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

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0
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1answer
16 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
24 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 ...
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1answer
42 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
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1answer
22 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: $$ (x,y) = (1,1) + t (1,1), \space 0\le t \le 1 $$ ...
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0answers
13 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
30 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
43 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
60 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
35 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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1answer
53 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 ...
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2answers
50 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
96 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
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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
vote
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
56 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: ...
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2answers
83 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
62 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
56 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
52 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
49 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
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1answer
44 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
26 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
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1answer
76 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
31 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
votes
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
26 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
votes
2answers
46 views

Möbius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Möbius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
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0answers
34 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
46 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
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1answer
49 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
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3answers
41 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
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0answers
8 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
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2answers
35 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
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1answer
13 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} = ...
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0answers
21 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 ...
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0answers
5 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 ...
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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 ...
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0answers
21 views

Re-parametrization of triangle

Let $\beta(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a closed parametric curve. For example, let $\beta(t)$ is a circle. So one parametrisation may be $\beta(t)=(x(t),y(t))'$ where $x(t)=2\pi ...
0
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1answer
34 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
0
votes
1answer
24 views

Find an Equation of the Tangent Plane to the Given Parametric Surface at the Specified Point.

I am given $x=u+v$, $y=3u^2$, and $z=u-v$. I need to find the equation of the tangent plane at $(2,3,0)$. I understand that the equation of the tangent plane is ...
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0answers
24 views

Find all real numbers a such that equation 3^(x2+2ax+4a−3)−2=|(a−2)/(x+2)| Has exactly two different roots x1,x2 those belong to [−4;0]

Find all real numbers $a$ such that equation $${3^{(x^2+2ax+4a-3)}}-2=|{a-2 \over x+2}|$$ Has exactly two different roots $x_1,x_2 $ those belong to $[-4;0]$ Tried plenty different things to solve: ...
0
votes
2answers
27 views

How do you find a parametric representation for a specific surface?

I trying to find a parametric representation of the plane which goes through the origin and contains the vectors $i-j$ and $j-k$. I found the cross product for these vectors and found that the formula ...
4
votes
2answers
43 views

How do you parameterize a circle?

I need some help understanding how to parameterize a circle. Suppose the line integral problem requires you to parameterize the circle, $x^2+y^2=1$. My question is, if I parameterize it, would it ...
0
votes
3answers
30 views

Express angular position of the Earth as a function of time

Say I have for example the Earth orbiting the Sun (in circular orbit) and I want to express angular position (in radians) as a function of time. Would I be correct in thinking that $2\pi/t$ does the ...
1
vote
1answer
20 views

Where Am I Going Wrong on this Curvature Problem?

Massively deviating from the given answer on this one, have no idea what's going on. I'm sure that I'm using the correct formulas, but the answer provided is very different from what I come up with. ...
0
votes
0answers
24 views

Shannon entropy on the simplex

The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And ...
0
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0answers
20 views

Parameterized Curvature Problem

Massively deviating from the given answer on this one, have no idea what's going on. Find the curvature of the following function: x = 5cos(t) ; y = 4sin(t) ; t = $pi$/4 Formula for curvature is k ...
0
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0answers
18 views

Horizontal and Vertical Tangents of Limicons

I feel like I am over thinking this problem, and am probably just confusing myself... So I need to find the values of $t$ where the equation $r=a+b\cos(t)$ has horizontal and vertical tangents, for ...