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

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5
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1answer
77 views

Can (x(t), y(t)) generate a surface? If so, can the surface be continuous?

Intuitively, the parametric equation $z = (x(t), y(t))$ seems to only be able to generate one-dimensional objects, i.e. curves. However... Let $x(t)$ be "the odd-indexed digits of the real number ...
2
votes
2answers
98 views

Parameter values that make function values side lengths of a triangle

I have been trying to solve the following problem for more than a week without any success. Given the function: $$f(x)=\frac{x^2+mx+4}{x^2+x+4}$$ Find all possible values of the parameter $m$ such ...
1
vote
1answer
146 views

How can I re-write an equation (or system of equations) in parametric form?

For the equation $y = 3x$ I need to re-write $x$ and $y$ in terms of a variable $t$. How can I find the value of each variable in terms of $t$?
2
votes
1answer
153 views

How to find a parametric equation?

I want to find an equation for a race track, so I could get the position of a point with respect to time. Let's say I have this track and here are a few points on it: Could it be possible to model ...
0
votes
1answer
190 views

parametrizing quarter of a circle

I am given the circle whose equation is: $(x-\frac{1}{2})^{2}+(y+\frac{1}{2})^{2}=\frac{1}{2}$. So, the coordinates of the origin of the circle are: $(\frac{1}{2},-\frac{1}{2})$ and the radius of the ...
1
vote
1answer
339 views

converting a parametric R5 vector into a Cartesian form

How do you solve a problem like this. I'm completely stumped. it seems like there should be an easy solution but I'm obviously over looking it. any help would be greatly appreciated.
2
votes
2answers
213 views

Constant velocity of a sine function

I am defining the location of an object based on the sine function. The position of the object at s seconds along the x-axis is defined as x=s and its position along the y-axis is defined as y=sin(x). ...
1
vote
2answers
382 views

Find the area bounded by the parametric curve…

Find the area bounded by the parametric curve $x = \cos(t)$, $y = e^t, 0 < t < \pi/2$, and the lines $y = 1$ and $x = 0$. I do not even know where to start with this problem. I know that I need ...
2
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3answers
2k views

How to find a parametric equation for the tangent line to the curve of intersection of the cylinders?

How can i find a parametric equation for the tangent line to the curve of intersection of the cylinders $x^2 + y^2 = 4$ and and $x^2 + z^2 = 1$ at the point $P_0(1,\sqrt{3}, 0)$?
0
votes
1answer
161 views

How do we find the length of the line (parametric curve)?

A curve in the $xy$-plane is given parametrically by $$x(t) = e^{2t}, \quad y(t) = e^{2t} \sin(2t), \quad t \in [0, \pi/2].$$ What is the length of this curve? Ok, actually I know what to do, ...
-1
votes
1answer
106 views

Converting parametric equations in a numerical equation

Is it possible convert this parametric equations in a numerical equation? $$ \begin{cases} \displaystyle x(t)=tv_0\cos(\theta)\\ \displaystyle y(t)=tv_0\sin(\theta)-\frac{1}{2}gt^2+h \end{cases} $$ ...
1
vote
0answers
56 views

Maximum value for parameter

I am facing the following problem: A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, ...
1
vote
1answer
3k views

Equation for making a circle in 3D space

I have a 3D space with axis $(x, y ,z)$ and I can make a circle in the $xy$-plane. To make a circle in the xy-plane I currently use spherical coordinates $(r, \theta, \phi)$ where $r = 1$, $\theta = ...
1
vote
1answer
442 views

Parameterize a straight line using polar coordinates… without angle.

