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Questions tagged [parametric]

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

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Convert Parametric Polynomial Equation to Cartesian - Always Possible?

Let $p(t),q(t)\in\Bbb R[t]$. Do there always exist nonconstant $f(t),g(t)\in\Bbb R[t]$ such that $f(p(t))=g(q(t))$? I started by looking at a tricky example $$x=t^5+2t^3+3t, y=t^6+t^4+t^2$$where there ...
user108580's user avatar
1 vote
2 answers
39 views

Significance of tangent indicatrix

I'm reading a proof of the Four Vertex Theorem, and the proof introduces the notion of a tangent indicatrix. The precise text is as shown above. Can someone explain the intuition behind the tangent ...
DC2974's user avatar
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0 answers
41 views

What is this formula? Area of Jordan curve

I came across the following area formula in do Carmo's book: I tried searching online for where this formula came from, but I couldn't find anything that matched this. Does anyone have a name for the ...
DC2974's user avatar
  • 111
1 vote
1 answer
101 views

Parametric Equation of a unit circle when the angle between $x$-axis and $y$-axis is not $90$ degrees

I know in regular Cartesian coordinates the parametric equation for a unit circle is $x=\cos(\theta)$, $y=\sin(\theta)$, and if the $x$ coordinates are stretched by an amount $a$, and the $y$ ...
Anders Gustafson's user avatar
2 votes
0 answers
49 views

Parameterization by arc length: the concept

Can someone let me know if my understanding of parameterization by arc length is correct? If we have a regular parameterized differentiable curve $\alpha: I \rightarrow \mathbb{R}^3$, it is ...
DC2974's user avatar
  • 111
1 vote
2 answers
65 views

Determining the Equation of a Parabola in a Three-Dimensional Plane

I have a plane with three points: P1 (x1, y1, z1), P2 (x2, y2, z2) and P3(x3, y3, z3). These points represent a parabola where P3 is the vertex and the other two points are the ends of the parabola. ...
sanjog karki's user avatar
2 votes
1 answer
67 views

Calculating Slip Between Two Curves at Their Contact Point in Space

Question: Calculating Slip Between Two Curves at Their Contact Point in Space I want to calculate the slip between two curves at their contact point in space as they move and/or stretch. The curves ...
Oday Allan's user avatar
0 votes
3 answers
66 views

How can we draw $x(t)=3\sin(t) \sin(3t),y(t)=3\cos(t) \cos(3t)$? [closed]

I tried to take multiple values for $t$ but this was a painful procedure. I wondered if there is any way to make to graph $(x,y)$ faster. However, every way I tried failed to work. For example, ...
JOHN BOURAS's user avatar
0 votes
1 answer
33 views

Number of solutions in the positive octant $\mathbb{R}^3_+$ of a polynomial system of equations

Important edit: apparently my conjecture was wrong (see the answer I posted). In any case, I would still be very interested in methods that allows me to find/estimate the number of solutions in the ...
Carlos Santi Toledo's user avatar
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0 answers
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Determine the Domain Extents of an Arbitrary Bounded Cylinder in Cylindrical Coordinates

More Description I'm trying to determine the domain extents of a cylinder with an arbitrary orientation and size in cylindrical coordinates. I need the domain extents, or the maximum and minimum ...
Signal11's user avatar
1 vote
1 answer
31 views

Is there a way to derive the Polar Curve Area Formula using Parametrics?

I just finished up Calc BC, and one formula that my teacher never really went into the derivation of was how the area of a polar curve is given by $A=\frac12\int_\alpha^\beta r^2d\theta$. One that I ...
Aidan Hyde's user avatar
0 votes
2 answers
24 views

Converting a set of parametric coordinates to Cartesian coordinates

I have the parametric set of coodinates P = $(\dfrac{1+at^{2}+6at}{4},\dfrac{3at^{2}+2at+3}{4})$ , where $a = \dfrac{1}{16}$, after solving a problem involving the locus of a point. I wish to convert ...
Bongo Man's user avatar
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1 answer
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How to modify off-center circle in polar coordinates so that input angle has a linear relationship with angle on circle?

I have a circle translated horizontally in polar coordinates described by the equation: $$r\left(\theta\right)=d\cos(\theta)+\sqrt{r_{0}^{2}-d^{2}\sin^{2}(\theta)}$$ where $d$ is the horizontal ...
R. Toy's user avatar
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1 vote
1 answer
46 views

Number of solutions of parametric equation $\sin\left(a(\sin x+\cos^2x)\right)=0$ for $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}$

I am trying to find the number of solutions $N(a)$ of the following parametric equation: $$\sin\left(a(\sin x+\cos^2x)\right)=0,$$ where $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}.$ What I have ...
lorenzo's user avatar
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0 answers
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Insights for Outcome Function Involving Multiple Interdependent Variables

I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
blizzard16's user avatar
1 vote
1 answer
76 views

Convert pair of parametric trig equations to $y=f(x)$ form

My apologies if this already has an answer, I've spent some time looking but haven't found anything that (to me) looked directly applicable. I have a set of parametric equations describing a periodic ...
SteveP's user avatar
  • 11
0 votes
1 answer
84 views

