Questions tagged [parametric]

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

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Is $-\sin(t)\cos(t)$ a parabolic function of $-\sin(t)+\cos(t)+1$?

Suppose we have the following functions with shared parameter $t$: $$x(t) = -\sin(t)+\cos(t)+1$$ $$y(t) = -\sin(t)\cos(t)$$ When we plot them together as a planar curve we can see what appears to be ...
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Does there exist $a,b,c\in \mathbb{R}$ s.t. $\cos(t)\sin(t)t = a \exp \left\{ \left( \frac{\cos(t)+\sin(t)+t-b}{c} \right)^2 \right\}$?

Suppose that we have two functions of a shared parameter $t$: $$x(t) = \cos(t)+\sin(t)+t$$ $$y(t) = \cos(t)\sin(t)t$$ When I plot $x(t)$ against $y(t)$ I get the sense that the graph might be 'half-of-...
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How to determine the equation of a conical helix from any point on the cone to the base of the cone?

Can anyone help me to determine the parametric equations of a helical cone from any arbitrary point A on the cone such that the helix starts at point A and finishes at the bottom of the cone.
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8 votes
2 answers
410 views

Find a rectangular equation from a parametric equation. Why is my approach wrong?

Question I have to find the rectangular equation for $x = \dfrac{t+1}{t}$ and $y = \dfrac{t - 1}{t}$. If I solve for $t$ in terms of $x$, I get $t = \dfrac{1}{x - 1}$, and I substitute this into $y$, ...
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When are two space curves with same image equivalent?

Let $I_1, I_2$ are two non-degenerating intervals of $\mathbb{R}$ and, let $\gamma_j : I_j \to \mathbb{R}^n,\quad j=1,2$ be two parametrized regular $\mathcal{C}^r$-curves with same trace (image in $\...
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3 votes
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Finding the vertex of the parabola parameterized by $p(t)=P_0+P_1t+P_2t^2$ for vectors $P_0, P_1, P_2$

A parabola can always be described in parametric form by position vector $p(t)$, $p(t) = P_0 + P_1 t + P_2 t^2 $ where $P_0, P_1, P_2$ are vectors in $2D$ or $3D$. I would like to prove that the ...
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Do a line and capsule intersect if mininum distance between the two lines is less than the capsule radius

In $R^3$ space say I have a capsule shape defined by two end points $a$ and $b$ and radius $r$. And a line define by parametric equation $p+tq$. I can also define a line that goes through the capsule ...
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Is there a closed formula for Vonnegut's "chronosynclastic infundibulum-ated" Winston Niles Rumfoord?

In The Sirens of Titan, a major plot point surrounds the astronaut Winston Niles Rumfoord who purposefully steered his spaceship into a "chronosynclastic infundibulum" and consequently was ...
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Find the surface area of cylinder between the intersection curves and cone, eliptic paraboloid.

The parametric equation of the eliptic paraboloid: $${r}(u,v) = \left(u\cos(v); u\sin(v); \dfrac{1}{4}u^2-\dfrac{21}{4}\right)\qquad 0 \leq v \leq 2\pi,\,\,0 \leq u \leq +\infty$$ The parametric ...
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For the parametric equation $ x = 3 \left( \theta - \sin \theta \right) , y = 3 \left( 1 - \cos \theta \right) $ find the derivative

Apologies if this question is a bit too easy for everyone. For the curve defined by the parametric equations $ x = 3 \left( \theta - \sin \theta \right) , y = 3 \left( 1 - \cos \theta \right) $ for $ ...
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Why Bezier Curve is not undefined at t=0 and t = 1? [duplicate]

Sorry,this might be a dumb question, but I couldn't really understand it. If we define Bezier Curves as: $B(t) = \displaystyle\sum_{i = 0}^{n} P_i\binom{n}{i}t^i(1-t)^{n-i}$ when t and i are zero it ...
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Equivalent parametric solution sets

I have this question that I've never seen before, I've only ever learned about how to find the parametric version of a solution set, but I've never learned how to change it in any way... I will type ...
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Continuity of an integral with parameter-dependent domain of integration

I have an integral of the form: $$g(s)=\int_{D(s)}f(x)dx,$$ where $x\in\mathbb{R}^n$, $s\in\mathbb{R}$, $D(s)$ is a parameter-dependant subset of $\mathbb{R}^n$, $f$ is a "nice" function (i....
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"applying" a function to another

So I have no idea if this has a name but my goal was to make a formula for kind of "bending" a fuction, say f, by another, g. I managed to figure out the parametric equations for it but can'...
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How do I calculate second derivative of involute of circle? I found its first derivative is tantheta but I have no clue how to proceed further

