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

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1answer
764 views

Find 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 a vector equation and parametric equation of 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 ...
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2answers
21 views

Radius of a wheel based on parametric equations

I am working on a question and I don't have the slightest idea where to begin. Any nudge in the right direction would be very helpful. Here is the question: A bicycle wheel has radius R. Let P be ...
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4answers
34 views

Eliminate $t$ to give an equation that relates $x$ and $y$

I am having problems understanding how to solve the following parametric equation. I have achieved an answer, but am unsure if my answer is correct or not. Eliminate t to give an equation that ...
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0answers
49 views

Constant parametric curves in terms of $x$ and $y$ [closed]

When parameters $\theta, \phi $ are constant for $$ x = \phi \cos\, \theta , \, y = \theta \cos \phi , $$ express them in terms of $x$ and $y$. EDIT1: The corresponding second plot ...
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1answer
6 views

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
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1answer
24 views

Parametric Curve Tangent Equations

Let a curve be given in the parametrized form by: $r(t) = (2\cos t, 2\sin t), 0 \leq t \leq 2\pi$ Find the equations of the tangents to the curve at each of its points $(X_0, Y_0)$. Having gone ...
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2answers
31 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
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1answer
23 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
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0answers
91 views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing." Basically, I mean graphing some function with respect to some other function, rather than with respect to the x-axis. To be more specific, for ...
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1answer
2k views

Find a parameterization for the circle of radius 2 in the xy-plane, centered at the origin, clockwise

Find a parameterization for the circle of radius $2$ in the $xy$-plane, centered at the origin, clockwise. I know to use $2\cos(t)$ and $-2\sin(t)$ but I'm not sure what to do after that
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0answers
29 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
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3answers
34 views

Find the coordinates of the point where the normal cuts the curve again problem

Find the equation of the normal to the curve $x=2\cos\theta$, $y=3\sin\theta$ at the point where $\theta=\frac{1}{4}\pi$. Find the coordinates of the point where this normal cuts the curve again. ...
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2answers
18 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
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0answers
13 views

Finding Extrema of a Function Restrained by a Parametrized Surface

Where on the parametrized surface $r(u,v)=⟨u^2,v^3,uv⟩$ is the temperature $T(x,y,z)=12x+y−12z$ minimal? Find all local maxima, local minima or saddle points. I know that one has to insert $r(u,v)$ ...
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1answer
23 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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2answers
301 views

Line integral around intersection of sphere and plane

The unit sphere is intersected by the plane x + y = 1. Find the line integral of F = around the intersection. $\int\int\nabla$x$F\cdot$ n dA the unit normal vector is easily found by looking at the ...
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1answer
23 views

Parametric equation for a circle centered at $(-5,-4)$ with a radius of $4$: Why is $t$ negative?

The parametric equations for $x$ & $y$ are as follows: $$x=-5+4 \cos (-t)$$ $$y=-4+4 \sin (-t)$$ My question is: Why is $t$ negative in this case? Thanks for any help.
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2answers
25 views

What is the implicit form of $x=cos(t),y=-3+cos(2t)$? [closed]

I know I have to use the properties of the trigonometric functions but I don't know which of them would help me get the answer.
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0answers
517 views

Helix around helix parametric equation?

I know the parametric equation for a $3D$ helix is: $x = R \cos t$ $y = R \sin t$ $z = h t$ Can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix ...
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2answers
27 views

Find point(s) of intersection between a line and a circle whose radius is parameterized by the same variable as the line

Let's assume we have a line: $$\begin{align} x&: x_0 + v_xt, \\ y&: y_0 + v_yt \end{align}$$ and a circle $$\begin{align} x&: X_0 + kt\cos(s), \\ y&: Y_0 + kt\sin(s).\end{align}$$ ...
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3answers
37 views

How a formula is developed

The rule for converting line equations to parametric equations is: $$\frac{(x-x_1)}{a} =\frac{(y-y_1)}{b} =t$$ I would like to know how this was developed. Thank you.
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1answer
1k views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
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1answer
303 views

Ray Disk intersection

So if I have a ray parameterized as $O + tD$ where $O$ is the origin, $D$ is the direction and $t$ is the parameter variable and a flat circular disk with a center point $P$ in 3D space and a radius ...
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1answer
49 views

Shannon entropy on the simplex

The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And ...
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1answer
30 views

Find the end points of a line segment in 3D space

I have a line segment in 3 dimensional space (x,y,z), and I want to find the 2 endpoints of this line segment. Is there a systematic way of doing this? To be specific, I have the line described by ...
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2answers
336 views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
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0answers
23 views

Finding twice-differentiable function of x of a parametric curve when dx/dt = 0

We're working on finding tangents of parametric curves and I feel like this problem isn't as hard as I'm making it out to be, but I am completely stumped. I am given this information: Given ...
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1answer
35 views

Finding an equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$

How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface. While reading the solution of ...
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2answers
2k views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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1answer
142 views

parabola in homogeneous coordinates

So if I have the parabola $Y = X^2$, how do I go about representing this homogeneously? I know I can parameterize it as $F(t) = (t, t^2)$, but then what? The reason I ask is because I have a ...
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1answer
324 views

Parameterizing an ellipse

Given the ellipse $(x-1)^2 + \frac{y^2}{4}= 1$, parametrize the curve in polar coordinates. I've forgotten something very basic here. Can someone help get me started?
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0answers
15 views

Find intersection between conotur point list and a line

Given: List of points representing a closed contour Task: Choose a random point on the contour and shoot a ray inside the contour and determine where the ray intersects the contour. This needs to be ...
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1answer
109 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
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2answers
2k views

distance between parametric line and a point $(4,3,s)$

I've tried solving this problem every way I know how and I just can't get it. I've looked at similar problems of this type, and I still cannot get an answer that seems right. Parametric Equations: ...
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0answers
14 views

Finding the points of the curve in $R^3$ where the tangent is perpendicular to a line

Given the line L defined by $y=2z,x-z=3$, find the points of the curve $C$ given by $x(t)=-4t^2,y(t)=\frac{7}{6}t^3,z(t)=t+3$ where its tangent is perpendicular to L. Progress so far: $ x - z = 3 ...
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0answers
23 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
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4answers
169 views

Parametric to implicit form of a curve

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$" How can I clear $t$ to arrive at the implicit equation?
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1answer
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Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
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1answer
58 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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1answer
20 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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1answer
391 views

Find the area of the surface obtained by rotating the curve of parametric equations

Rotate about the $x$ axis $x = 2t-2/3t^3$ $y = 2t^2$ $0 \leq t \leq 1$ I did the integral of $\sqrt{(2-2t^2)^2+(4t)^2}$ and got $(2x(x^2+3))/3$ and then I did the integral of $2\pi 2t^2 ...
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1answer
42 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
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1answer
23 views

Slope of a Parametrized Curve

Say that we have the parametrized curve $x=e^{3t}, y=te^{-t}$. What would be the slope of this at the point $(1,0)$ and also on which points on the curve would the curve be horizontal? What I have ...
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2answers
22 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
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0answers
28 views

Is it possible to turn the parametric equation of a line in 3 dimensions into the general equation?

I Know it is impossible to do so since the parametric equation for a plane is the intersection of $2$ planes.For example: $x$ $=$ $\frac{-5}{4t}+\frac{1}{4}$; $y=\frac{3}{4t}+\frac{5}{4}$; $z=t$ ...
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Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: ...
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2answers
92 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
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1answer
60 views

Show that $Y^2-X^3\mid f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R}[X,Y]$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
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1answer
33 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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1answer
36 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...