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

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2
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0answers
349 views

Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space. When say quadratic ...
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1answer
70 views

Determine if a point is contained in the circle in 3d space

I have a problem where I need to determine if a point is contained in the area of a circle in 3d space. For my circle, I have the radius (R), the position of the center (C) and a normal vector to the ...
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1answer
117 views

Are these two parametric equations equivalent

I have two parametric equations that I suspect to be equivalent. I know I need to find a bijection map between the two to find whether they are, but I'm not sure how to go about doing so. The two ...
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1answer
35 views

Under what conditions does integrating the normal vector along a boundary make no sense?

So suppose you have an open, simply-connected, and bounded subset $D$ of $\mathbb{R}^2$ with the boundary $\partial D$. I am interested in the integral of the normal vector along the boundary, i.e., ...
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2answers
699 views

Parametric equations of cycloid on a Ramp

A small wheel of radius r is situated at the top of a ramp having an angle θ = π/3 rad as it appears in the figure below. At t = 0 the wheel is at rest and then it starts to rotate clockwise in the ...
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1answer
78 views

Some questions about parametric integrals

1) What is the error in the following calculation ? $\int_{0}^{oo} \frac {sin(px)}{x}dx$=$\frac {\pi}{2}$ derivating by p at both sides $\int_{0}^{oo} cos(px)dx$=0 But the second integral does not ...
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2answers
327 views

Show that the equation of the folium of Descartes in terms of $x$ and $y$ is $x^3+y^3=axy$

I'm given that the parametric equations are $x=\frac{at}{1+t^3}$ and $y=\frac{at^2}{1+t^3}$ and that $a>0$ Here's my attempt at a solution: Find $x^3$ and $y^3$ in terms of $t$.. ...
2
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0answers
116 views

Parametric equation of cycloid on Ramp [closed]

A small wheel of radius r is situated at the top of a ramp having an angle θ = π/3 rad as it appears in the figure below. At t = 0 the wheel is at rest and then it starts to rotate clockwise in the ...
0
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1answer
476 views

Assignment: Find the number of parameters in the general solution to a system of linear equations

This is a question given in an assignment I'm working on: If the coefficient matrix $A$ in a homogeneous system of 33 equations with 28 unknowns is known to have rank 12, how many parameters are ...
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1answer
39 views

Can we represent a symmetric curve by a parameter with symmetry?

Question : Can we represent the following curve $C$ by one parameter $t$ as $x=f(t),y=g(t),z=h(t)$ with symmetry? The curve $C$ in the $xyz$ space is defined as $$\begin{cases} x^2+y^2+z^2=1 ...
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1answer
288 views

Derivation of parametric equations of a hyperbola

Can somebody please show me how to derive the parametric coordinates of a hyperbola from $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ without just substituting them in? Thanks
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2answers
681 views

find area of the region $x=a\cos^3\theta$ $y=a\sin^3\theta$

Find the area of the region enclosed by $x=a\cos^3\theta$ and $y=a\sin^3\theta$ What steps should I take in order to find the area?
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1answer
408 views

Find equation of tangent line

Find the equation of the tangent line at parameter values $\theta=\pi/6$ and $\theta =5\pi/4$ to the cycloid given by $$x(t)=r\theta-r\sin \theta$$ and $$y(t)= r-r\cos \theta$$ with $\theta\in ...
0
votes
1answer
313 views

Find parametric equations

Find parametric equations for a particle moving two full revolutions clockwise around a circle of radius 2 centered at (3,-1). in other words give equations for x(t) and y(t), and specify the time ...
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1answer
82 views

$\frac{dy/dt}{dx/dt} \text{ at } t = a \text{ or } \lim_{t \to a} \frac{dy/dt}{dx/dt} \text{?}$

Take an example of parametric equation: \begin{cases} x = t^3\\ y = t^6 \end{cases} Obviously the formula $\displaystyle \left. \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \right.$ does not work at $t=0 ...
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votes
6answers
307 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
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1answer
115 views

