The parametric tag has no wiki summary.
0
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1answer
11 views
Ray Disk intersection
So if I have a ray parameterized as $O + tD$ where $O$ is the origin, $D$ is the direction and $t$ is the parameter variable and a flat circular disk with a center point $P$ in 3D space and a radius ...
0
votes
1answer
26 views
Find position on surface of a lens
If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
-1
votes
0answers
19 views
Flux integrals, parameterization
let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2
parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi
(thi is the symbol of I with the circle in the ...
0
votes
5answers
37 views
Parametric equations for given line
How would you find the parametric equations for:
1) a line through $(3,1)$ and $(-5,4)$.
2) a segment joining $(1,1)$ and $(2,3)$.
Can anyone show me the steps of doing it cause the way my textbook ...
3
votes
2answers
56 views
Why do we need to find the intersection between these lines?
We have the functions
$$ x = -1 + 2 \cos(t)$$
$$ y = 3 + 2 \sin(t)$$
They give P's orbit
with $t$ on $\left[0, \dfrac{3}{2} \pi\right]$
Find (to 2 decimal places accurate) for which values of t ...
6
votes
0answers
52 views
Find a parametric formula to $n=(a^2+1)(b^2+1)$ in three distinct ways
I mentioned that the number $4420$ is expressible in the form $(a^2+1)(b^2+1)$ (where $a,b$ are positive integers) in three distinct ways,here is a list of these numbers:
...
0
votes
1answer
35 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
votes
1answer
41 views
Find the point of intersection of the line and surface
I have an odd problem with no solution. I am completely lost on how to solve this.
Problem:
Find the coordinates of the point(s) of intersection of the line $x = 1+t$, $y = 2+3t$, $z = 1-t$ and the ...
1
vote
3answers
42 views
Equation that always has a single solution, obtainable only by numerical methods.
I am looking for a parametrized equation (i.e. a class of equations), which has the following properties:
It has the form of $f(x) = 0$, where $f(x)$ is an increasing or decreasing (i.e. monotone) ...
1
vote
0answers
39 views
Tangent Vectors in a Surface
As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface $ \mathbf{x}(u^1,u^2) ...
5
votes
2answers
34 views
Parametrizing a given line and equations
1) Parametrizethe given line contraining the points (3,2) and (-5,6).
2) Find the parametric equations for the segment joining the given points (2,3) and (5,5) where $0\leq t \leq 1$.
...
1
vote
1answer
23 views
Is a parameterization defined to be surjective and/or injective?
A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know ...
3
votes
1answer
58 views
Find the second derivative ${{{d^2}y} \over {d{x^2}}}$ in terms of t when $x = 3 - 2{t^2}$ and $y = {1 \over t}$
This is my attempt:
$\eqalign{
& x = 3 - 2{t^2} \cr
& y = {1 \over t} \cr
& {{dx} \over {dt}} = - 4t \cr
& {{dy} \over {dt}} = - {t^{ - 2}} = {{ - 1} \over {{t^2}}} ...
1
vote
1answer
40 views
Parametric to Implicit ( {x(t),y(t)} --> P(x,y) == 0 )
I have this parametric equations:
$x(\theta) = r Cos(\theta) - \frac{v_{0}^2Cos(\theta)Sin(\theta)}{g}$
$y(\theta) = \frac{v_{0}^{2}Cos^2(\theta)}{2g} + r Sin(\theta)$
This is for $\theta \in ...
1
vote
1answer
18 views
position question from velocity and given point.
A particle moves along the $x$-axis so that at any time $t\geq 0$, its velocity is given by $v\left(t\right)=\sin\left(2t\right)$. If the position of the particle at time $t = \frac{\pi}{2}$ is $x = ...
0
votes
3answers
63 views
Explanation of the area under the curve given by a parametric equation
My textbook says the area under a graph is given by:
$\smallint ydx$
And it then goes on to say by the chain rule:
$$\smallint ydx = \smallint y{{dx} \over {dt}}dt$$
Could someone explain to me how ...
2
votes
1answer
44 views
Find the cartesian equation of: $y = \sin (2t)$ and $x = \cos (t)$
$\eqalign{
& y = \sin (2t) \cr
& x = \cos (t) \cr} $
therefore:
$\eqalign{
& \sin (t) = {y \over {2\cos t}} \cr
as: & \cos (t) = x \cr
& {\rm{ sin(t) = }}{y \over ...
