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

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118 views

Parametric Equations: Find $\frac{\mathrm d^2y}{\mathrm dx^2}$.

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
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1answer
443 views

distance between parametric line and a point (4,3,s)

I've tried solving this problem every way I know how and I just can't get it. I've looked at similar problems of this type, and I still cannot get an answer that seems right. Parametric Equations: ...
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0answers
37 views

Can i formulate any curve with parametric equation?

Can i formulate any curve with parametric equation ? if not, so what kind of curves can be explained with parametric equations ?! Thanks in advance
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0answers
52 views

symmetric ranges of curve division on an oloid

taken from wikipedia , I drew an Oloid by using the functions with ; ; and with ; ; If I use any equal spaced range of t(x) on ; I will end up with an unequal range t(y), which ...
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1answer
72 views

area under a parametric curve problem

so I have a parametric curve, x = cos(t) y = sin(2t) I found that I need the area from 0 to pi/2. put this into an integral in terms of t I get $$ -\int_0^{\pi/2}sin(2t)sin(t)dt $$ But in my ...
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2answers
86 views

Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
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0answers
271 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
63 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
91 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
495 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
73 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
215 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 $t$.. ...
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0answers
67 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 ...
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1answer
379 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
222 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
530 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
326 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 ...
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1answer
169 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
79 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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6answers
263 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
102 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 ...
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2answers
77 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: ...
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0answers
228 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
103 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
90 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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0answers
16 views

Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
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3answers
193 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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1answer
756 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 ...
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2answers
172 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
440 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
47 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 ...
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95 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
2k 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
57 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
315 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. ...
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2answers
186 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. ...
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35 views

Strange attractors - groups other than four?

I'm using the following code to generate strange attractors: ...
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1answer
252 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 ...
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1answer
107 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
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2answers
104 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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0answers
63 views

Rheotomic surfaces parameterization?

Are there parameterizations for rheotomic surfaces? Or, am I stuck with implicit formulas and marching cubes for plotting points? Are there special cases where the surfaces are parameterizable? Here ...
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1answer
137 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 ...
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1answer
2k 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 ...
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2answers
445 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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91 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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68 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?
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3answers
1k 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
430 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 ...