Finding area using green's theorem

A friend of mine gave me a variant of the goat problem, which is the following: If a goat is tied to a circular fence of radius $10$ feet with a rope of $20$ feet, how much land can the goat roam? I ...
Kamal Saleh's user avatar
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0 answers
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Writing the equation of a family of circles touching two circles

I know that the equation of a circle passing through the intersection of two circles is $S_1+\lambda S_2=0$ and I know that the equation of a circle passing through the intersection points of a line ...
Cognoscenti's user avatar
0 votes
2 answers
58 views

Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions.

the problem Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions. the idea I tried applying the quadric formula and got that $...
IONELA BUCIU's user avatar
2 votes
1 answer
65 views

Is it "ok" to see constants as a "section" in $n$ dimentional space to get to lower dimentionality space?

Given that in $y=5x$, $5$ is a constant (let's call it $a=5$) in a 2d graph. I thought that it could be seen as a simplification of a more general case of some 3d space where $y$ is a function of both ...
Igor's user avatar
  • 91
3 votes
0 answers
55 views

How should I keep arc length equal between multiple points on a parametric curve?

I made this thing in desmos: https://www.desmos.com/calculator/na9sehjskk The distance between points changes depending on the speed of the points. Is there a way to keep the distance between them ...
aaaaaaaa1234564's user avatar
0 votes
0 answers
17 views

Modelling with parametric volume of revolution,

Modelling question I think the curve is only the part of the cross-section from Q to R. I understand how to do the integration for part b of the question but can't figure out how I would find a and b....
terry cruise's user avatar
2 votes
1 answer
58 views

Parameters behind non-symmetric Lissajous loop?

I'm trying to guess what kind of two signals can create this kind of Lissajous curve: However I can't figure out what are the parameters that break the symmetry of the curve. The relative phase ...
ValientProcess's user avatar
1 vote
0 answers
32 views

Can we set the components of a parametrized equation to be vectors?

Assume I have a polynomial function $y=f(x)=ax^3+bx^2+cx+d$ for $a,b,c,d \in \mathbb{R}$. In order to investigate the curve, I parametrize the equation as such $(t,x(t),y(t))$. Now, If I were to ...
Perfectoid's user avatar
2 votes
1 answer
74 views

Intersection of a spheroid with a plane

Given the following equation for a spheroid, $$ x^2 + y^2 + \frac{z^2}{k^2} = a^2 $$ and plane, $$ ux + vy + \frac{wz}{k^2} = a^2 $$ I need to find the parametric equations $x(t), y(t), z(t)$ for ...
Chris-Al's user avatar
  • 265
2 votes
3 answers
117 views

Find ellipses constrained by 2 points and their respective tangents. Parametric form.

The goal is to programmatically draw a path between the 2 points that follows a conic curve if one exists. Approximations and iterative methods are acceptable. Let's start with a simple, parametric ...
evn's user avatar
  • 21
0 votes
0 answers
16 views

Software for Exportable NURBS surfaces from Parametric Equations $x=f(u, v), y=f(u, v), z=f(u, v)$ (Must be Suitable for Engineering)

The title pretty much says it all. Is there any software out there that lets you input 3D parametric equations without having to go to the trouble of writing a bunch of code and then lets you export ...
Johnny's user avatar
  • 11
3 votes
2 answers
156 views

Determine the length of the rod that can be inscribed in a cuboid

Question You have a cuboid of dimensions $2a \times 2b \times 2c $. I want the find the (maximum) length of the right circular cylindrical rod of radius $r$, that can inscribed in the cuboid. Use ...
Quadrics's user avatar
  • 24.3k
1 vote
0 answers
20 views

Can I reparameterize this parametric curve to have constant velocity with closed-form expressions?

Question Summary I have a 2D parametric curve defined by two functions, $f_x(t)$ and $f_y(t)$, and by several parameters that adjust the overall shape of the curve. As-is, if $t$ varies at a constant ...
Lawton's user avatar
  • 1,861
2 votes
2 answers
73 views

When is it OK to take powers of the equation of a curve?

Isnt raising curve equations to powers "dangerous" in general? Take $$ x=y $$ if you square that, you get $$x^2 = y^2$$ The new equation contains the points described by $ x=y $, and $ x=-y $...
Yeslin Sequeira's user avatar
0 votes
0 answers
42 views

Maximizing Speed on an Elliptical Path

I'm currently delving into a problem that involves analyzing the motion of a particle moving along an elliptical path in the plane, and I could really use some help figuring out a couple of aspects. ...
tarek hankir's user avatar
1 vote
1 answer
54 views

Need Help with Particle Motion Analysis in $\mathbb{R}^2$

I'm currently working on a problem related to particle motion in the plane $( \mathbf{R}^2 )$ and could use some guidance. The particle moves along a curve, and at any given time $( t )$ seconds after ...
tarek hankir's user avatar
1 vote
0 answers
39 views

How to find the first intersection point of 2 systems of symbolic equations?

Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do... Imagine 2 line segments randomly defined in a $2D$ room. The ...
proj786's user avatar
  • 11
0 votes
2 answers
76 views

How do I find the corner shape of the bounding box of a smooth curve of constant width?

Given the functions $$ p(θ) = \frac{S}{2} × \frac{\cos\bigl(n × (θ - α)\bigr)}{n^2 - 1} + \frac{W}{2}\\ \begin{align} X(θ) = \cos(θ) × &p(θ) - \sin(θ) × p'(θ) - \left(p\left(\frac{0}{2}\right) - \...
Lawton's user avatar
  • 1,861
1 vote
0 answers
45 views

Parameterization of area between $y=x+\cos x$ and $y=x+\sin x$

In the $xy$-plane a function is given: $f(x,y)=x+y$. Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$. a) Make a ...
Zert44's user avatar
  • 51
0 votes
1 answer
40 views

Finding the distance of the intersection point of two conics to $(-3,2)$

Consider the intersection of the curves $x^2+y^2+6x-24y+72=0, x^2-y^2+6x+16y-46=0$. Determine the sum of the distances of their intersection points and $(-3,2)$. My first thought looking at this ...
Cognoscenti's user avatar
0 votes
0 answers
17 views

Graphing dy/dx of a parametric equation.

Consider the first quadrant of a circle. We can represent the first quadrant of a circle as: $y_1 = \sqrt{1-x^2},$ such that $0\leq x \leq 1 \\$ and in parametric terms as: $\left(\frac{1-t^{2}}{1+t^{...
idk's user avatar
  • 125
0 votes
0 answers
32 views

Wrong intuition about arc length over a surface

I was trying to writ metric tensor over a surface $M$ starting with the usual arc length, but I find some unclear steps. Consider a parametric curve in $\Bbb R^n$ $$\gamma = \{ \mathbf x(t), t \in [...
Turquoise Tilt's user avatar
1 vote
1 answer
53 views

Proof of the Cycloid Parametric Equation

One of the steps of deriving the equations for the parametric curve of a cycloid is the following: Here we establish that the distance PT is equal to the distance OT, which then (alongside other ...
Agustin G.'s user avatar
0 votes
0 answers
42 views

Proving the curvature of a plane curve is equal to that of a space curve

Let $\gamma : (a,b) \rightarrow \mathbb{R}^2$ be a regular curve. Let $\iota : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the map \begin{equation}\iota\left(\begin{pmatrix}x \\y\end{pmatrix}\right) = \...
spooleey's user avatar
  • 456
1 vote
1 answer
56 views

Implicit equation of all points that a circle that traces along a 2d parametric curve.

I want to find an implicit equation that contains points that fall within a circle that has an origin that follows a 2d parametric curve, which would look like you painted a circle along that curve. I ...
Allan J.'s user avatar
  • 121
0 votes
1 answer
53 views

How can you find multiple answers for vectors a and v in parametric vector form given by the Cartesian equations?

For example you have the cartesian equation: (x-2)/-2 = y/3 = (z-1)/3 One possible choice for a is the vector <2, 0, 1>. How can you find a different ...
Markus H's user avatar
0 votes
0 answers
109 views

parametric reflection of one curve across another

In this other question the user asks for a parametric curve and "imposing" one curve on another. You can find a demonstration here. I have been meaning to use the tangent to answer a similar ...
vallev's user avatar
  • 296
2 votes
2 answers
63 views

How to parametrically express a parametric plane curve of a 3D plot as an axis in different 2D plot?

Preface Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$): $$x=x\left(t\right)$$ $$y=y\left(t\right)$$ In addition, there is a scalar function $f\left(x,y\right)$ defined ...
Dave's user avatar
  • 31
0 votes
1 answer
88 views

Solve for t: x = t cos(a) - sin(a) f(t) [closed]

I was trying to generate a method of rotating a function f(x) by angle a $y = f(x)$ is the Cartesian equation, which can be represented in parametric form in terms of $t$ as $$x = t \;\; \text{ and } \...
Zuctiv Azenci's user avatar
0 votes
1 answer
56 views

Parametric equation of inward pointing half sin waves

Parametric equation of inward pointing half sin waves I can create a circle in red and I can create a sin wave that goes around a circle in green. Parametric equation: ...
Rick T's user avatar
  • 467
2 votes
1 answer
81 views

orienting a point in polar coordinates along a particular unit vector

I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle. I wish to parameterize this circle and ...
Stan Shunpike's user avatar
3 votes
1 answer
122 views

Shortest distance between vertex of a circular cone and a quarter of its conical helix

I was given with the question below: ...
kingking's user avatar
  • 118
1 vote
0 answers
40 views

Parametric prolate epicycloid modelling and integration

I was trying to model an epicycloid for my math assignment but none of the parametric equations I found ended up helping me model it on desmos. One of the more prominent equations I found on the ...
TheShadowSider101's user avatar
0 votes
0 answers
69 views

Is there any hope to find solution of the integral $\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx $?

Is there any hope to find a closed-form solution to this integral $$\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx=? \qquad\textstyle{with}\qquad a,b>0 \qquad\textstyle{and}\qquad a,b\in\...
math2021's user avatar
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