Parametric equations of this involute curve are given as 𝑦 = 𝑎(sin 𝜃 − 𝜃 cos 𝜃) and 𝑥 = 𝑎(cos 𝜃 + 𝜃 sin 𝜃).
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Equation of surface

Suppose we have a surface f(x, y) =4, 0<x<4, 0<y<4. If we start rotating the surface with constant angular velocity in the z axis by 180 degrees what will be the equation of each plane ...
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How to get the coefficients in a parametric cubic function

Let's say I have 4 points with x and y coordinates. And I want to determine the parametric cubic function: $x(t) = a_x t^3 + b_x t^2 + c_x t^3 + d_x$ and $y(t) = a_y t^3 + b_y t^2 + c_y t^3 + d_y$ So, ...
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2 votes
1 answer
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Cycloid of Ceva - going from polar to parametric curve

Ceva Cycloid polar coordinates form is: $$ r = 1 + 2\cos(2\phi) $$ I found that the relation between polar and Cartesian coordinates can be expressed: $$ x = r\cos\phi, y = r\sin\phi $$ I need to ...
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Guideline for study of parametric curves

I apologize if my question is a bit silly. What is the proper way to study and plot parametric curves of the form $\vec{r}(t)=(x(t),y(t))$ like the following one? There is a related question here How ...
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How do I find x(t) and y(t) at the particle given dy/dt and dx/dt?

So, I was given $ \frac{dx}{dt}=4cos(t) $ and $ \frac{dy}{dt}=sin(t) $ at $ 0\le t \le \frac{π}{2}$ I was told that the particle is at the origin at t = 0, and to find x(t) and y(t), the position of ...
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Why does parameterizing a curve with its length yield $|g(T,T)|=1$?

Let $l(t)=\int^t_{t_0}|T|(t')dt'$, where $T$ is a tangent vector to some curve $C(t)$. Why does setting this function as a parameterization of the curve $C$, hence letting $l(t)=\psi C(t)$, imply $|T|^...
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Why do we need tangent vector unequal to zero for smoothness of a vector function?

]1 My textbook gives this definition of smoothness of a $\vec r(t)$ on an interval $I$ of $t$. Why do we need $\vec r'(t) \neq\vec0$ on $I$?
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Expressing parametric equations of a spiral in terms of X coordinate

I'd like to draw the spiral as a bitmap graphics. Given a simplified parametric form of an Archimedean spiral for x and y as functions of t: ...
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Proper way to study parametric curves

What is the proper way to study curves defined parametrically? For the sake of convenience, say the curve defined by the set of equations: \begin{equation} \begin{cases} x(t)&=t^4+4t\\ y(t)&=t^...
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Intersection of two parametric surfaces (governing equation)

in order to solve the interaction between two parametric surfaces (represented as Bezier oder B-Splines) i need to "solve" the non linear equation system. As both surfaces are depended in ...
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How can I find the acceleration vector of a particle that sweeps out equal area in equal time?

I am given the parametric equations of the path in which the particle travels. $x=\frac{c\cos t}{1+e\cos t}$ and $y=\frac{c\sin t}{1+e\cos t}$, where $c$ is a constant and $e$ denotes the ...
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2 votes
1 answer
30 views

Comparing a parametric curve lying in the plane to a curve $y = f(x)$

How can one compare if a parametric curve lying in the Cartesian plane lies above or below a curve given by $y = f(x)$ on some interval? Specifically, how can one show that the cycloid given ...
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Given the initial conditions of a satellite determine its orbital motion

I'll be honest it has been a while since I took diffeq. That said I figure it shouldn't be too hard to to solve this. $$ P_0 = (Longitude,Latitude,Altitude) \\ V_0 = (Speed,InclinationOffEast,...
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Parametric Equations on $x= 1 + \sin (\pi t),$ $y= 3 \sin (\pi t)$

Consider a particle following the parametric equations \begin{align*} x &= 1 + \sin (\pi t),\\ y &= 3 \sin (\pi t). \end{align*}a) Give a precise description of the graph of these parametric ...
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1 answer
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When converting to parametric equation, should plus-minus sign be used?

I'm following a tutorial which says this: Convert the following equation to a pair of parametric equations for $x$ and $y$ in terms of $t$: $$y=x^2+3$$ Step 1 - Set $t$ equal to $x^2$: $$t=x^2$$ Step ...
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Parametrising the intersection of a double cone and a plane?

As the title states, I am struggling to parametrise the hyperbola resulting from intersection of a double cone and a plane. The equation of the cone is given as $z^2=x^2+y^2$ and the equation of the ...
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3 votes
0 answers
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How to determine the properties of a self intersecting curve (parametric functions) : number of loops and of passages through a point?

Let's define a parametric vector $(f(t),g(t))$. If there a way to determinate how many times this function will come back to a certain point ? For instance here is the plot of $(f(x),g(x))=\left(-2 ...
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Simple curve constructed from a parabola - can it be expressed explicitly?