I had to parameterize a straight line with starting point in $A=(-3,7)\\ $ and endpoint in $B=(4,1)$. My idea was to use the equation for the line that goes through two points. That is: $$ \frac { ...
1
vote
1answer
99 views

Parametric Equation of a Particle Movement inside a Vortex in a Rectangular Box

I am trying to simulate the movement of a particle in a vortex in a rectangular box, I am currently using an ellipse but that causes the particle to collide with the walls more that I want. The ...
1
vote
2answers
270 views

Finding Tangent line from Parametric

I need to find an equation of the tangent line to the curve $x=5+t^2-t$, $y=t^2+5$ at the point $(5,6)$. Setting $x=5$ and $y = 6$ and solving for $t$ gives me $t=0,1,-1$. I know I have to do ...
1
vote
1answer
577 views

Parametric Equation

Let $P_1$ be the plane through the origin containing the vectors $[1,2,-1]$ and $[0,1,1]$. Let $P_2$ be the plane through the point $(1,1,1)$ parallel to the vectors $[-1,2,2]$ and $[3,4,-2]$ I know ...
0
votes
1answer
40 views

Represent sorting position by a parametric form

Given a set of random integers {0,5,100,65,...,0,1,2}, is there a mathematical method existing to construct a parametric form $f$ (the number of parameters $<<$ the number of integers) so that ...
0
votes
1answer
134 views

Parametric Linear Program: Continuous Solution?

Consider the parametric linear problem $$ x^*(\theta) := \min_{Y , \ Z } \left\| Z \right\|_1 $$ $$ \text{sub. to: } \ \theta A + B Y = \theta C Z.$$ where $Y \in \mathbb{R}^{m \times s} $, $Z \in ...
2
votes
1answer
185 views

Regular parametrization of a curve

Let $\gamma : \left\{ \begin{array}{ccc} \mathbb{R} & \to & \mathbb{R} \\ t & \mapsto & (t^2,t^3) \end{array} \right.$ and $\Gamma= \gamma(\mathbb{R})$. Because of the singularity at ...
3
votes
1answer
3k views

find length of curve of intersection

I have come to a dead end on a problem and I need someone to tell me either if I did it correctly, or how to fix it if I did not. This is Stewart Calculus 7th edition, problem 13.3.12. Here is the ...
1
vote
1answer
441 views

parameterization of helical torus

A Helix is parameterized as $\langle R \cos(t), R \sin(t), \alpha t\rangle$ and one can visualize it as "wrapping" around a cylinder of radius R. I would like to accomplish the same thing but wrapping ...
1
vote
1answer
142 views

Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
0
votes
1answer
69 views

Probability distribution for a function of a random variable

I have the distribution of X with respect to parameter t vaying between 0 and 1. However, in nature, parameter t is not uniformly distributed. It has a known probability distribution. What is ...
3
votes
1answer
522 views

Summation of an Arithmetic, Parametric Sequence

I'm trying work on my ability to break complex patterns down, and in this case I'm trying to model the denominators of Lacsap's Fractions: I managed to get the sequence that represents the ...
1
vote
3answers
653 views

Write each pair of equations as a single equation in $x$ and $y$.

Write each pair of equations as a single equation in $x$ and $y$. a)$\begin{cases} x=t+1 &\\ y=t^2-t & \\ \end{cases}$ b)$ \begin{cases} x=\sqrt[3]{t}-1 &\\ ...
0
votes
1answer
169 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
1
vote
2answers
242 views

A circle on the plane [duplicate]

Possible Duplicate: Parametric Equation of a Circle in 3D Space? I know that, for example, if a circle is on a plane with counter-clockwise orientation, and with center $(a,b)$ and radius ...
2
votes
2answers
389 views

Distance between point and a spiral

I'm trying to work out an algorithm where, given the equation for a spiral in polar coordinates, $r(\theta)$, and a point rectilinear coordinates, $P(x,y)$, I can work out the minimum distance between ...
0
votes
1answer
244 views

Is this valid parametric equation to create control points for a helix in 3D space?