0
votes
1answer
22 views
Parametric Equation along a line segment
I am having some trouble understanding how to determine the parametric equation of a line segment between A(1,1) and B(-1,1).
I did some research and came across the following relation:
...
1
vote
4answers
76 views
Some general question about parametric equations
My textbook doesn't explain this very well, what I want to know is the purpose of parametric equations, what is a parameter? what is the advantage of these equations over a function y=f(x), what do ...
1
vote
0answers
28 views
Vector Tangent to Curve of Intersection
I am having problems solving this.
Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$.
I'm able to do this kind of thing using ...
1
vote
3answers
43 views
How to take parametric equations (x, y) to create a derivative formula?
I always thought that if I take the derivative of the y and x equation and divide y' by x', then that would be the derivative in formula form.
Is this correct?
0
votes
1answer
24 views
How to check the visibility of these three points?
For question d part i, I have calculated the distances from $Q$ to $P_1$ and $P_2$ respectively and found $P_1$ to be closer with a distance of root $6$, with $P_2$ having a distance of root $24$.
...
1
vote
1answer
40 views
What is the normal form for this line?
I have calculated the parametric form of a line as: $L = P_1 + tP_1P_3 = <2,2,0> + t<1,2,2>$.
If I am given a point $ K = <1,-1,-1>$, how would I show the normal form of plane $E$ ...
1
vote
1answer
35 views
Parameterization of the surface a torus
For a calculus question I have I need to parameterize the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the z-axis (with $0<a<b$).
I've had a go at ...
4
votes
2answers
68 views
Circumference parametrization
Let $C=\{(x,y)\in \Bbb R^2: (x-x_0)^2+(y-y_0)^2=r^2\}$ and let $\varphi :[0,2\pi]\to \mathbb{R}^2$, $\theta \mapsto (x_o+r\cos \theta, y_0+r\sin \theta)$, with $r>0$.
I'm trying to prove that ...
0
votes
0answers
33 views
parametric equation derivative question: can someone help me understand this question?
I am given $x$ and $y$ coordinates in parametric form with equations...
$x(t)$ and $y(t)$.
The questions asks to calculate $f'(x)$ for when $x = x(2\pi/5)$.
Now am I first to calculate the ...
0
votes
1answer
32 views
Parametric problem: do these 2 comets collide. Am I solving this correctly?
$\text{comet1} = x_1(t), y_1(t)$
$\text{comet2} = x_2(t), y_2(t)$
set $x_1(t) = x_2(t)$ and solve for $t$. Since $t$ had a square, I had 2 possible values for $t$ ($t_1$ and $t_2$).
substitute ...
3
votes
2answers
114 views
Finding surface area of a cone
I will describe the problem then show what I tried to solve it.
I need to find the area of the cone defined as follows:
$$z^2=a^2(x^2+y^2)$$
$$0\leq z\leq bx+c$$
where $a,b,c>0$ and $b<a$.
...
4
votes
1answer
28 views
Parametric plots: Determine if 2 comets collide at a given time. Am I solving it correctly?
There are $2$ comets
comet 1 $(x(t), y(t))$,
comet 2 $(x_1(t), y_1(t))$
I need to determine if these two comets collide. From reading my steps below, is this the proper way to solve this?
$1.$ set ...
0
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0answers
13 views
Can one characterize which surfaces are capable of being described by a closed-form parameterization?
Speaking intuitively, I can visualize a lot of surfaces in my mind; but it seems that some of the ones I can imagine are not capable of being described by the 'usual suspects', i.e., elementary ...
0
votes
2answers
33 views
The parametric form of a line
For the parametric representation of a line L with the following points, is my answer correct:
P1 = <2,2,0>, P2 = <0,-2,-4>, P3 = <3,4,2>
Is this correct: X = P1 + s.P1P2 + t.P1P3 = ...
0
votes
0answers
34 views
$\frac{dy}{dx}$ of a parametric curve
Given $x = sin^2(t)$, $y = cos^2(t)$, I need to find $\frac{dy}{dx}$ in every non-singular point of the curve.
So $\frac{dy}{dt} = -2sin(t)cos(t)$ and $\frac{dx}{dt} = sin(2t)$.
To find the ...
1
vote
1answer
40 views
Parametric motion question
What exactly happens when both $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}x}{\mathrm{d}t}$ equal zero?
I know that if $\frac{\mathrm{d}y}{\mathrm{d}t} =0$ then its a vertical tangent with ...
1
vote
2answers
47 views
Finding the coordinates of a point of intersection from a pair of parametric equations.
