# Tagged Questions

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

108 views

### Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
18 views

### Injective parametrization of a curve. ( piecemeal $C^1$)

$\gamma:$[0,1]$\to R^2$ is an injective parametrization of a curve $\Gamma$, which is piecemeal $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: Show that for every $n\in N$ there ...
36 views

### Converting parametric equations with trigonometric functions into Cartesian form

Ahoy, I am having trouble with a computer-based assignment and the question is as follows: $$x = 2\cos^5 t, \quad y = 2 \sin^5 t$$ Write these in Cartesian form, $F(x,y) = c$. I understand ...
49 views

### Parametric Curves and Tangents

I am struggling with a question regard parametric curves and finding tangents to them but something is going wrong somewhere in the process and I cannot figure out why. The question asks: consider ...
68 views

### Consider the parametric curve given by: $x=3\cos(t)$, $y=t^{3/2}$.

The question asks to find the equation of the tangent to this curve at the point $t=\pi/4$. I've determined $$\frac{dy}{dx} =(\frac{dy}{dt})/(\frac{dx}{dt}) = -0.222$$ Have I got the right idea? ...
50 views

### Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then ...
33 views

### Parametric parabola

I was given my Math C assignment today and the moment I looked at question 1 I knew I had no idea what to do. This is the graph I was given: I was asked to provide an equation for the curve however ...
60 views

### All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
45 views

### Looking for a family of astroids

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the ...
87 views

### 3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
38 views

### Finding answers to system of equations

Let's say we have such a system structure of equations: ...
87 views

### Parametric equation of clock hands

I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to ...
79 views

### Does every algebraic variety admit a local parametrization at every non-singular point?

I am reading a text in which the first sentence of the proof of a theorem is: Let $X(t)=X(t_{1},\ldots,t_{n})$ be a local parametrization of the algebraic variety $X$... I guess that every ...
52 views

### Calculate parametric bounds of a circle in a 2D quadrilateral

Given a 2D quadrilateral defined by the points $(p0, p1, p2, p3)$ and a circle centered at $c$ with a radius of $r$, I want to find a quad in the parametric space of the outer quad that tightly bounds ...
38 views

### Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
73 views

### Finding the parametric form of a standard equation

I need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did this ...
32 views

### Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
28 views

### equation of tangent line and parametric equation of tangent line (are they equivalent)

I am not sure about the steps to finding parametric equations of tangent lines and was wondering if these statements are equivalent Is there a difference if I am asked to find: equations of ...
41 views

### Is there an algorithm for parametrization of equations?

In this and this Math.SE questions askers wanted to parametrize their equations. It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate ...
38 views

### Parametric equation with image of a function.

Find all values of $a$ for which the image of the function $$y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}$$ contains $[2, 3]$. Now, I've already transformed it to ...
15 views

### Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
36 views

### Eliminating the parameter?

How would you eliminate the parameter where the x coordinate is in terms of t, but the t is squared. x= 3t - $t^2$ y= t + 1 I know to solve for y as a function of x, but I'm not sure how to do so ...
48 views

### How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t}$$ $$y=\sin(t) \cdot e^{-t}$$ Show that $dy/dx =tan(t-\pi/4)$. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
105 views

### How to parametrize the curve $y^2 = x^2(x+3)$

I am emberassed to ask this, but I couldn't find a way. I want to write the curve $y^2 = x^2(x+3)$ as $$y=f(t) \quad \quad x=g(t) \quad \quad t \in \mathbb R$$ I guess I have to do something like ...
50 views

### Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
47 views

### How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
3k views

### What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
111 views

### Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
418 views

### Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
26 views

### Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
26 views

### How to write explicity a curve on $S^n$?

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$. I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the ...