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

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8 views

Find intersection between conotur point list and a line

Given: List of points representing a closed contour Task: Choose a random point on the contour and shoot a ray inside the contour and determine where the ray intersects the contour. This needs to be ...
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0answers
12 views

Finding the points of the curve in $R^3$ where the tangent is perpendicular to a line

Given the line L defined by $y=2z,x-z=3$, find the points of the curve $C$ given by $x(t)=-4t^2,y(t)=\frac{7}{6}t^3,z(t)=t+3$ where its tangent is perpendicular to L. Progress so far: $ x - z = 3 ...
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0answers
16 views

Parameterization of an ellipsoid in spherical coordinates

\begin{align*} 25x^2+16y^2+z^2=1 \\ \frac{x^2}{4^2} + \frac{y^2}{5^2} + \frac{z^2}{20^2} = \frac{1}{20^2} \end{align*} The spherical coordinates are defined as, \begin{align*} x &= \rho ...
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1answer
96 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
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0answers
19 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
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4answers
163 views

Parametric to implicit form of a curve

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$" How can I clear $t$ to arrive at the implicit equation?
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1answer
14 views

Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
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1answer
54 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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1answer
19 views

Conic-Sections, Ellipse, Parametrics, Normals and Tangents [closed]

Find the equation of the tangent and normal to the ellipse defined parametrically by: $x=5\cos\theta$ and $y=3\sin\theta$ at the point where $\theta=\pi/4$.
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1answer
13 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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1answer
37 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
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1answer
23 views

Slope of a Parametrized Curve

Say that we have the parametrized curve $x=e^{3t}, y=te^{-t}$. What would be the slope of this at the point $(1,0)$ and also on which points on the curve would the curve be horizontal? What I have ...
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2answers
22 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
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0answers
28 views

Is it possible to turn the parametric equation of a line in 3 dimensions into the general equation?

I Know it is impossible to do so since the parametric equation for a plane is the intersection of $2$ planes.For example: $x$ $=$ $\frac{-5}{4t}+\frac{1}{4}$; $y=\frac{3}{4t}+\frac{5}{4}$; $z=t$ ...
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0answers
10 views

Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: ...
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1answer
32 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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1answer
23 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
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1answer
36 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and ...
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2answers
28 views

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior. my works I'm stuck in next step
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2answers
19 views

Parametric equation question showing minimum value of d^2

for the equation $d^2 = (1-a)^2t^2 + 18(1-a)t +117$ Show that when $a = 2$, the minimum value of $d^2$ is attained when $t=9$. I set $a=2$ to get $d^2 = t^2 - 18t + 117$ should i now just run it ...
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2answers
16 views

Rearranging this equation

This is based on a parametric equation problem. We have two ships A and B at $(-2,at +1)$ and $(4, t+10)$ respectively. I need to show that $d^2 = (1-a)^2t^2 +18(1-a) t +117$ using the distance ...
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2answers
36 views

What's the parametric equation for the plane through a point (x,y,z) perpendicular to (a,b,c)?

Find the parametric vector and Cartesian equations for the following planes: a. The plane thru point $(2,1,-2)$ perpendicular to vector $(-1,1,2)$. b. The plane thru the three points $(2,2,-2)$, ...
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2answers
28 views

Curl of a vector field.

Let S be a piecewise smooth oriented surface in $\mathbb{R}^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
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2answers
20 views

2 lines passing Q and R meets at the mid-point,

Consider the straight line whose parametric equation is $$(x, y) = (1, 1)+ t(12,−1)$$ Show that the above line and a line passing Q and R meets at the mid-point. $Q = (5, 5)$ and $R = (9,−4)$ How ...
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3answers
64 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish ...
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0answers
25 views

Find the parametric equations of the plane given the following pieces of information

Find parametric equations of the plane that passes through the point $$P (- 2, 1, 7)$$ and is perpendicular to the line whose parametric equations are $$x = 4 + 2t , y=-2+3t, z =-5t$$ Here is my ...
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1answer
22 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 ...
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1answer
62 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 ...
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1answer
31 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 ...
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1answer
32 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
30 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
23 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 ...
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1answer
7 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
37 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 ...
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2answers
31 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
47 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 $\overline{cd}(t)$ ...
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1answer
22 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 ...
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2answers
45 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 : ...
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1answer
14 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 ...
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88 views

Very difficult surface integral

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
9 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
28 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
40 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
60 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
45 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 ...
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1answer
25 views

Need help with parametric parabola

so 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 (http://imgur.com/nRXOlJy). I was asked to provide an ...
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1answer
35 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
42 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
66 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
35 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...