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

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5
votes
1answer
32 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 ...
3
votes
3answers
4k views

Parametrisation of the surface a torus

For a calculus question I need to parameterise the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the $z$-axis (with $0<a<b$). I've had a go at this, ...
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 $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
0
votes
1answer
88 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
1
vote
2answers
72 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
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
0
votes
1answer
42 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 ...
0
votes
1answer
2k 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 ...
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
2answers
358 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 ...
14
votes
6answers
9k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated ...
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
0answers
995 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 ...
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
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: $$ ...
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 ...
0
votes
1answer
4k 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
0
votes
1answer
570 views

Reverse direction of parametric equation

For the graph $y = \sqrt{x}$ the normal parametric equations would $x = t^2$ and $y = |t|$. However, the direction for that graph would be going from infinity to zero when $t \leq 0$ and zero to ...
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 ...
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 ...
1
vote
2answers
180 views

Catenary equation in 3D

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is lowest point of the catenary curve. I only know z-coordinate of this third point. I need to find ...
2
votes
2answers
4k 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 $r⃗ (7)=(1,4,7)$, ...
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 ...
-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 $t$.
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
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 ...
0
votes
1answer
448 views

Parametrization of the intersection of a cone and plane.

EDITED with new progress updates. As the title states, I'm trying to parametrize the intersection of a cone and a plane. The equations are: $z^2 = 2x^2+2y^2$ and $2x+y+3z=4\implies ...
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 ...
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 ...