Parametric surfaces - Parameterization of torus

A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This ...
0
votes
2answers
126 views

calculate velocity using parametric functions

if i have the following parametric functions where time is m/s : x = 8 t y = -5 t2 + 6 t and i want to find the initial velocity can i do the following: ...
2
votes
0answers
263 views

Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
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3answers
149 views

Trouble with caculus problem, parametric equations, I don't know what I'm doing wrong

When a mortar shell is fired with an initial velocity of v0 ft/sec at an angle α above the horizontal, then its position after t seconds is given by the parametric equations $x = (v0 \cos \alpha)t$ , ...
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1answer
96 views

Parallel Curves to a Parabola

I have been modelling parallel curves to a parabola and realise if the parallel curve to a parabola is offset enough then the curve will overlap. I came across this research paper to explain why a ...
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3answers
322 views

Equation of a Circle from parametric functions of sin and cos

Given: x = 2 cos (t/2) y = 2 sin (t/2) How do we find the equation of the circle? I know that x^2 + y^2 = 1, where x = cos(t) y = sin(t) so x^2 = (2 cos (t/2))^2 y^2 = (2 sin (t/2))^2 How do ...
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votes
1answer
986 views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
0
votes
2answers
209 views

Line integral around intersection of sphere and plane

The unit sphere is intersected by the plane x + y = 1. Find the line integral of F = around the intersection. $\int\int\nabla$x$F\cdot$ n dA the unit normal vector is easily found by looking at the ...
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1answer
516 views

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. <1+2$\sqrt{t}$ ,$t^{3}$-1 , $t^{3}$+1 > at point P(3,0,2)
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2answers
49 views

Possible $x$ values for a parametric equation

$x=\sin t $ and $y=3\cos 2t$ over the interval $-\pi/2 \leq t \leq \pi/2$. I know how to eliminate $t$, but I was asked to determine the possible $x$ values for the parametric equation and the ...
1
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0answers
107 views

Splitting parametric curve equation into two ranges

I am examining the speed of motion on curves and in the textbook i am reading , the example was showing that using a parametric equation for an ellipse will result in a regular motion (as expected) , ...
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4answers
3k views

Cardioid: Converting from parametric form into cartesian form

I'm trying to find the cartesian equation of the curve defined by the parametric equations: x=2cost-cos2t, y=2sint-sin2t I feel stumped. How can I go about this?
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2answers
61 views

Testing the multivariable chain rule

Let $x = t$, $y = t^2$, and $z = xy$. Then $$\frac{\partial y}{\partial t} = 2t$$ and since $t = x$, $$\frac{\partial y}{\partial x} = 2x = 2t.$$ Everything is consistent. But now consider using ...
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1answer
431 views

Prove parametric equations trochoid

I have to show that the parametric equations of a trochoid are: $x = r\theta - d\sin\theta$ and $y=r-d\cos\theta$ where r is radius and d is the distance between center of the circle and a point P. ...
2
votes
2answers
298 views

Find parametrizations for Circles and Ellipses

a) The portion of the circle $x^2 + y^2 = 4$ traversed clockwise from $(-2,0)$ to $(0,2)$ b) The part of the ellipse $(x^2)/(4) + (y^2)/(9) = 1$ that lies above the line $y = 0$, traversed clockwise. ...
2
votes
1answer
1k views

How to Parametrise a Parabola?

What is the method of find the parametric equations for all types of parabolas? And in both directions? So if I had 2 points: Parametrise from A to B Point $A = ( \frac{3}{\sqrt2} , 9 )$ and Point ...
0
votes
1answer
119 views

Circle wrapped on a cylinder

I'd like to know the parametric equations of a circle(r) wrapped on a cylinder(R). $x(t)= r\times cos(t)$ $y(t)=?$ $z(t)=?$ Which are the parametric equations? Thanks
0
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2answers
141 views

What is the $uv$ pair, or $uv$-plane, exactly?

Maybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the ...
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1answer
175 views

Finding parametric and non parametric equations?

need help to solve this equation really stuck on it. The plane through the point $(1, -2, 4)$ and containing the line given by $(x, y, z)$ = $(3, -2,1)+t(2, 1, -3)$ anyone help me out? thanks a ...
2
votes
1answer
4k views

Finding parametric and non parametric equations of a plane?