A curve is given by:
$$x = 2t + 3 $$
$$y = t^3 - 4t$$
The point $A$ has parameter $t = -1$.
Line l is a tangent to the curve at $A$.
Line l cuts the curve at point $B$.
Find the value of $t$ at ...
2
votes
2answers
65 views
Parametric equation for plane with $\langle 0,1,1\rangle + s.\langle 1,0,-1\rangle$ and $\langle 0,0,-3\rangle + t.\langle 2,1,2\rangle$
In $3$-dimensional space, two lines $l_1$ and $l_2$ are given parametrically as follows:
$$
X = \langle 0,1,1\rangle + s.\langle 1,0,-1\rangle \text{ and } Y=\langle 0,0,-3\rangle + t.\langle ...
2
votes
2answers
44 views
Implicit form of a parametric surface
Let $\Sigma$ be the surface in $\mathbb{R}^3$ parametrized by
$$ (u,v) \mapsto \Big(\;p_X(u,v),\; p_Y(u,v),\; p_Z(u,v)\;\Big), $$
where $p_X, p_Y, p_Z$ are polynomials. Is there a standard way to ...
2
votes
1answer
52 views
How to interpret this task?
I have a task given to me in my homework I can not figure out what asks of me. The task is worded like this:
A curve in a plane is given by
$$
x(t) = 3(t - \sin(t))
$$
$$
y(t) = 3(1 - \cos(t))
$$
...
1
vote
1answer
64 views
Proper methods of solving parametric equations
I'm learning parametric equations in this section. Although I understand why the following works, I'm having difficulty understanding why the method employed for solving it is the correct one.
I'm ...
1
vote
1answer
61 views
How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation
Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral :
$$
\iint_D e^{\frac{y-x}{y+x}}
$$
a) by transforming to polar coordinates
b) by using the ...
3
votes
2answers
63 views
Having trouble solving question involving parametric equations
I have been given the following:
$$y = a \cdot \cos^3t$$ $$x = a \cdot \sin^3t$$
$$0 \leqslant t \leqslant {\frac\pi2}$$
I am supposed to show that the mean value of $y$ over the interval ...
0
votes
0answers
27 views
1
vote
2answers
74 views
Eliminating parameters to obtain surface equation
Given the following vector equation, how do I eliminate the parameters $u,v$ to get an equation of a surface in rectangular coordinates?
$$\vec{r}(u,v)=3u\cos(v)\hat{\imath} + 4u\sin(v)\hat{\jmath} ...
0
votes
1answer
43 views
Line integrals of vector fields
Consider the vector field:$$\vec G = \left(\frac y{x^2+y^2}, \frac {-x}{x^2+y^2}\right)$$ compute $\int_\Gamma \vec G$ where $\Gamma$ is the proportion of a parabola $y=a(x-1)^2$ from (1,0) to (2,a). ...
0
votes
1answer
32 views
Find parametrics equations of a line
Consider the line in $R^2$ that is given by the equation $d_1x_1 + d_2x_2 = c$ for numbers $d_1, d_2$ and $c$ in $R$ where $d_1$ and $d_2$ are not both zero. Find parametric equations of the ...
1
vote
1answer
49 views
Representing A Plane Curve By A Vector Valued Function
I am given the function $x^2+y^2=25$, and I am suppose to write this as a vector valued function.
I have always been awful at these sort of problems, even with parametric equations, which requires ...
1
vote
0answers
44 views
Computing the surface area of a (piecewise) polynomial parametric surface
I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases.
Case ...
0
votes
0answers
47 views
Torus equation in terms of tangent
So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
0
votes
1answer
38 views
3D Surface parametrization basics
I'm studying 3D rendering: I have a surface and the points on the surface are given by some function f such that $p = f (u, v)$
Since I'm a newbie this is unclear to me: how can a function of u and v ...
1
vote
2answers
44 views
Parametrizing this curve
How can I parametrize the trajectory
so that it is a smooth path $h:[-1,1]\rightarrow \mathbb{C}$?
I think that I should use $$h=\left\{\begin{array}{ccl}t+i |t|&:&-1\leq t \leq 0\\
? ...
0
votes
1answer
29 views
parabola in homogeneous coordinates
So if I have the parabola Y = X^2, how do I go about representing this homogeneously? I know I can parameterize it as F(t) = (t, t^2), but then what?
The reason I ask is because I have a 3*3 matrix ...
