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

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0
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1answer
28 views

Can you give a general expression for the rate of rising water poured in an object?

Given an 3-d object defined by a set of parametric equations (x, y, z), can you write a formula expressing the rate that a liquid rises as it's poured into this object at a constant flow rate? Assume ...
0
votes
2answers
44 views

Removing parameter $t$ from $z$-axis

How do I remove the parameter $t$ from the $z$-function in the following: $$\begin{align}x&=a\cos{t}-a\\ y&=a\sin{t}\\ z&=nt\end{align}$$ (where $n,a$ are arbitrary coefficients) So far I ...
3
votes
2answers
163 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 $$ x'(t)^2+y'(t)^2=1 $$ ...
0
votes
1answer
69 views

How can I increase the resolution of a 3d plot in Sage to see all the details of the noodle? [closed]

I want to plot my noodle with a high resolution to see all the details. How can I achieve this with Sage? Here is my parametric, piecewise function: ...
2
votes
3answers
85 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 ...
1
vote
2answers
50 views

Find a parametrization of the intersection curve between surfaces

Find a parametrization of the intersection curve between the surfaces $−3x^2+2z=10$ and $4x^2+10y^2=5$. You should parametrize such that $y=k\sin(t)$ for some constant k. The answer should be in ...
1
vote
1answer
117 views

Gradient of a rational Bezier curve

I'd appreciate help working out the gradient of a rational Bezier curve $C = (\,x(t) \,, \,C_y(t) \,)$. I know that the gradient $g$ of a the parametric curve is $$ g(t) = \left( \frac{dy(t)}{dt} ...
1
vote
1answer
78 views

Find the shaded areas A1, A2, A3

I think i know how to find the angles KAG and KAH that's what i did: (this is the picture from my assignment sheet) then i have to find the shades areas A1,A2 and A3 but i don't know how to ...
0
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0answers
17 views

Derivative of a parametrized vector on a nonfixed basis

Suppose a curve defined by a vector parametrized through the variable $u$, and expressed on a non-fixed base, like the polar coordinates base. You derive it with respect to that parameter. What ...
1
vote
1answer
271 views

Coordinate of intersection between line and square

TL;DR given a square and a point $p$, I need the intersection between the perimeter of the square and a ray cast from the center of the square through point $p$. This is my approach so far, but I will ...
1
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1answer
84 views

sketch the line segment whose parametric equations

Sketch the line segment whose parametric equations are $x=2+t, v=t^2-1, t∈[0,3]$ That's what i did $t=x-2$ $v=(x-2)^2-1$ $v=x^2-4x+3 $ $v=(x-3)(x-1)$ $x=3,1$ and that's my sketch I am not ...
0
votes
1answer
151 views

Find a vector equation and parametric equations or the line in R^3 that passes through the point (1,2,-3) and is parallel to the vector u=(4,-5,1).

Find a vector equation and parametric equations or the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
2
votes
1answer
50 views

Quick Parameterization Help

I've been trying to paramaterize $$x^2+y^2+z^2=9,\ x^2-y^2=3$$ but haven't had any luck. I was thinking to let $x=\sqrt{3}\ \sec(t), y=\sqrt{3}\ \tan(t)$ to complete the identity on the right ...
0
votes
1answer
172 views

Find a vector equation and parametric equations of the line in $\mathbb{R}^2$ passing through the origin and is parallel to the vector $\vec{u}=(2,3)$

anyone can help me? :< Are there any equations that I could use in this question? I am so confused. I only know how to do the question if it changes "parallel" to "perpendicular" because I only ...
0
votes
2answers
194 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...
0
votes
1answer
18 views

Parametrizing curve with not only one peak

I obtained experimental data (thermal analysis) and need to parametrize the resulted curves for modeling. An example of two curves obtained: I tried to use a Weibull distribution, but since I ...
1
vote
1answer
118 views

Circle On Sphere

Sorry if this sounds too silly but my math skills are very poor and I just need this problem fixed. I made this graphic with geogebra 3D and it was quite easy there but I don't know how to write ...
2
votes
2answers
68 views

When does this parametric curve cross itself?

