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

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31
votes
3answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
10
votes
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: ...
8
votes
5answers
12k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
7
votes
3answers
3k views

Parametrization of a line

This is a very basic question, and its funny that I'm able to solve more advanced problems like this, but I was presented with a basic one and got stumped. I have the equation $$y=-\frac{3}{4}x+6.$$ ...
6
votes
6answers
3k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated ...
6
votes
3answers
229 views

Curvature of the image of a curve projected onto a surface

(Adding a bounty since I need more details than I have so far) Given a point $$ s_{0}=S(u_{0},v_{0}) \;\;\;\; (S:\mathbb{R}^{2}\to\mathbb{R}^{3}) $$ and a point $$ c_{0}=C(t_{0}) \;\;\;\; ...
6
votes
2answers
267 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...
5
votes
1answer
76 views

Can (x(t), y(t)) generate a surface? If so, can the surface be continuous?

Intuitively, the parametric equation $z = (x(t), y(t))$ seems to only be able to generate one-dimensional objects, i.e. curves. However... Let $x(t)$ be "the odd-indexed digits of the real number ...
5
votes
2answers
133 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 $0\leq t \leq 1$. ...
5
votes
0answers
130 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.
4
votes
3answers
1k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
4
votes
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 ...
4
votes
2answers
416 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
4answers
985 views

Arc Length Problem

I am currently in the middle of the following problem. Reparametrize the curve $\vec{\gamma } :\Bbb{R} \to \Bbb{R}^{2}$ defined by $\vec{\gamma}(t)=(t^{3}+1,t^{2}-1)$ with respect to arc length ...
4
votes
2answers
443 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 ...
4
votes
1answer
276 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 ...
4
votes
1answer
69 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 ...
3
votes
5answers
2k views

Length of $r=3\sin(\theta)$

I have a general understanding of calculating arc length, but this one's a real curve ball. So, I need to find the exact length of $r=3\sin(θ)$ on $0 ≤ θ ≤ π/3$ So the way I've thought of ...
3
votes
2answers
507 views

Converting $x=\frac{1}{2}\cos\theta\;;\;\; y=2\sin\theta $ to Cartesian form

How can we transform these parametric equations to Cartesian form? $x=\frac{1}{2} \cos\theta, \quad y=2\sin\theta \quad\text{ for}\;\;0 \leq \theta \leq \pi$
3
votes
2answers
260 views

Parametrization of curve length in D dimensional space. How is it done?

Sorry, its been a while and my calculus was never good. This is really a very elementary question which I am unable to un -complicate from its shroud of notation. My difficulty is how does this ...
3
votes
3answers
3k views

How to convert a plane (e.g. $4x - 3y + 6z = 12$) into parametric vector form?

I can convert something in the 2nd dimension fine, but I'm having difficulty with something like $4x - 3y + 6z = 12$. Any help? EDIT: Solve using only algebra, no matrices yet.
3
votes
2answers
70 views

Parametrization of the lemniscate

All over the net it is stated that the parametrization of the lemniscate with Cartesian equation: $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto ...
3
votes
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 ...
3
votes
1answer
160 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}}} ...
3
votes
1answer
3k views

find length of curve of intersection

I have come to a dead end on a problem and I need someone to tell me either if I did it correctly, or how to fix it if I did not. This is Stewart Calculus 7th edition, problem 13.3.12. Here is the ...
3
votes
1answer
45 views

Area of part of parametric function

I need to get area of function: $x= 2\sqrt{2}\cos ^3 t$ and $y= 4\sqrt{2}{\sin ^3 t}$, but only the part when $x\geq1$. How can I do that? I know that area of full function would be $$S= \int_a^b ...
3
votes
1answer
46 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
3
votes
2answers
155 views

Arc length paramatrizations satisfy original system of differential equations?

Say we have a system of differential equations $$ \begin{cases} x'''(t)+f(t)x'(t)=0\\ y'''(t)+f(t)y'(t)=0 \end{cases} $$ on an interval $[a,b]$, along with the restriction that $$ x'(t)^2+y'(t)^2=1 $$ ...
3
votes
2answers
92 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 ...
3
votes
1answer
460 views

Summation of an Arithmetic, Parametric Sequence

I'm trying work on my ability to break complex patterns down, and in this case I'm trying to model the denominators of Lacsap's Fractions: I managed to get the sequence that represents the ...
3
votes
1answer
315 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...
3
votes
1answer
1k views

Equation-driven smoothly shaded concentric shapes

Background Looking to create interesting video transitions (in grayscale). Problem Given equations that represent a closed, symmetrical shape, plot the outline and concentrically shade the shape ...
3
votes
1answer
30 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
3
votes
1answer
299 views

How are the parametric equations describing the cupid curve derived? [duplicate]

No doubt as some people have already seen, today morning wolfram posted the best valentine ever. The graph depicting cupid with its arrow and floating hearts around it involves something like 6 pages ...
3
votes
2answers
738 views

Plotting parametric equations in gnuplot

I am trying to plot the following parametric equation in gnuplot: fx(t) = -35*cos(t) + 65*cos(-.35*t) ...
3
votes
2answers
38 views

Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
2
votes
4answers
597 views

Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$

I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too. $$x=2t-4t^3$$ $$y=t^2-3t^4$$ I can guess that the curve is a polynomial equation so in principle I can write this as $$w_1 ...
2
votes
3answers
756 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( ...
2
votes
1answer
110 views

Parametric Equation of a Circle Using a Line

Consider the unit circle $$ x^2+y^2=1. $$ How can I parametrize it using the line $y=m(x+1)$, where $m$ is its slope?
2
votes
3answers
4k views

Parametric form of a plane

Can you please explain to me how to get from a nonparametric equation of a plane like this: $$ x_1−2x_2+3x_3=6$$ to a parametric one. In this case the result is supposed to be $$ x_1 = 6-6t-6s$$ ...
2
votes
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 ...
2
votes
2answers
288 views

Curve arc length parametrization definition

I did some assignments related to curve arc length parametrization. But what I can't seem to find online is a formal definition of it. I've found procedures and ways to find a curve's equation by ...
2
votes
3answers
70 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...
2
votes
2answers
47 views

Searching for a probability distribution appropriate for my task

I'm making a game (not important), but I'd like to have real probability distribution function (instead of classical dice notation). I like the normal distribution, but I would like to also shift the ...
2
votes
6answers
222 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
2
votes
1answer
266 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 ...
2
votes
2answers
246 views

parametric curves, parameter and integration

I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between ...
2
votes
2answers
102 views

Converting $x = \sin \frac{t}{2}, y = \cos \frac{t}{2}$ to Cartesian form

How can we transform these parametric equations to Cartesian form? $$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$$
2
votes
2answers
2k views

Finding Where A Parametric Curve Intersects Itself

The problem I am working on is to find the where the curve intersects itself, using the parametric equations. These are: $x=t^2-t$ and $y=t^3-3t-1$ For the graph to intersect itself, there must be ...
2
votes
3answers
15k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...