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

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1answer
598 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
636 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
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2answers
285 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
489 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 ...
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1answer
129 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
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1answer
2k 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 = ...
31
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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) ...
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1answer
356 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 ...
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2answers
1k 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} + ...
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1answer
298 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
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0answers
820 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. ...
0
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1answer
97 views

How to find the parametric equations for: $zx + zy - xy = 0$

I'm trying to find the parametric equation for $zx + zy - xy = 0$ or equivalent $z = \frac{xy}{x+y}$ but couldn't find any hint in the web neither in a couple of calculus books for this particular ...
0
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1answer
36 views

Reparametrization of angles

Why is it true that $(\cos (a \theta +b), \sin (a \theta +b), (c \theta +d))$ for $\theta \in [\theta_1,\theta_2]$ can always be written as $(\cos \alpha \theta' , \sin \alpha \theta', \beta ...
0
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1answer
183 views

Detect “cusp” in parametric curve

I'm using the word "cusp" informally here, I apologize if there is a formal definition for it. What I'm looking for is a point where the derivative is non-continuous, I think. I have a a sequence of ...
4
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4answers
983 views

Arc Length Problem

I am currently in the middle of the following problem. Reparametrize the curve $\vec{\gamma } :\Bbb{R} \to \Bbb{R}^{2}$ defined by $\vec{\gamma}(t)=(t^{3}+1,t^{2}-1)$ with respect to arc length ...
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2answers
325 views

Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation?

I have the following parametric equation: $$x=t^2-2t$$ $$y=\sqrt{t}$$ I'm interested finding the area of the region bounded by this curve and the y-axis (i.e. $0 \leq t \leq 2$). We have: ...
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1answer
306 views

Solving Parametric Equation: Multiple coefficients of trigonomic functions

How can I solve: $ x = 16 \sin^3(t) \\ y = 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t) $ I've derived $t = arcsin(\frac{x^\frac{1}{3}}{16^\frac{1}{3}})$ from the first equation but I am still unsure ...
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1answer
442 views

Solve x = sin(t) for t

How can I solve: $(\frac{x}{16})^{\frac{1}{3}} = \sin(t)$ for t?
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1answer
537 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
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1answer
271 views

Doubt with parametric and symmetric equations

In the line through $P(0, 0, 0)$ and is perpendicular to $x=y-5$, $z=2y-3$, when we solve the equations and get the symmetric equations in order to find the vectors $V_1$ and $V_2$, why the normal ...
3
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1answer
315 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...
1
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2answers
80 views

For what values of $b$ does this function lack extrema?

I need to find the range of values of the parameter $b$ for which the function below has no extrema. $$ \frac{b}{3}\,8^x + (2) 4^x + (b+3) 2^x + b \ln(2)$$ In the beginning I thought it'd be ...
1
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1answer
3k views

Find a parametrization of the curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$ and use it to compute the area of the interior.

I have the following homework question that I know the answer to $(3\pi/8)$, however, I don't understand how to get this answer. The question: Find a parametrization of the curve $x^{\frac{2}{3}} + ...
2
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2answers
3k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
2
votes
1answer
726 views

Equation for control point distance for fixed-length cubic Bézier path (with specific constraints)

A particular Stack Overflow question asks how to construct a specific cubic Bézier path of constant length. I have experimentally determined the ideal distances of the control points from the nearest ...
0
votes
1answer
437 views

3D parametric equations with polar coordinates

I'm currently studying for my calc 2 midterm and came across this and it completely lost me. I'm not even completely sure where to begin with it. Any ideas? Put $\langle x[r,t],y[r,t],z[r,t] \rangle ...
2
votes
1answer
159 views

Finding Angle of Elevation to hit X, Y

My ultimate goal is to find the angle of elevation necessary to launch a projectile from the origin to (x,y) with initial velocity V and under gravitational acceleration g. Wind resistance is ignored. ...
2
votes
3answers
15k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
0
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2answers
41 views

Parameterization of the negative half of the y-axis

I need to parameterize the negative half of the y-axis in spherical and cylindrical coordinates. I know what spherical and cylindrical coordinates are, just not sure where to start to parameterize the ...
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3answers
154 views

Would this not take a ridiculously long calculation? (Surface area of parametric surface)

One of the question in my homework asks to verify that the surface are of $ \mathbf{r} = a(1+\cos\phi)\sin\phi \cos\theta \mathbf{i} + a(1+\cos\phi)\sin\phi \sin\theta \mathbf{j} + ...
8
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5answers
11k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
0
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1answer
5k views

