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

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0
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1answer
96 views

finding a vector valued function for the intersection of two shapes

I have a problem for my cal 3 class to find a vector valued function for the intersection of these two equations. $4x^2+4y^2+z^2=16$ and $x=z^2$ so i know that the first equation is a ellipsoid ...
4
votes
1answer
119 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
0
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1answer
41 views

The area of surface obtained by rotating a rectifiable curve

Let $\Gamma :X=\gamma(t),a\leq t\leq b$ be a rectifiable parameterized curve in the $(x,z)$-plane of $R^3$, which means $\gamma:[a,b]\to R^3$ is a $C^1$-mapping with $\gamma(t)=(x(t),0,z(t))^T$ and ...
1
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1answer
86 views

How to parameterize a straight line?

Why does the straight line from $(x_1,+y_1,+z_1)$ to $(x_2,+y_2,+z_2)$ become $r(\vec t)=(1-t)(x_1,+y_1,+z_1)+t(x_2,+y_2,+z_2)$ for $0 \leq t \leq 1$?
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0answers
34 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
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0answers
46 views

Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
0
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1answer
17 views

Derivative of the magnitude of a parametric function

I am trying to show that $d/dt$ $|r(t)|^2 = r(t)*r'(t)$, where $r(t)= <x(t), y(t), z(t)>$ and $r(t) \neq 0$. I first tried using the fact that $|r(t)|^2 = (x(t))^2+(y(t))^2+(z(t))^2$ and then ...
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3answers
40 views

Parametrization of $x^2+y^2-ay=0$

I am to find the circulation of $$y^2 dx + x^2 dy$$ along the (counterclockwise) path $$\Gamma : x^2+y^2-ay = 0$$ both with and without using Green's theorem. Apparently, $\Gamma$ is supposed to ...
0
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2answers
84 views

Parametrization of an intersection cylinder ellipsoid

I'm trying to parametrize the surface given by the equations : $$\frac{x^2}{2}+\frac{y^2}{2}+z^2=1$$ and $x^2+y^2=y$. I found this function : $f:[0,1] \times [0,2\pi] \to \mathbb{R}^3$, $$(r,x) ...
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1answer
68 views

Find parametric line between two 2D line segments that is an exact distance from a point

Given two 2D line segments, $\overline{ab}$ and $\overline{cd}$, and a point $p$, I would like to find a scalar value $t$ such that the line segment between $\overline{ab}(t)$ and $\overline{cd}(t)$ ...
0
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1answer
26 views

In parametric equations , how can the resulting equation after eliminating $T$ , consist of points not on the original set of equations?

I'm doing my Math level $2$ SAT subject test and there is a problem in the book that says "The resulting equation of eliminating $t$ may consist of points not on the graph of the original set of ...
2
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2answers
69 views

Find the plot of $y=1+\cos t$, $x=\sin^2t$.

I'm trying to find the plot for the following : $$y=1+\cos t, x=\sin^2t$$ I'm trying to get ride off variable $t$. This is what I done for some reason is incorrect : ...
0
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1answer
22 views

Finding the coordinate at time $t$ of a line determined by the points $(x1,y1), (x2,y2)$

I have the problem here, I create a program that clipping a line with the input (x1,y1,x2,y2). but the algorithm only explain until I get ...
9
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1answer
188 views

Evaluating the Surface Integral $\iint_{x^3+y^3+z^3=a^3} \frac{\bf{x}}{||\bf{x}||} \cdot d\bf{S}$

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}) \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
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0answers
19 views

Injective parametrization of a curve. ( piecemeal $C^1$)

$\gamma:$[0,1]$\to R^2$ is an injective parametrization of a curve $\Gamma$, which is piecemeal $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: Show that for every $n\in N$ there ...
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2answers
43 views

Converting parametric equations with trigonometric functions into Cartesian form

Ahoy, I am having trouble with a computer-based assignment and the question is as follows: $$x = 2\cos^5 t, \quad y = 2 \sin^5 t$$ Write these in Cartesian form, $F(x,y) = c$. I understand ...
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3answers
51 views

Parametric Curves and Tangents

I am struggling with a question regard parametric curves and finding tangents to them but something is going wrong somewhere in the process and I cannot figure out why. The question asks: consider ...
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3answers
69 views

Consider the parametric curve given by: $x=3\cos(t)$, $y=t^{3/2}$.

