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

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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
29 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
24 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
357 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 ...
0
votes
0answers
27 views

Rotating a 2d equation in a 3d space? [closed]

I'm attempting to rotate a shape taken from a parametric equation. I'm doing this inside a Java program, so it automatically adjusts the variables as I need. I have: $t = \pi$ but t decreases in ...
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
25 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
18 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
976 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
44 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
158 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
29 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
24 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
563 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
174 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
3k 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
38 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
37 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
37 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
46 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
70 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
439 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
26 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
43 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? $$\frac{4t-1}{\cos(t)}$$
2
votes
2answers
977 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 ...
3
votes
1answer
66 views

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: ...
0
votes
2answers
35 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
41 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
63 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 ...