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

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170 views

Find the parametrization of the intersection of two surfaces.

I'm having trouble figuring this problem out since the $3x^2 + z^2$ is throwing me off. Especially because of the $z$. Find the parametrization for the curve of intersection between the cylinder ...
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1answer
26 views

Understanding Parameters

My textbook (New Tertiary Mathematics, Volume 1 Part 1, Pure Mathematics: The Core, by C Plumpton & P S W Macilwaine) introduces Parameters in the following manner: "The coordinates of a point on ...
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32 views

Prove that z(t) and z~(t) are admissible parametrizations of the same smooth curve

Prove that $$z(t)=t+it^2, 0\leq t \leq1 $$ and $$\tilde{z}(t)=tan\Gamma+itan\Gamma, 0 \leq \Gamma \leq \frac{\pi}{4}$$ are admissible parametrizatiions of the same smooth curve. Do the above ...
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7 views

Tangential Angle (Para)

Could anyone offer some insight into this question? I understand that where it's 0 on the angle because it's a minimum point from para differential. I also understand that at $t=0$ it's parallel to ...
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2answers
29 views

If $x=\cos t,y=\cos(2t+\pi/3)$ find an analytical relation between $x$ and $y$.

I'm having a bit of trouble figuring this out. At the moment this is the near solution I have: $$y=\frac12(2\cos^2 t-1)-\sqrt{3}\sin t\cos t.$$ I should be just about to solve it but find myself ...
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0answers
146 views

Trying to find the volume of a 3D torus shape that I made

After playing around with 3D parametric equations on my calculator (modifying the equations of a standard torus), I came across a shape that I like. The equations are: $$x=(2+\sin t)\cos u$$ ...
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23 views

Confused by Parameterizations and Coordinate Conversions

So I have a few questions regarding parameterizations and coordinate conversion. Ever since dealing with parametric equations last semester I have felt like I have never truly understood ...
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182 views

Parametric integration negative area?

I know there is a question very similar to mine already here Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation? , but I am still a bit ...
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1answer
41 views

Is this differential equation correctly calculated as having as its solution this parametric family of curves?

Consider the following family of curves is parametric form: $$x(t) = a(\cos{t} + t\sin{t})$$ $$y(t) = a(\sin{t} + t\cos{t})$$ Where $a\in\mathbb{R^+}$ is a constant, and $t\geq0$. Find ...
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32 views

Parametrization of the intersection of the surfaces

How do I find a vector function that represents the curve of intersection of the sphere $(x+\sqrt 2/2)^2+(y+\sqrt 2/2)^2+(z-\pi/4)^2=4$ and the plane $x-y+\sqrt 2 z-\pi\sqrt 2/4=0$ ? Of course, I know ...
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1answer
33 views

Counterexample: Different curves

I am a TA in an introductory course to multivariable calculus. As defined in class two curves are said to be equal if their images are equal. Now a problem in their problemset was to prove that two ...
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23 views

Find the line tangent to the parametric curve $\left\langle t^3-1,t^4+1,t \right\rangle$

Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only. I am given a parametric curve with the ...
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1answer
106 views

How to calculate area of curved surface in specific region

i don't know how to use this site. this question is my first. please see the image uploaded in this page.
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3answers
234 views

What function could describe this GIF animation?

I found this image on Beautiful Mathematical GIFs Will Mesmerize You and this GIF really caught my attention. From what I see, it's a 2D circle morphing into the 3D sphere. What function could ...
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1answer
43 views

Parametric representation of a solid trapezoid

Question: Define a parametrical representation of a solid trapezoid as shown in the following figure: I came up with a solution by combining representations of the left rectangle and the right ...
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3answers
168 views

Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and$ Q\equiv(3,−3,1)$

Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and $Q\equiv(3,−3,1)$ for each of the given conditions on the parameter $t$ If $\vec r(3)=P$ and $\vec ...
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0answers
131 views

Given the equation of a cylinder $ x^2+z^2=1,$ find the parametric and locus form of the curve of intersection with plane

Given the equation of a cylinder $x^2+z^2=1,$ describe the curve of intersection between the cylinder & the planes z=x & y=x in the parametric form & the form F(x,y,z)=0. I am so lost ...
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2answers
62 views

How do solve this equation with dot product?

