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

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5
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1answer
40 views

For what $n$ is $\sum_{i=1}^\infty \frac{\cos (it)}{i^n}$ bounded and why doesn't a sine behave the same way?

I've been looking at a parametric curve $$\pmatrix{X \\ Y}=\pmatrix{\sum_{i=1}^N \frac{\cos (it)}{i^n} \\ \sum_{i=1}^N \frac{\sin (it)}{i^n}}$$ where, for the plots below, $N$ runs from $1 \rightarrow ...
0
votes
1answer
12 views

Is this a correct parametrization of a rectangle on the complex plane?

$z = 3 + i(2t - 1), t \in [0,1) \\ z = 3 - 6(t-1) + i, t \in [1,2) \\ z = -3 + i(1 - 2(t-2)), t \in [2,3) \\ z = 6(t-3) - 3 - i, t \in [3,4]$ I parameterized a rectangle with vertices at ...
0
votes
1answer
6 views

What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
1
vote
2answers
73 views

How to compute $\int_0^2(1+4t^2+9t^4)^{1/2}\text{d}t$?

The original question was: find the length $\ell$ of the curve $\gamma$ given the parametric equations: $$x=t~~~~~ y=t^2~~~~~ z=t^3 $$ from $t=0$ to $t=2$
0
votes
1answer
16 views

Rectangular Hyperbola - Eliminating the Parameter

Question: The point P (2p,2/p) lies on the rectangular hyperbola C with equation xy = 4. (a) Find the equation of the normal to C at P. The normal at P meets C again at the point Q. The mid-point ...
1
vote
2answers
40 views

Speed of a parametric function?

I know speed = |velocity| Why is speed of parametric defined as $$speed = \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$ How is this derived? What is the principle here? Is ...
0
votes
2answers
43 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [closed]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
0
votes
1answer
31 views

parametric equations, finding the range of t

When parametrizing a curve how doe we obtain the range of $t$? For example lets say we have the parametrization: $x(t) = 1+3t$ and $y(t) = 2+5t$. How do we find the range of t? $t\to[?,?]$
0
votes
0answers
22 views

Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a ...
0
votes
0answers
26 views

Parametrisation of the curve after a short time

I am trying to wrap my head around this differential geometry problem. I am given velocity V with components in the principle normal and binormal directions. Then I am given an approximation of the ...
0
votes
0answers
21 views

Calculating the Constant of Integration in Parametric, Vector-based Equations

I'm having trouble finding the constant of integration in parametric, vector-based equations. Given an equation: $$ a(t)\ =\langle \cos(t),\ \sin(t)\rangle $$ and $$ \int\ a(t)\ dt\ =\langle 0,\ ...
0
votes
1answer
26 views

Sketching a parametrised cone and a geodesic lying on it.

I just started a new module at University and I am having some trouble with parametrisation. I am given a parametrisation of a geodesic lying on a cone in notation $r(t)=x(t){\bf i}+y(t){\bf ...
1
vote
1answer
19 views

Show two parametrizations to be equal

Given the two curves \begin{align*}&\mathcal{C}\left\{\begin{matrix}u = t\\v = t\end{matrix}\right., & t\in [0,1]\\ \\ &\mathcal{C'}\left\{\begin{matrix}u = t^3\\v = ...
1
vote
1answer
45 views

Angle between position and velocity vectors is constant?

Is there a name for such a curve or can this even happen? I know when the velocity vector, $\mathbf{x'}$, and position vector, $\mathbf{x}$ are always orthogonal $\mathbf{x}(t)$ parametrizes a circle ...
0
votes
1answer
160 views

Find a parametrization of a hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$

Find a parametrization of the hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$ where $a,b$ are real numbers. I'm not sure about my answer: $$y \begin{pmatrix} -1\\ 1\\ 0\\ 0 ...
1
vote
1answer
27 views

Find the parametrization of the curve resulting from intersection of two surfaces

The question reads as follows: Find a parametrization of the curve resulting from the intersection of the surfaces: $z = x^2 - y^2$ and $z= x^2 +xy - 1$ My attempt: (Use y = t as a parameter, so ...
1
vote
0answers
31 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of a sinus modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ ...
0
votes
0answers
17 views

Parametric integration of curves

If you were to integrate this curve between $t=\pi$ and $t=\frac{5\pi}{6}$ what area would you be working out? Also if you integrated this curve between $t=\pi$ and $t=\frac{2\pi}{3}$ what area ...
2
votes
1answer
19 views

Calculating the surface area of revolution for parametric equation.

