cos 2\pi t)'$. Then my question: Is the derivative ...
0
votes
1answer
10 views

Question about second derivatives of parametric equations?

Consider the parametric curve given by $$x=4+t^2,\, y=−10t^2−10t^3$$ For $\dfrac{dy}{dx}$, I found $-5(2+3t)$ For $\dfrac{d^2y}{dx^2}$, I keep getting $\dfrac{d}{dt}\dfrac{-5(2+3t)}{2t} = ...
2
votes
0answers
18 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
0
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0answers
4 views

Underlying connections between parametric surfaces and co-ordinate transforms

I've recently been learning about parametric surfaces and surface integrals involving various co-ordinates systems. I was wondering, is there a fundamental connection between parametrization and ...
0
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0answers
6 views

Partial differential version of $ G(y,m) = G( \frac{y}{m} + m^{1/4}, m^{1/2}) $

I figured there must be a relation between partial differential equations and parametric equations like the wave equation in physics. I was working on something and wondering if anyone could tell me ...