Find the points where the curve given parametrically by$$\mathbf{r}(t)=\left(2+\cos\frac{3}{2}t\right)\left(\begin{matrix}\cos t\\\sin t\end{matrix}\right)$$crosses itself. So, I understand that ...
2
votes
1answer
55 views

There exists a constant arc length parametrization

I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. ...
0
votes
1answer
510 views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a ...
1
vote
1answer
101 views

Tangent Line of a Parametric Curve

Deduce the equation of the tangent line to the curve defined by the equations x=cosh(t), y=sinh(t), and z=ct I have somewhat of a good grip on the definition of a ...
2
votes
1answer
61 views

Find slope of a curve without calculus

Is it possible to find the slope of a curve at a point without using calculus?
1
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0answers
133 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
1
vote
0answers
37 views

Line integral, Parametrization

I have this line $A=\{(x,y) \in R^2 : y^2+4x^4-4x^2=0\}$ , $(x>0)$ I parametrized it like that : $b(t) = (t, \sqrt{4t^2- 4t^4})$. And my $F$ is $F(x,y) = (x+y,-x)$. But when I calculate my ...
2
votes
2answers
57 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 ...
0
votes
2answers
97 views

Cartesian equation of $ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $

I have this parametric equation: $$ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $$ and I have to obtain the Cartesian equation. Any ...
1
vote
1answer
36 views

can the derivative of a closed complex contour at any point be zero?

If C is a closed contour in the complex plane parametrized by z(t)=u(t)+i*v(t), can there be any point where z'(t)=0?
1
vote
1answer
55 views

Parametrize $|x|+|y|+|z|=1$

How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize. I think we may need to divide it into 8 pieces and consider them respectively.
1
vote
1answer
978 views

find parametric equations for the path a particle that moves along the circle $x^2+(y-1)^2=4$

Find parametric equations for the path a particle that moves along the circle $$x^2+(y-1)^2=4.$$ In the manner describe a) One around clockwise starting at $(2,1)$ b) Three times around ...
0
votes
2answers
34 views

Show that the parametric equation $ x=x_1+(x_2-x_1)t , y=y_1+(y_2-y_1)t$

Can anyone help me to solve this? Show that the parametric equation $ x=x_1+(x_2-x_1)t $ $ y=y_1+(y_2-y_1)t\ $ with $(0\le t\le 1)$ describe the segment that joint the point $P_1=(x_1,y_1)$ and ...
0
votes
2answers
2k views

how to convert this parametric equation into a Cartesian equation.

I did not know how to answer this question Sketch the curve by using the parametric equation to plot points. indicate with arrow the direction in which the curve is traced as t increases $x=t^2+t$, ...
1
vote
1answer
54 views

Finding the length of a parametric curve

$$x=\frac{t^2}{2} \text{ , } y=\frac{(2t+1)^{3/2}}{3} \text{ , } 0 \le t \le 20$$ The formula for the length of a parametric curve is $L=\int_{a}^{b}\sqrt{[f'(t)]^2+[g'(t)]^2}dt$. Taking the ...
0
votes
1answer
373 views

Find Equation of a Perpendicular Line Going Through a Point

I have the following parametric equation for line g: $$ x=3t\land y=-7+5t\land z=2+2t $$ I have to find the equation of a line perpendicular to $g$ and going through point $Q(3,-2,4)$ which lies on ...
-1
votes
1answer
56 views

Write $y=\cos x -7$ in parametric equation [closed]

How do I write $y= \cos x-7$ in a parametric equation? I'm not sure how to do this.
0
votes
1answer
51 views

