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

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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 ...
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3answers
79 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, ...
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1answer
27 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, ...
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3answers
65 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 $(0,0,\sqrt{2})$ ...
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29 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 ...
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1answer
26 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 ...
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1answer
22 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 ...
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1answer
42 views

Second degree parametric inequalities

I am asked to solve the parametric inequality and to find for which values of a every x is a solution. $$ (a+5)x^2 - 2x(a+1) + 2a - 4 \ge 0 $$ So in order every x to be a solution to the ...
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0answers
33 views

Cycloid angular parameter solution to an ODE for density fluctuations

I'm just reading over some Cosmology notes and there is a little ODE solve that I am not quite understanding. I have an equation of the form: $$ \ddot{R}=-\frac{GM}{R^{2}} $$ Integrating gives: $$ ...
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1answer
21 views

Length and revolved surface area of parametric curve

Can someone verify this for me?
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1answer
58 views

Creating a Parametric Equation

The problem is as follows: "In each of the following cases, you will be asked to write down a family of parametric curves that have the property that at $$t = 1$$ we have $$x'(t) = y'(t) = 0$$ but the ...
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22 views

Convergence of a sequence depending on parameter

My math teacher gave us this exercise for homework: Given $ a \in [-3, \infty), $ a given sequence $(x_n),$ with $ x_n = \left(\frac{a^n+2}{3^n+4}\right)$ Determine $a$ so that the sequence ...
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2answers
84 views

On the complete solution to $x^2+y^2=z^k$ for odd $k$?

While trying to answer this question, I was looking at a computer output of solutions to $x^2+y^2 = z^k$ for odd $k$ and noticed certain patterns. For example, for $k=5$ we have $x,y,z$, $$10, 55, ...
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2answers
52 views

Set the parametric equation of an arc with two points

Like the title says, I am looking for a method to find the parametric equation expressed in the form $\vec{r}=...\vec{i}+...\vec{j}$ of the arc that connects the points (2,0) and (1,2). I am asking ...
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1answer
19 views

Calculate the work of the force generated by an electric charge in movement

We know that the force generated by an electric charge that is located at the origin, on a charged particle at a point $(x,y,z)$ of position vector ...
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1answer
21 views

Moving of 5 units on a curve

If I am at a point (0, 0, 3) and a move of 5 units in the positive direction of $t$ on a curve definite by $$ \begin{align*} x &= 3\sin(t) \\ y &= 4t \\ z &= 3\cos(t)\\ \end{align*} $$ ...
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1answer
44 views

Domain and range of locus formed from parametric equations

\begin{cases} x = t^2 + 2t\\ y = 4(t+1)^2 \end{cases} Determine the cartesian equation of the locus? What is the domain of the locus? Note: I have found the cartesian equation: $y=4x+4$ I am just ...
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1answer
33 views

Parametrization of a cylinder that is parallel to x axis

The answer is no it does not matter. The surface is $y^2+z^2=4$, I parametrized it so: $\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$ But Pauls Outline works through the ...
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1answer
91 views

find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R^2}$

...for the curve $C$, which goes along the circle of radius $3$, from the point $(3,0)$ to the point $(-3,0)$, and then in a straight line along the $x$-axis back to $(3,0$). So I set the half-circle ...
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1answer
33 views

Eliminate the parameter to find a Cartesian equation of the curve.

I got $y=x^\frac{7}{2}$ as the cartesian equation for the following parametric equations, but it is showing up as incorrect. Can anyone explain to me what I did wrong? Thank you!
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1answer
29 views

Compute the length of a parametric curve.

It seems like I am not using the good process to compute the length of a given parametric curve. I am not sure if it's inside my calculations or if the steps I use are not correct. The equation of ...
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0answers
8 views

Find a parametric representation for an implicit function f(x,y,z)=0

I encountered a very interesting implicit function: $ z \exp \left[ (x-0.5-e^{z-y})^2+y^2-0.2z+3 \right] = \sin \left[ (xz-0.5)^2+2xy^2-0.1z \right]$ I wonder if there is any ways to find a ...
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1answer
32 views

Length of the curve $ \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \;\; 0 \le t \le \frac{\pi }{6}\;\;$

The length $\;L\;$ of the curve C given by $\displaystyle \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \displaystyle \;\; 0 \le t \le \frac{\pi }{6}\;\;$ is found by ...
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2answers
61 views

How to handle this curve? [closed]

I started with differentiation of all three coordinates of this parametrically given curve. I want to show that the respective curve has related equation of the plane and also to prove that it a ...
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1answer
35 views

Tangents to Parametrized Curves

I need to find the equation for the line tangent to the curve: $$ x = 2\cos t, y = 2\sin t, t = \frac \pi4 $$ I also need to find the value of: $$ \frac{d^2y}{dx^2} $$ I'm not too sure what to do.. ...
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2answers
53 views

Find Cartesian equation from Parametric Equations Including Sec and Tan

Need to find the cartesian equation from: $$ x = sec^2t - 1 , y = tan t, -\frac\pi2 \lt t \lt \frac \pi2 $$ With sin and cosine I use the unit circle, but I don't know what to do with sec and ...
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1answer
34 views

Finding Cartesian Equation giving Parametric Equations

The equations are: $$ x = cos(\pi - t), y = sin(\pi - t), 0 \le t \le \pi$$ I don't really understand what to do. On the last problem I had: $$ x = cos2t, y = sin2t, 0 \le t \le \pi $$ and I just ...
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3answers
257 views

Show that the curve has two tangents

I'm a little stuck on a math problem that reads as follows: Show that the curve $x = 5\cos(t), y = 3\sin(t)\, \cos(t)$ has two tangents at $(0, 0)$ and find their equations What I've Tried $ ...
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0answers
23 views

Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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0answers
44 views

Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
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1answer
20 views

Finding c in parametric quadratic equation

I tried searching before posting this, but couldn't find anything. I have this parametric* equation : $ x^2 + 54x + 5a^2 = 0 $ They are asking me to find the values of a for which the roots of the ...
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2answers
26 views

Express the region $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ as the union of cubic curves

Let $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ I know the family of curves $\gamma(t)=x^2+tx^3$ belong to D, for $t\in [0,1]$. It is true that for every $(x,y)\in D$ there exist a unique ...
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1answer
19 views

$\sin\left(ax + \frac{\pi}{6}\right)$, find $a$ with given slope

I was given a function $$y = \sin\left(ax + \frac{\pi}{6}\right)$$ In the point $x = \frac{\pi}{12}$, the slope of the tangent line of that point is $\frac a2$. I need to find $a$ if it's given that ...
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40 views

Mapping of lines/circles in complex plane by linear transformation

The problem So for the line i tried to use the functon g(t)= = a + tv (where t is real and v paralel to the line going through a) and plug it into the given function and from there i got it into a ...
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2answers
171 views

Finding arc length parametrization of a parabola

Suppose we have a parabola of equation $y = x^2$ in a given Cartesian coordinate system. An obvious parameterization of it is the system $x = t$, $y = t^2$, but there are infinite other possibilities, ...
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1answer
43 views

Calculating line integral on an intersection of two surfaces

I need to calculate the **absolute value ** of the integral: $$ \oint_C (4z+2xy)dx + (x^2+z^2)dy+(2yz+x)dz $$ where $C$ is the intersection of the surfaces: $z=\sqrt{x^2+y^2 }, x^2+y^2 = 2y$ . Will ...
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3answers
47 views

How to use parametric equations for line integral?

The question is to find the work done in moving a particle in a force field: $$\overrightarrow F = 3x^2i+(2xy-y)j+3k$$ along the straight line from (0, 0, 0) to (2, 1, 3) So, work done $=\int ...
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0answers
42 views

How can i translate a parametric equation to cartesian

The parametric equations are : $$ x=16\sin^3(t)$$ and $$y=13\cos(t)-5\cos(2t)-2\cos(3t)-\cos(4t) $$ with $t$ from $-\pi$ to $+\pi$ so I'm new to this kind of equations and i really don't know how to ...
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1answer
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Parametric equations - finding A

The curve with parametric equations x = a(t-2), y = at² + 2 (where a≠0), meets the y-axis at the point (0,5). (a) Find the value of the constant a. (b) Hence determine whether the curve meets the ...
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Vectorial function representation

I'm trying to solve the following excersice: Given the following equations, represent by a vector function: $y=x^2+1$ from $(3,10)$ to $(-1,2)$ $y=4-x$ with $x\in [-2,3]$ ${x^2\over{25}} + ...
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2answers
988 views

Find the angle between two planes using their normal vectors

The angle between two intersecting planes is defined to be the angle between their normal vectors. Find the angle between the planes $x – 2y + z = 0$ and $2x + 3y – 2z = 0$. Find the parametric ...
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1answer
47 views

Normal to a parametric curve: $x=2t+3$, $y=2/t$ [closed]

A curve is given by the parametric equations $x=2t+3$, $y=2/t$. Find the equation of the normal at the point on the curve where $t=2$. I honestly do not understand how to do this question.
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Why is the parameter (t-1) in this example?

Example 1 on this page: http://mathinsight.org/parametrized_curve_tangent_line_examples Why do they use $(t-1)$ in the last step ($l(t)=c(1)+(t−1)c′(t_0)$)? Why not just use $t$?
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2answers
23 views

Finding cartesian equation from Parametric equation.

A curve $C$ is defined by the equations $$x=\frac{1+t}{1-t}$$ $$y=\frac{1+t^2}{1-t^2}$$ where $t$ is a real parameter. I found the $\frac{dy}{dx}=\frac{2t}{(t+1)^2}$. How to prove that $C$ has ...
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1answer
40 views

Transforming a sawtooth into a sinus with one parameter

Can you help me in finding the analytical expression of a function $f_\alpha(\theta)$, with one parameter $\alpha=(0,1)$ by which one can continously transform a sawtooth curve into a sinus? With ...
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1answer
48 views

Find the equation of the locus of R if the chord $PQ$ passes through $(0,a)$

The parabola is $x^2=4ay$ Information given: Points of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola $x^2=4ay$. The equation of the tangent at P is $y=px-ap^2$ The ...
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34 views

Finding the locus of a point R

The two points $P(2ap,ap^2)$ and $Q(2aq,aq^2)$ are on the parabola $x^2=4ay$ The equation of the tangent to $x^2=4ay$ at an arbitrary point $(2at,at^2)$ on the parabola is $y=tx-at^2$. The tangents ...
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1answer
50 views

Prove that $p^2+pq+2=0$

The information given is that a point $P(2ap,ap^2)$ on the parabola $x^2=4ay$. The normal to the parabola at P intersects the parabola again at $Q(2aq,aq^2)$. O is the origin of the graph. The ...
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0answers
24 views

Show that TU is perpendicular to the axis of the parabola.

The parabola is $x^2=4ay$ Show that TU is perpendicular to the axis of the parabola. Information given: Points of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola ...
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2answers
44 views

How to set control points for spline curves

I've written a program that calculates points on spline curves (including Hermite, Bezier, and B-splines) and plot the curve on the screen (the curve is plotted on an html canvas using javascript). ...