I solved a problem using a method that's completely different from the mark scheme and I got the right answer, but I'm unsure whether or not it's just some coincidence. Here's the question: The ...
0
votes
2answers
46 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
-1
votes
1answer
39 views

Solving matrix using Gaussian elimination and a parameter

$\begin{bmatrix} x_{1} & 2x_{2} & & & ax_{5} & x_{6} & = & -2 \\ -x_{1} & -2x_{2} & & & (-1-a)x_{5} ...
1
vote
1answer
38 views

How to find a number of integral solutions (all $x$)

If $A$ is between $[1..9000]$ $$A*X = 1 \pmod{9000}$$ All parameters are integers. I have found some solutions: $$A = 6907, X = 43,$$ $$A = 7111, X = 991$$ But I don't know how to find all $x$. I ...
0
votes
1answer
23 views

Convert to cartesian?

How would I convert $X(t)=\cos(t)a+\sin(t)b$ to cartesian, where $a=(3,3)$ and $b=(-1,1)$. I tried saying $x(t)=3\cos(t)-\sin(t)$ and $y(t)=3\cos(t)+\sin(t)$ but I am stuck on how to remove the $t$.
-2
votes
1answer
26 views

Finding point of intersection between 2 parameterised lines

Given the problem of finding the intersection of 2 parameterised lines L1: $x=2-t ; y=1+t$ and L2: $x=2+t ; y=4+t$. Recovering original eqns $y=3-x$ and $y=2x$ yields the correct answer of ...
0
votes
1answer
26 views

Finding a system of equations that defines a line $r$

I'm trying to get a system of equations that defines the following line $r$. Here is its parametric equation $r: (1,2,3)+t(3,1,2) \mid t \in \mathbb{R}$ To find the two equations I calculate the ...
1
vote
3answers
45 views

express $\frac{\sin 3a}{\sin a}$ with only $\cos a$

How can I express $\frac{\sin 3a}{\sin a}$ while using only $\cos a$? Thanks in advance
1
vote
0answers
49 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
2
votes
2answers
71 views

How to prove parametric equation of a ellipse

The parametric equation of a ellipse is $$x=a \cos t\\y=b \sin t$$ It can be viewed as $x$ coordinate from circle with radius $a$, $y$ coordinate from circle with radius $b$. How to prove that it's ...
1
vote
0answers
28 views

Collinearity of three points on a curve.

In the realm of elliptic curves, the collinearity of three points is of a fundamental importance because this condition allows us to define on the curve a law of Abelian group, the study of which is ...
0
votes
2answers
33 views

How to solve this parametric linear equation?

How to solve this parametric linear equation? I need to find all numbers for $\alpha$ with which has a single, infinity or none solution. $$ \left[\begin{array}{rrr|r} \alpha & 1 & 0 ...
1
vote
2answers
38 views

How to use parametric equation/trigonometric identity to show an ellipse?

I have the equation $16x^2+25y^2=400$, and the parametric equation $(x,y)=(5\cos t, 4\sin t)$. If I plug in the parametric equation into the first equation, I end up with the trigonometric identity ...
2
votes
6answers
102 views

If $a^2 + b^2 = 1$, show there is $t$ such that $a = \frac{1 - t^2}{1 + t^2}$ and $b = \frac{2t}{1 + t^2}$

My question is how we can prove the following: If $a^2+b^2=1$, then there is $t$ such that $$a=\frac{1-t^2}{1+t^2} \quad \text{and} \quad b=\frac{2t}{1+t^2}$$
0
votes
2answers
54 views

Calculus problem of finding the equation of a line.

Find the equation of a line that passes through the origin, with positive slope, and its tangent to the parabola given by :$ y = x^2 - 2x + 2$ My approach to this problem was to differentiate the ...
0
votes
2answers
59 views

Parametric Trig Functions

A closed curve in the $(x, y)$-plane is represented by the functions $$x(θ)=\frac12(\cos \theta +\sqrt2 (\sin \theta))$$ $$y(θ)=\frac12(− \cos \theta +\sqrt2 (\sin \theta))$$ where the parameter ...
2
votes
2answers
44 views

How to find the points in which a given curve intersects itself?

