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

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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
241 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
76 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
56 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
731 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
305 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
90 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
120 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
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1answer
265 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 ...
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1answer
303 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 ...
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1answer
56 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 ...
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2answers
79 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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0answers
46 views

Approximately space-filling parametric curves from unit line to n-cube

I am trying to find the solution of TSP in 2D Euclidean space using parametric curves of two interpolation polynomials constructed so that at times $t_k$ the curve passes through point $p_t$. See my ...
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1answer
53 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 ...
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1answer
149 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
129 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.
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1answer
179 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 ...
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1answer
274 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
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1answer
93 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 ...
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1answer
53 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 $C$. ...
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1answer
69 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 $\ ...
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3answers
135 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) = ...
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3answers
51 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 ...
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1answer
33 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 ...
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1answer
108 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 ...
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2answers
52 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 ...
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1answer
104 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
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1answer
36 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 ...
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5answers
94 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 ...
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2answers
72 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
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1answer
91 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: ...
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1answer
302 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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1answer
109 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 ...
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3answers
77 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) ...
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1answer
99 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) ...
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2answers
121 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 $0\leq t \leq 1$. ...
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1answer
59 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 ...
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1answer
153 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}}} ...
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1answer
70 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 ...
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1answer
81 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 = ...
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2answers
174 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 ...
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1answer
1k 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 ...
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1answer
65 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: ...
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4answers
257 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 ...
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0answers
81 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 ...
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3answers
46 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?
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1answer
28 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 $24$. ...
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1answer
70 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 $E$ ...
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1answer
554 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
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2answers
108 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 ...