Finding an equation and parametric description given 3 points

Let m be the plane through (0,1,1), (0,1,0) and (-2,-1,-1). This concept has always confused me: How would I find the equation and parametric description given just these points?? I think the ...
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0answers
28 views

MLE estimation of parameters, converting normalized observations to integers and back

I am fitting a model's parameters to grouped data by maximizing the likelihood equation: $L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$ $\theta$ is the vector of parameters. $n_i$ is ...
6
votes
3answers
229 views

Curvature of the image of a curve projected onto a surface

(Adding a bounty since I need more details than I have so far) Given a point $$ s_{0}=S(u_{0},v_{0}) \;\;\;\; (S:\mathbb{R}^{2}\to\mathbb{R}^{3}) $$ and a point $$ c_{0}=C(t_{0}) \;\;\;\; ...
0
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2answers
1k views

How to find a smooth parametrization of a Curve

In order to solve a line integral, I need to establish a smooth parametrization of the curve over which it is supposed to be integrated. The curve, $D$, is the intersection of the surfaces $x^2 + ...
1
vote
1answer
95 views

Probabilistic ordering

I want to characterize the probabilistic ordering of some (random) variables without going into a parametric from of the variables themselves. I couldn't easily find any theory for this and I am not ...
3
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1answer
1k views

Equation-driven smoothly shaded concentric shapes

Background Looking to create interesting video transitions (in grayscale). Problem Given equations that represent a closed, symmetrical shape, plot the outline and concentrically shade the shape ...
3
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2answers
259 views

Parametrization of curve length in D dimensional space. How is it done?

Sorry, its been a while and my calculus was never good. This is really a very elementary question which I am unable to un -complicate from its shroud of notation. My difficulty is how does this ...
3
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3answers
3k views

How to convert a plane (e.g. $4x - 3y + 6z = 12$) into parametric vector form?

I can convert something in the 2nd dimension fine, but I'm having difficulty with something like $4x - 3y + 6z = 12$. Any help? EDIT: Solve using only algebra, no matrices yet.
3
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5answers
2k views

Length of $r=3\sin(\theta)$

I have a general understanding of calculating arc length, but this one's a real curve ball. So, I need to find the exact length of $r=3\sin(θ)$ on $0 ≤ θ ≤ π/3$ So the way I've thought of ...
6
votes
2answers
267 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...
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3answers
218 views

Express $z$ in terms of $x$ and $y$, i.e., find $z= f(x,y)$

I've been banging my head against the wall for a while now: $x = s^2 - t^2$ $y = s + t$ $z = s^2 + 3t$ Express $z$ in terms of $x$ and $y$.
1
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1answer
443 views

Parameterization of an implicit function

I'm trying to find the area of an irregular domain that is bounded by $x = c$, $y = c$, and $c = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$, where A can vary in the range [-1,1], and x and y are only ...
2
votes
4answers
595 views

Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$

I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too. $$x=2t-4t^3$$ $$y=t^2-3t^4$$ I can guess that the curve is a polynomial equation so in principle I can write this as $$w_1 ...
6
votes
6answers
3k 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 ...
1
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2answers
367 views

Finding a quadratic Bézier curve of length $l$ between two points

I have two points $P_1$ and $P_2$ in the plane. For each of the points, I have two vectors $v_1$ and $v_2$. I want to find a quadratic Bézier curve from $P_1$ to $P_2$ of length $l$ leaving $P_1$ in ...
7
votes
3answers
3k views

Parametrization of a line

This is a very basic question, and its funny that I'm able to solve more advanced problems like this, but I was presented with a basic one and got stumped. I have the equation $$y=-\frac{3}{4}x+6.$$ ...
2
votes
2answers
692 views

Reparametrizing a curve in terms of the arc length

We want to reparametrize the curve $$\displaystyle \vec{r}(t)=<t^3+1, t^2-1, \frac{\sqrt{5}}{2}t^2>$$ in terms of the arc length measured from the point t=0 in the direction of increasing t. ...
2
votes
1answer
177 views

Parametric Equations for a Hypercone

The n-dimensional cone, with vertex at the origin, central angle, $\alpha$ and central axis in the direction of the unit vector $\xi$ is defined to be all those points, $x\in {R^n}$ whose dot product ...