$$[(2,-7)+t(2,10)]\cdot n_1=0$$ $$\text{Solving for $t$, we find }t=\dfrac5{12}$$ $n_1=(1,1)$ or $(-1,-1)$. How does $t=\dfrac5{12}$?
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40 views

Understanding the definition of “smoothly equivalent”

$\underline{Definition}:$ The two curves $C_1:z(t), a\leq t\leq b$ and $C_2:\omega(t), c\leq t\leq d$ are $smoothly\ equivalent$ if there exists a $1-1\ C^1$ mapping ...
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1answer
50 views

Determine a parameterization for the line which is tangent to the curve at t=2

(1) A curve is given by the function $$r(t)=(t^3 -3t^2 +2t +4)i + (13-5t)j +(t^2 -t-3)k$$ Determine a parameterization for the line which is tangent to the curve at $t=2$ I started by solving for ...
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24 views

Expressing Curvature of a Polar Function in Terms of its Derivatives

I could use a little guidance with this question: Consider the curve $r=f(\theta)$, where $f$ is any twice differentiable function. Determine an explicit formula for the curvature $\kappa$ in terms ...
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2answers
221 views

Finding arc length parametrization of a cycloid

Find an arc length parametrization of the cycloid with parametrization r(t)= . I took the derivative and found the speed to be sqrt(2(1-cost))but now I'm unsure how to integrate that to get s. How do ...
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2answers
178 views

Catenary equation in 3D

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is lowest point of the catenary curve. I only know z-coordinate of this third point. I need to find ...
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1answer
42 views

Intersection of two parametric equations

This is a super basic question I'm sure but I can't figure it out and that's so frustrating. I must, in this homework problem (yes it is homework, so please do not give away the answer but rather make ...
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2answers
1k views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
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2answers
59 views

Converting from Parametric to Cartesian?…

I've been working on converting from parametric equations into cartesian form, but can't figure out this problem? $$x=(t^2+1)/(t^2-1)$$ $$y=(2t)/(t^2-1)$$ How do I covert that to Cartesian? Any help ...
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1answer
77 views

trigonometric parametrization

I am trying to figure out a pattern. I will start with examples. $$\text{Let } PD(\text{Set } A):= \text{Parametric Description of }A$$ $$ A:=\{(x,y)\in \mathbb R ^2|x^2+y^2 =1 \} $$ $$PD(A): ...
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1answer
299 views

Finding Arc Length Parametrization

Find an arc length parameterization of the line $y=6x+7$. Confused on how to start with x's and y's.
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3answers
78 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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1answer
2k views

Convert vector parametric equation to general form

Given the equation of a plane $x$ is $$x(s,t)=(0,1,1)+s(1,0,1)+t(2,1,-1)$$ How can I convert this equation into the general form $$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$$ Thank you.
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1answer
29 views

Trajectory of a particle for $t\ge1$ given $r(t)$ for $0\le t \le 1$.

I have a question on the process for which to solving this question. It is a homework question, and I already have the answer, but I am not sure on the correct process to attaining that answer. The ...
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1answer
23 views

Intersection of two parametric lines

This is not a question on my homework, just one from the book I'm trying to figure out. They want me to find the intersection of these two lines: \begin{align} L_1:x=4t+2,y=3,z=-t+1,\\ ...
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Parametrisation of surface

Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$. I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure ...
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0answers
20 views

figuring out parametric equation of a moving dot of specific velocity along acurve

I currently need to model a dot moving along an arbitrary curve given it's velocity, initial point, and $y=f(x)$ form of equation. I vaguely remember from my high school teaching that it will possibly ...
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1answer
66 views

Two questions on parametric equations, vectors, and planes.

I have two questions regarding parametric equations that I am struggling with. Question 1: a) Give a parametric equation for the line passing through $(-1,-2,3)$ and $(1,5,-2)$. b) Give the ...
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1answer
41 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 ...
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1answer
46 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? ...
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1answer
26 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?
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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 ...
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1answer
39 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 ...
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1answer
192 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 ...
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2answers
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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 ...
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2answers
159 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 ...
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1answer
18 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. ...
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1answer
18 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 ...
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54 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
59 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 ...
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1answer
25 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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1answer
163 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 ...
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4answers
49 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 $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$. ...