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

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1answer
20 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
0
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1answer
20 views

At what extent I can use trigonometric functions and properties with parametric curves?

I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the ...
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2answers
29 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...
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2answers
19 views

Explanation of how to go from polar to parametric equations.

I was wondering how you can make a polar equation parametric, and I just don't get it. My book says that for $r = f(\theta)$, $x = f(t) \cos t$ and $y = f(t) \sin t$, but this makes absolutely no ...
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3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
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1answer
25 views

parameterize the following functions [closed]

Help determining the parameterized solution of the following functions $$a) { \left( x-2 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 }=4\quad if\quad 1\le y\le 3$$ $$b) \frac { { \left( x+3 \right) ...
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2answers
18 views

Parametric equation for intersection of curve

Here's the three part question: A) Find parametric equations for curve which is the intersection of the cylinder $x^2 + z^2 = 1$ and the plane y = -x. B) Show that the curve lies on the surface $x^2 ...
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0answers
40 views

Parametric equation of a circle with given radius and starting point

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v = 0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ ...
1
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1answer
32 views

Parametric equation of a circle given starting point.

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v=0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ and ...
2
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1answer
38 views

Parametric & Trigonometry

$$x=7\sin(t)+\sin(7t)$$ $$y=7\cos(t)+\cos(7t)$$ How would I solve this one out? I have to simplify the two enough to graph it. I squaring the two and adding them together, but I hit a roadblock: ...
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0answers
20 views

Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
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2answers
45 views

Parametric Equations

$x=3\sin^3t$ $y=3\cos^3t$ How would I even begin to work out this one? I'm supposed to graph it, but I have no clue what how to even start it.
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0answers
15 views

Partial derivative of straigh-line parametrized integral

I would like to evaluate the following $$ F(\mathbf{r}_1,\mathbf{r}_2) = \int_0^1 ds~f(\mathbf{r}_1 + (\mathbf{r}_2 - \mathbf{r}_1)s) $$ where $\mathbf{r}_{1/2} = (x_{1/2} , y_{1/2})$, i. e. a ...
2
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1answer
38 views

Improper parametric arc length

The first thought I had to solve this problem was using the integral, $$ \int_1^\infty \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\: \:dt $$ Once you solve for the derivatives ...
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0answers
12 views

How can I find the projection of a sphere onto a direction vector subject to constraints?

Given a direction defined by a unit vector, I can find the projection of a point on that direction by using the dot product. The projection interval of a set of points is the max and min values of the ...
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0answers
35 views

After removing the parameter from $x=\sec \theta$ and $y=\cos\theta$, why does the domain become $|x|\geq1, |y| \leq1$?

For the parametric equations $x=\sec \theta$ and $y=\cos\theta$ with initial domain $0\leq\theta\lt\frac{\pi}{2}$, $\frac{\pi}{2}\lt\theta\leq\pi$, I understand that you arrive at $y = \frac{1}{x}$ ...
1
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1answer
24 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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2answers
30 views

Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
2
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0answers
26 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
0
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1answer
34 views

Help Needed Changing Parameter

Given that $r(t)=(4(\sin(t)−t\cos(t)),4(\sin(t)+t\sin(t)),(3/2)t^2)$ is a vector-value position function. Find the arc length function $s$. I need to change the parameter before deriving to calculate ...
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0answers
44 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
0
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1answer
241 views

Is this valid parametric equation to create control points for a helix in 3D space?

Is this a valid way to compute new points that are on a helix and if not what is it wrong? The Cartesian coordinates of each new helix control point could be described by the following ...
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1answer
739 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 ...
2
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1answer
406 views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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1answer
55 views

How to find the parametric equation of $x^y=y^x$ without Lambert W function?

This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos ...
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1answer
335 views

Shortest distance between a 3D parametric surface and a point

Right now I'm working on a library for finding the distances between objects in Lua. I've had some trouble finding the distance between a point and a bounded plane. I'm using these parametric ...
2
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2answers
2k views

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where ...
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2answers
50 views

Multivariable calculus - scalar field

I don't know how to solve this problem. Determine if $\mathbf{F}$ is or not the gradient of a scalar field. If it is find the corresponding potential function f. $\mathbf{F}(x,y,z)= 3y^4 ...
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2answers
85 views

Find Vector and Parametric Equation

I'm having some trouble finding answers to these problems. When i try to find help online, all i find are (x,y,z) problems and I'm simply looking for a PreCalculus (x,y) problem solving technique: ...
0
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1answer
26 views

How would I parametrise a straight line?

