Tagged Questions

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

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0
votes
1answer
266 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
321 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
593 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
103 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 ...
1
vote
1answer
56 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
votes
1answer
234 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 $t= -1$ ...
5
votes
0answers
158 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.
0
votes
1answer
504 views

Intersection of cubic bezier curve and circle

Let $B$ be a cubic Bézier curve with control points $P_0,P_1,P_2,P_3 \in \mathbb{R}^2$, and $C$ be a circle with center $P_C$ and radius $r$. How can I find all intersections of $B$ and $C$? Is ...
4
votes
1answer
339 views

Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
2
votes
1answer
113 views

Constructing shortest interpolation curve from points in $\mathbb R^2$ with parametric equations.

Assume we are given a set of $n$ points from $\mathbb R^2$, $(x_1,y_1),(x_2,y_2)\dots(x_n,y_n)$. We want to construct a path connecting all these points using a pair of parametric equations ...
1
vote
1answer
68 views

Parametric representation of $\sqrt{x^2+y^2}\le z \le 2$

just wondering how to parametrize this. Question is: Let $C$ denote the conical region $\sqrt{x^2+y^2}\le z \le 2$. Find a parametric representation $\mathbf{x}(u,v)$ for $S$, the surface of $C$. ...
1
vote
1answer
77 views

Polar parametrization surface intersection

here is my problem: I need some help, i need the parametrization of the intersection of this two surfaces: $\ z^2= x^2+y^2 $ $\ (x-1)^2+y^2=1 $ Well, i can do it with cartesian equations $\ ...
0
votes
3answers
201 views

Create animation of parametric plot

I would like to create an animation of parametric plot. With a moving point on curve. The parametric functions are: x(t) = ...
1
vote
3answers
55 views

Is what I'm doing valid?

Find the POI of the following two planes: $$\pi_1: -3x + 3y + z + 6= 0$$ $$\pi_2: 3x - y + 2z - 2 = 0$$ I started by isolating "$z$". $$\pi_1: z = 3x - 3y - 6$$ $$\pi_2: z = \frac ...
1
vote
1answer
43 views

Existence of Lipschitz reparametrization

Suppose we are given a continuous path, $$\gamma:[0,1]\rightarrow (X,d)\text{,}$$ in a metric space $(X,d)$. When we deal with differentiable enough paths in Riemann manifolds we can give a ...
0
votes
1answer
177 views

Parametric simultaneous equations

I stumbled on this one a few days ago and I'm probably missing something obvious... I basically need to solve those parametric equations for the other coordinate $(x,y)$ other than the point ...
0
votes
2answers
66 views

Ray-Lens Intersection

So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector. I also have a spherical lens with aperture ...
0
votes
1answer
131 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
37 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 ...
0
votes
5answers
170 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
73 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 ...
10
votes
1answer
92 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
402 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
462 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
78 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
1answer
113 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
211 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
89 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
209 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
84 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
109 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
2answers
513 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
2k 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
87 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
460 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
98 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
50 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
30 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
78 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
2k 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
128 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
1answer
58 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 ...
4
votes
2answers
534 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
152 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 ...
1
vote
0answers
18 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
54 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 = ...
1
vote
0answers
44 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
41 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
2k 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 ...