Hi there having some trouble with this. The plane through the three points $(5, 4, -8)$,$(1, 6,-3)$ and $(7,-2,5)$ so I then converted it to $(5, 4, -8) + s(-4, 2, 5) + t(2, -6, 13)$ then ...
0
votes
2answers
733 views

Conditions for a smooth parametric curve

A curve defined by $x=f(t), y=g(t)$ is smooth if $f'(x)$ and $g'(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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1answer
97 views

Turning points of a Bezier curve

I would like to find those points where a Bezier curve $\mathbf{C} = [(t(u) , P(u)]^T$ has zero gradient. Following the chain rule, I have tried the following $$ \frac{dP}{dt} = \left( \frac{dP}{du} ...
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1answer
84 views

What is parametrics and Parametrization of equations?

I always read about parametric equations and parametrization of equations, but what is that anyway? how can I tell the difference between a Parametric equation and a 'normal' one?
2
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3answers
2k views

Parametric equations, eliminating the parameter $\,x = t^2 + t,\,$ $y= 2t-1$

$$x = t^2 + t\qquad y= 2t-1$$ So I solve $y$ for $t$ $$t = \frac{1}{2}(y+1)$$ Then I am supposed to plug it into the equation of $x$ which is where I lose track of the logic. $$x = \left( ...
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1answer
537 views

Parametric Equations of an Oblique Circular Cone

I am trying to determine the parametric equations for a specific shape of an oblique circular cone with no success. Exhaustive web searchs and many texts have not been fruitful as regards ...
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3answers
116 views

Parametrization of Circle in 3D

I'm given the vector valued function (supposedly a circle) $r(t) = (3\cos t, 4\cos t, 5\sin t)$. However, I cannot see immediately how this is a circle. How do I verify that it is? I also have a ...
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1answer
280 views

Reverse direction of parametric equation

For the graph $y = \sqrt{x}$ the normal parametric equations would $x = t^2$ and $y = |t|$. However, the direction for that graph would be going from infinity to zero when $t \leq 0$ and zero to ...
2
votes
1answer
338 views

Sphere parameterization with 6 patches

I am looking for a parameterization of the sphere with 6 patches, like in http://www.image.ucar.edu/staff/rnair/research09.html and the inverse of this parameterization. As well, I would need a ...
0
votes
1answer
631 views

Parametric Equations rotation of axes

For old coordinates $(x,y)$ the new coordinates $(u,v)$ are related like this: $x = u\cos(\theta) - v\sin(\theta)$ $y = u\sin(\theta) + v\cos(\theta)$ So would it be correct to say that to rotate ...
2
votes
1answer
106 views

Parametric Equations (Basic) - Cartesian equation of curves

$x = 2 \cos t$, $y = 2 \sin t$, $0 \le t \le 2\pi$ Find the Cartesian equation of the curves. Please help i know it's basic but my problem is that $2 \cos t$ doesn't equal $1 - \sin^2 t$ and if it ...
2
votes
2answers
103 views

NURBS, parametrized curves and manifolds

Let's start with the definitions: A parametrized curve is a map $γ : (α,β) → R^n$ , for some $α,β$ with $−∞ ≤ α < β ≤ ∞$. A NURBS curve is defined by $C(u)=\sum_{i=1}^n R_{i,p}(u)\mathbf{P_i}$ as ...
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1answer
57 views

Cut edge between two parametric surfaces

I want to make a model of an ultrasound field, that impinges on a test object. The shape of the sound field can be simplified as a cone and the test object is cylindric. I used the following ...
0
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1answer
250 views

Parametric equations, Exponential Function.

Consider the curve defined by the parametric equations $x=t^2 +t-1$ and $y=te^{2t}$ i) Show that $dy/dx =e^{2t}$ ii) Hence show that the tangent to the curve at the point on the curve where $t= -1$ ...
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0answers
166 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.