Is this a valid way to compute new points that are on a helix and if not what is it wrong? The Cartesian coordinates of each new helix control point could be described by the following ...
1
vote
2answers
137 views

Stuck on space curves for vector valued functions

I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point. How would you sketch the graph for the parametric equations: $x = \cos t$, $y = \sin t$, ...
2
votes
1answer
187 views

Motion on a parametric surface

Please excuse what will surely turn into a long rambling question, full of incorrect terminology. I'm trying to figure out the mathematics of moving on a parametric surface - that is, for some ...
0
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0answers
877 views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...
1
vote
1answer
644 views

Using Parametric Equations to Define the Position of an Object in Motion

What is the intuition behind the equations for the parametric equation applications below? Rather than memorize the formulas for a quiz i'd like to gain a deep understanding of them.
1
vote
1answer
98 views

Parametric equation involving exponents and e

How do you go about solving this problem? For each plane curve given below, find a rectangular equation. State the appropritate interval for $x$ and $y$. $x(t) = e^{5t}$, $y(t) = e^t$, $t \in ...
1
vote
1answer
189 views

Parametric representation of rectangular form in terms of parameters $\rho$ & $\theta$

I need to represent the cone $z=\sqrt{3x^2+3y^2}$ parametrically in terms of $\rho$ and $\theta$ where $(\rho,\theta,\phi)$ are spherical coordinates. Attempt. I tried using: ...
3
votes
3answers
5k views

Parametric form of a plane

Can you please explain to me how to get from a nonparametric equation of a plane like this: $$ x_1−2x_2+3x_3=6$$ to a parametric one. In this case the result is supposed to be $$ x_1 = 6-6t-6s$$ ...
0
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2answers
1k views

Give the parametric eqns of the line intersecting the planes

Give the parametric equations of the line of intersection of the planes $$4x + 2y + 2z = -1$$ and $$3x + 6y + 3z = 7$$ Also, give the equation of the plane that passes through the point $(2,-1,4)$ ...
0
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1answer
646 views

Switching a non parametric equation to a parametric equation of a plane

More than the result (that I already have printed in my book), I'd be interested in the procedure to switch from a non parametric to a parametric equation of a plane in the Euclidean space. Here is ...
2
votes
2answers
704 views

Parametric question of the curve $x^2 + y^2 + 2x - 4y = 0$?

What is the parametric form of the curve above? If I had to solve it, what I would say is that the first step is to complete the square. However, where would I go from there?
2
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2answers
2k views

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where ...
2
votes
2answers
342 views

Curve arc length parametrization definition

I did some assignments related to curve arc length parametrization. But what I can't seem to find online is a formal definition of it. I've found procedures and ways to find a curve's equation by ...
0
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1answer
592 views

Newton's Method, and approximating parameters for Bézier curves.

I've been wanting, for quite a while now, to polish up some source code I wrote for approximating arbitrary Bézier curves to given series of points. I managed to accomplish quite a bit, but I hit a ...
1
vote
1answer
140 views

Range of Parameters and Integral Evaluation

I am currently working on understanding this problem and am in need of some assistance. I'll let you know what I've done so far, and then hopefully someone will be able to help me. So the problem is ...
0
votes
1answer
3k views

How to write parametric equations for a given polar equation?

I'm doing an extra credit problem for math, we haven't learned too much on this topic. The instructions are: Write parametric equations for the given polar equation. The problem is: $r = ...
35
votes
3answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
1
vote
1answer
380 views

What Parametric Equations are required to move along a circle while moving left?

I'm working on a program where I can set objects along arbitrary parametric paths. Moving left is easy: X = x - dT(V) Y = y ...
0
votes
2answers
2k views

How do I change this parametric equation: $x=t+1/t, y=t^2 + 1/t^2$ into a Cartesian equation?

I've just started parametric equations on my own & I am a bit confused on how to convert this parametric equation into a Cartesian equation. $$\begin{array}{rcl} x=t + \frac{1}{t}, y= t^{2} + ...
0
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1answer
307 views

Partial Derivatives and Chain Rule

I have a line $L\subset\mathbb{C}^n$ which is parametrized by $x_1=a_1t, x_2=a_2t,\dots, x_n=a_nt$, a function $f(x_1,\dots,x_n)$, and I want to look at the restriction of $f$ onto $L$. This is just ...
2
votes
0answers
929 views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...