Apologies in advance for my lack of knowledge with *tex. Hi everyone and thanks for any sort of help! I am given the following parametric curve: $(t^2\cos t, t^2\sin t,t^2), \text{where} -2\pi \le ...
0
votes
0answers
12 views

Parametrized curve with adjustable plateau

I am trying to create a parametrized curve. Basically I want a monotone curve through $(0, 0)$ and $(u, 1)$ with a plateau at $(ru, s)$ with $u\gt 0$ and $r,s\epsilon[0; 1]$, so my constraints are ...
0
votes
1answer
28 views

How to represent 2D lines (i.e. on $x$ and $y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric $(x,y,z)=f(u,v)$

How to represent 2D lines (i.e. on $x$ and $y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric $(x,y,z)=f(u,v)$ Hi, I've been working on a Simplex problem and would like to ...
2
votes
3answers
36 views

Find the derivative $dy/dx$ from the parametric equations for $x$ and $y$

Let \begin{cases} y=2t^2-t+1 \\ x=\sin(t) \end{cases} find $\frac{dy}{dx}$ Is this all that I need to do? $$\frac{4t-1}{\cos(t)}$$
0
votes
2answers
36 views

Parametric Equations. Find $\frac{dy}{dx}$ in terms of $x$

Find $\frac{dy}{dx}$ in terms of $x$ if the parametric equations of a curve are given by $x=e^{\sqrt{4t}}$ and $y=\sqrt{e^{6t}}$. My attempt, I found $\frac{dx}{dt}=\frac{e^{2\sqrt{t}}}{\sqrt{t}}$ ...
0
votes
1answer
17 views

Parametric Equations (Concavity)

The question is: A curve is defined by the parametric equations $$ x = t^2 + a $$ $$ y = t(t-a)^2 $$ Find the range of values for t in terms of a where the function is concave up? What I have ...
0
votes
0answers
41 views

I don't understand these directional vectors

I'm currently practicing for my final calculus exam in 20 days time, it has a vector section, here is my problem. When I need to find these parametric equations for either lines of planes, I need a ...
0
votes
2answers
43 views

Path of a cycloid

In this question, it's said that the path of a cycloid can be given as this parametric equation: $$\begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*}$$ and is shown here: ...
1
vote
1answer
66 views

Parameterization of a torus

Given that the parameterization of a torus is given by: $x(\theta,\phi) = (R + r\cos(\theta))\cos(\phi)$ $y(\theta,\phi) = (R + r\cos(\theta))\sin(\phi)$ $z(\theta,\phi) = r\sin(\theta)$ and the ...
1
vote
0answers
31 views

How do I detect collisions in a parametric equation$?$ (Lissajous curve)$?$

Let's say you have a simple parametric equation where $x= \sin(3t)$ and $y=\cos(7t).$ This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure. You can ...
-1
votes
3answers
48 views

Parametrization of a hypocycloid

How do I prove that a hypocycloid, which has equation $$x^{2/3} + y^{2/3} = a^{2/3}$$ can be parameterized by $$x = a\cos^3(\theta),\qquad y = a\sin^3(\theta)$$? The problem assumes that it is true, ...
0
votes
1answer
17 views

Parametric equation for $x²+4y² =1$, $x²+z^2=1$? ($y\ge 0, z\ge 0$)

In order to find this parametric equation, I would do: $$z=\sqrt{1-x²}\\x=\sqrt{1-4y²}\implies \\z = \sqrt{1-(1-4y²)} = \sqrt{1+4y²}$$ Then, if I choose $x=t$ as a parameter, I get: $$x=t, ...
3
votes
3answers
48 views

How to find parametric equation of the intersection $x²+y²+z²=2$ and $y=x$?

I know that if I substitute $y=x$ into $x²+y²+z²=2$ I get $$2x²+z²=2$$ which in some way gives me $$x²+\frac{z²}{2} = 1$$ which is an ellipse. My parametric equation goes from $(0,0,\sqrt{2})$ ...
0
votes
0answers
26 views

Parametrization of $x^2-y^2=1$; Why does the following only cover the “right-hand side”?

I have the parabola $x^2-y^2=1$. I am asked to first show that $\cosh(t)=x,\sinh(t)=y$ parameterizes the curve. This, I have done; substituting to the implicit function expression gives $1$. But I ...
1
vote
1answer
22 views

Find a Function That is in The Shape of the Given Boundary

John lives 2 miles north from a road, which is separated from John's house by a grove. If he walks from his house to the road along any straight line, the last mile of his walk is through the ...
1
vote
1answer
19 views

Different forms of representing curves

I am reading about mathematical representation of curves and have come across following points which I can't seem to understand : 1) Why Explicit representation cannot be used to represent closed ...