Direction of t (Vector Space)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x = e^{-t}\cos t, y = e^{-t} \sin t, z = e^{-t}; (1, 0, 1). $$ The ...
2
votes
1answer
56 views

Area of a band in $\mathbb{R}^2$

If I have a continuous, and smooth curve $\mathcal{C}$, length $\ell$, in $\mathbb{R}^2$ and at each point on the curve I were to draw a line segment, length $d$, normal to the curve centered at the ...
1
vote
1answer
468 views

Find all points of intersection of the curves $r^2=3\sin(2\theta)$ and $r^2=3\cos(2\theta)$

Find all points of intersection of the curves $r^2=3\sin(2\theta)$ and $r^2=3\cos(2\theta)$. Give your answers as ordered pairs in cartesian coordinates, in order of increasing radius and ...
1
vote
0answers
64 views

Beth needs to make a crossing in her canoe

I have a math problem that has me stumped. I cannot seem to find a good starting point for this, and am flying blind with no check values. Maybe it's end of semester fog, but I'm struggling with ...
0
votes
1answer
45 views

Paremetric surface revolved around y-axis

if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation $2\pi x\sqrt{(x')^2+(y')^2}$ and I'm given $$x=(2/3)t^{3/2}$$ $$y=2\sqrt{2}$$ and I got ...
1
vote
3answers
48 views

How to define a finite objects with parametric equations

I never had seen parametric equations, but while trying to learn line integrals through Wikipedia, quickly found these equations are remarkable. Some can represent things for which more normal ...
-1
votes
2answers
128 views

Identifiying the next point on the surface of a cube ( or 3D object )

I have a cube of unit length. Each face of the cube is divided into 10 x 10 equal segments. Consider an object of size equal to that of a segment moving through the surface of the cube ( or any 3D ...
0
votes
1answer
17 views

Proving that two function coordinates of a parametric curve equals 1

I am having difficulty with this question, Note: This is not homework, It is from a practice test that I am using to study Consider the curve: $x(t) = \frac{1-t^2}{1+t^2}$ ; $y(t) = ...
0
votes
1answer
139 views

Surface Area of a Parametric Curve

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. $$ ...
0
votes
1answer
52 views

When can I use a parameter in equation (of the a plane)

In my book there is an example: Find vector and parametric equation of the plane $x-y+2z=5$ Now, the solution is: solving for $x$ in terms of $y$ and $z$ yields $x = 5+y-2z$ and then using parameters ...
1
vote
0answers
18 views

Parameterisation Q

I'm looking to parameterise the expression $$f(r,\theta)=Usin(\theta-\alpha)(r-\frac{a^2}{r})-\frac{\Gamma}{2\pi}lnr-k=0$$ s.t. $z=re^{i\theta}$. I get a horrible expression if I parameterise for r. ...
0
votes
2answers
528 views

Finding vector and parametric equations provided only one point.

Normally to answer these questions I have a point and one or two vectors. However, for this one I only have a point. How can I concoct these equations provided there is limited information? Find ...
0
votes
0answers
19 views

How to find new co-ordinates for points on a line dragged as a bezier curve.

I have a line with a set of points. I captured the start point and the end point of the line and found two control points for a bezier curve using the linear parametric equation. I construct the ...
0
votes
1answer
24 views

Finding the velocity vector

Am I finding the equation of the slope of the tangent line at c(t)? $\frac{dy/dt}{dx/dt}$ = $\frac{2t}{3t^2-8}$
0
votes
1answer
147 views

Intersection of a parametric curve and a circle

Given a curve defined by a parametric equation $x(t)$ and $y(t)$, how might one calculate the point of intersection with a circle? The derivatives $x'(t)$ and $y'(t)$ are also available if they prove ...
1
vote
1answer
43 views

Integration without using parametrization .

I would like to integrate the following line integral without using parametrization . I wanted to integrate the following $$\int_C \frac{1}{z-a} dz$$ , where $C$ is a a curve along $|z-a| =r$ . ...