The question asks to find the equation of the tangent to this curve at the point $t=\pi/4$. I've determined $$\frac{dy}{dx} =(\frac{dy}{dt})/(\frac{dx}{dt}) = -0.222$$ Have I got the right idea? ...
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3answers
51 views

Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then ...
0
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1answer
35 views

Parametric parabola

I was given my Math C assignment today and the moment I looked at question 1 I knew I had no idea what to do. This is the graph I was given: I was asked to provide an equation for the curve however ...
2
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1answer
62 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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1answer
47 views

Looking for a family of astroids

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the ...
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0answers
99 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
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2answers
39 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...
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1answer
106 views

Parametric equation of clock hands

I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to ...
1
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1answer
86 views

Does every algebraic variety admit a local parametrization at every non-singular point?

I am reading a text in which the first sentence of the proof of a theorem is: Let $X(t)=X(t_{1},\ldots,t_{n})$ be a local parametrization of the algebraic variety $X$... I guess that every ...
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0answers
54 views

Calculate parametric bounds of a circle in a 2D quadrilateral

Given a 2D quadrilateral defined by the points $(p0, p1, p2, p3)$ and a circle centered at $c$ with a radius of $r$, I want to find a quad in the parametric space of the outer quad that tightly bounds ...
2
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1answer
40 views

Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
3
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2answers
118 views

Finding the parametric form of a standard equation

I need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did this ...
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0answers
35 views

Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
0
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1answer
30 views

equation of tangent line and parametric equation of tangent line (are they equivalent)

I am not sure about the steps to finding parametric equations of tangent lines and was wondering if these statements are equivalent Is there a difference if I am asked to find: equations of ...
1
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1answer
42 views

Is there an algorithm for parametrization of equations?

In this and this Math.SE questions askers wanted to parametrize their equations. It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate ...
0
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1answer
42 views

Parametric equation with image of a function.

Find all values of $a$ for which the image of the function $$y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}$$ contains $[2, 3]$. Now, I've already transformed it to ...
0
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1answer
15 views

Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
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2answers
40 views

Eliminating the parameter?

How would you eliminate the parameter where the x coordinate is in terms of t, but the t is squared. x= 3t - $t^2$ y= t + 1 I know to solve for y as a function of x, but I'm not sure how to do so ...
0
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3answers
48 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
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2answers
110 views

How to parametrize the curve $y^2 = x^2(x+3)$

I am emberassed to ask this, but I couldn't find a way. I want to write the curve $y^2 = x^2(x+3)$ as $$y=f(t) \quad \quad x=g(t) \quad \quad t \in \mathbb R$$ I guess I have to do something like ...
0
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1answer
55 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
1
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1answer
48 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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2answers
4k views

What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
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1answer
114 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
1
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1answer
542 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
1
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1answer
27 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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0answers
26 views

How to write explicity a curve on $S^n$?

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$. I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the ...
26
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2answers
372 views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two ...
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0answers
30 views

Fundamental Theorem of Calculus for Line Integrals

Use the Fundamental Theorem of Calculus for Line Integrals to compute $\int_C F*dr$ where $$F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy-2z)k$$ and C is the path from $(1,6,-1)$ to $(5,2,3)$ given by $x(t)=2t+1, ...
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2answers
80 views

Parametrization by arclength

I could not re-parametrize the curve r[s_] := {-(5 + 2*Cos[2*s])*Sin[3*s], (5 + 2*Cos[2*s])*Cos[3*s], 2*Sin[2*s]} neither by hand nor with Mathematica. Is ...
1
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0answers
29 views

Parametrization of arbitrary objects to display on an x-y-scope

I am trying to find an approach for general parametrization of an arbitrary geometric object or closed curve. Though I am not sure if I am on the right path with that. Basically I have an geometric ...
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0answers
30 views

How to write an equation with a denominator parameter less and greater than 1

I am writing a paper, but I do not know what is the most compact and exact formula to express this equation: $$\left| A_y(v) \right|^2 = \hat{q} v^\gamma \quad \text{where} \quad \left\{ ...
3
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0answers
41 views

Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...