# 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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### 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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### 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....
1 vote
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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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### 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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### 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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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: $4x^2 + y^2 + z^2 = ... • 711 0 votes 0 answers 21 views ### How to get the moment of the intersection between an accelerating object's parametric curve and a circle During a C# project of creation of a 2D collider engine I ended up falling into a problem that is way beyond my reach. One point with a given starting position, a given starting velocity and a given ... 2 votes 2 answers 47 views ### Describe the locus of$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$. My attempt ... • 1,073 0 votes 0 answers 40 views ### Solving Diophantine equation using parametric I'm learning about the parametric method to solve the Diophantine equation. But I don't know how to get$x,y,z$. For example: Let$m$and$n$be distinct positive integers. Prove that there exist ... • 55 0 votes 1 answer 23 views ### Finding a parametric equation for an implicit cartesian curve The title really says it all: I want to know how to find a parametric equation for a curve defined by an implicit Cartesian equation. I would imagine that this is impossible in general, but I'm ... • 401 0 votes 0 answers 37 views ### How do I plot in python a cone for which I have the vertex point, line that is the central axis and opening angle? I managed to get the equation of a cone for which I have the vertex, opening angle and central axis which the cone must revolve around. A sample is attached below for which they get the equation of a ... 0 votes 0 answers 28 views ### Given a table of values$x$,$y$and$z$, find a formula that expresses$z$as a function of$x$and$y$I'm not really versed in this kind of mathematics, but I ran into this problem while programming, so I find myself turning to the hivemind. (Thus I don't even know the proper tags to use here... Feel ... • 3 0 votes 2 answers 62 views ### How to tell that a parametric curve does not intersect itself? For example: if a curve is defined by$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 ... 0 votes 0 answers 33 views ### 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}...
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 ...