Some years ago I thought of a problem that I've returned to frequently but never been able to solve. Maybe it's impossible, I'm not a mathematician. I thought I should ask some experts rather than ...
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1 answer
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Prove a relation for radius of curvature, using parametric coordinates

I have to prove that for any curve, the radius of curvature, $\rho$ , the following relation is true where $x = f(t)$, $y = \phi(t)$, i.e., the parametric form is to be used $$\frac{1}{\rho^2} = \frac{...
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Area Differential of Parametric Surface

Given a parametric surface of the form $x_i=g_i(u_1,u_2)$, how would it be possible to express $dS$ (the differential of the surface area) in terms of $du_1$ and $du_2$? I know the final result but I ...
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Can you decompose any random parametric curve with Fourier series?

Imagine a random parametric curve in $\mathbb{R}^n$. Can the parametric equation always be expressed as a sum of sines and cosines, as a Fourier series? Does anyone know a good introduction to how to ...
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4 votes
2 answers
315 views

Collision of Ball in Triangle

A ball with position and velocity $(P_0,V_0)$ is in a triangle. Which side of the triangle it will hit? Calculations: The ball's motion is $$L( t) = P_0+t*V_0 = \ ( V_{0_{x}} t+P_{0_{x}} ,V_{0_{y}} t+...
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Parametric equations for 3D surface

I have an equation x² + y² + z³ - z² = 0 This is an alpha-loop rotated around the z-axis. Can someone please help me to convert this to three parametric equations for x, y, and z (varying for u and v)?...
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Is an implicit representation of 3D non-planar curve possible?

In a book that I am reading, Polygon Mesh Processing (page 1, last paragraph), the authors say this: [...] implicit definition is only available for planar curves, i.e., $\mathcal{C} = \{x \in \...
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What wrong am I doing while writing these parametric coordinates?

I kinda understand that it should be $-60^\circ $ in the second picture. But I am getting that after looking at the result. I can't exactly understand what wrong I am doing. PS: The length of the line ...
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Find the parametric equation for the tangent line to the intersection curve between an ellipsoid and a paraboloid?

We got various problems in this site asking for similar problems btw an ellipsoid and a plane. What if it's btw an ellipsoid and a paraboloid? I got the equation of both surfaces: $4x^2 + y^2 + z^2 = ...
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How to get the moment of the intersection between an accelerating object's parametric curve and a circle

During a C# project of creation of a 2D collider engine I ended up falling into a problem that is way beyond my reach. One point with a given starting position, a given starting velocity and a given ...
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2 votes
2 answers
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Describe the locus of $w$ if $w=\frac{1-z}{1+z}$ and $z=1+iy$ (i.e $z$ is a complex number that moves along the line $x=1$)

So I'm trying to solve the following problem: If $z=x+iy$, express $w=\frac{1-z}{1+z}$ in the form $a+bi$ and hence find the equation of the locus of $w$ if $z$ moves along the line $x=1$. My attempt ...
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Solving Diophantine equation using parametric

I'm learning about the parametric method to solve the Diophantine equation. But I don't know how to get $x,y,z$. For example: Let $m$ and $n$ be distinct positive integers. Prove that there exist ...
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Finding a parametric equation for an implicit cartesian curve

The title really says it all: I want to know how to find a parametric equation for a curve defined by an implicit Cartesian equation. I would imagine that this is impossible in general, but I'm ...
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How do I plot in python a cone for which I have the vertex point, line that is the central axis and opening angle?

I managed to get the equation of a cone for which I have the vertex, opening angle and central axis which the cone must revolve around. A sample is attached below for which they get the equation of a ...
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Given a table of values $x$, $y$ and $z$, find a formula that expresses $z$ as a function of $x$ and $y$

I'm not really versed in this kind of mathematics, but I ran into this problem while programming, so I find myself turning to the hivemind. (Thus I don't even know the proper tags to use here... Feel ...
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How to tell that a parametric curve does not intersect itself?

For example: if a curve is defined by $x=3\cos{t}-\cos{3t}$, $y=3\sin{t}-\sin{3t}$, $0\le t \le \pi$ then $\frac{dx}{dt}=-3\sin{t}+3\sin{3t}$, and $\frac{dy}{dt}=3\cos{t}-3\cos{3t}$ So, we can not say ...
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Area between 2 3D parametric curves?

Remotely similar to this but what if both curves ($r_1$ and $r_2$) are both 3D, both depend on one common parameter (let's say $t$), and both are on the same surface in $\mathbb{R}^3$ (e.g. $\mathbb{S}...
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How parametric deformation retraction formula is obtained (Hatcher's Algebraic Topology)

Hatcher Chapter $0$ Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian ...
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