If I want to parameterise a straight line and I have the equation, eg $y=2x+1$ and I also have two co-ordinates it passes through, would it ok to use the co-ordinates to parameterise in terms of $t$?
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2answers
32 views

“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...
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0answers
14 views

Finding the bounds of a parameterisation

So I need to understand how to find what in this case "t" runs between for the following co-ordinates so that I can find the work done - (1,0,0) to (5,6,8) Parameterisation - $x=1+t^2, y = 3t, z ...
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0answers
35 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
1
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1answer
19 views

What is the Implicitization Problem

Let $V$ be a subset of $k^n$ given parametrically as $x_1 =g_1(t_1,...,t_m) ...x_n=g_n(t_1,...,t_m)$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an ...
0
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1answer
31 views

x(u,v), y(u,v), z(u,v) parametric equations for a special cycloid

I'm trying to find out a 3d parametric equations for a cycloid I know that a cycloid is a 2d curve it is generated by a point on a rolling circle. but my circle is rolling around another circle both ...
0
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1answer
19 views

Sketching Parametrizations - how to get something more understandable?

So I have some parametric functions (of one variable) I'm trying to sketch. Generally I can do so by "reverse parametrizing" where I take $x(t)$ and make $t$ a function of $x$ and then substituting ...
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1answer
35 views

How to parameterise the curve $ x^2 = 4y, 3x^3 = 8z$?

As per title, I'm unsure how to parameterise the given curve? Are there different methods? I'm unsure about parameterisation in general, I just tend to remember specific formulas.
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0answers
15 views

Bezier Surface evalution

So the problem I'm having at the moment, is a thinking problem. I can draw a bezier surface (parametric surface) with 16 control points and if I evaluate S(u, v) I get a coordinate in the 3D space. ...
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0answers
35 views

Solving a complex exponential / logarithmic equation

I've found this interesting equation on the web: $$p-1 = (1 - e^{\alpha-\beta t})^{t+1}$$ which has to be solved for t, considering that the parametes: $\alpha, \beta, p$ are defined correctly. ...
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1answer
17 views

Is there a parametric form for a degenerate conic section?

With parametric form I mean a parametrization like $(\cos{t}, \sin{t})$ for a circle. A conic section has such a parametrization, but suppose it degenerates in 2 lines (ranges of points), is there a ...
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2answers
37 views

Parametric equation of a curve in $R^3$

How to find the parametric equation of the curve in $R^3$, which is the intersection of the sphere of radius $a>0$ centred at the origin, and the plane $x+y+z=0$? I've tried to start looking for ...
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1answer
110 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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1answer
56 views

Parametric Equation of sine wave helically wrapped around a cylinder

I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D). The parametric equation below gets me close to what I'm looking for, but not quite since the ...
0
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1answer
34 views

Prove the normal will be at constant distance form origin in this parametric function?

Given a function, $x = a(cos \theta + \theta \sin\theta])$, $y = a(sin\theta - \theta\ cos\theta)$, $a \in R$ Prove that the normal drawn on each point is at constant distance form the origin? If ...
0
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1answer
28 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
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1answer
38 views

Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
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1answer
24 views

Parametrization of a paraboloid part

Find the parametric equation of the surface $S$, where $S$ is the part of the paraboloid $z=x^2 + y^2 + 1$ bounded by the plane $z=2x+3$ My attempt The OXY projection of $S$ is $x^2 + y^2 + 1 = 2x + ...
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0answers
17 views

Parametrics Question Help Please [duplicate]

Would anyone be able to verify the answer of (iv) being $y=-a$? I assume since -a is a constant, the x value is irrelevant Thanks
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0answers
16 views

Calculate the convergence domain of parameter improper integral

$$\int^{+ \infty }_{1} x^{u} \frac{x + \sin x}{x - \sin x}dx$$ The answer is $u<-1$. I suppose we need to find the simplified equivalent form of it, but I stuck on my way.
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1answer
32 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...