Questions tagged [parametric]

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

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Find the parameterization of shadow cast on plane

Suppose there is a light source at $(0, 0, 8)$. Consider the line segment $r(t) = \langle 3-3t, 1+t, 2+2t\rangle, 0\le t\le1$. Find the shadow cast by the line segment on the plane $x+y-2z=8$ I have ...
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Is the Stokes-Carten theorem suitable for knotted manifolds

I am currently trying to discretise a torus knot onto a square lattice grid. My method of achieving this has been to first use the Torus knot parametric equation, then define a circle that follows the ...
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Intersections of the involute of a circle with the $x$-axis

I am looking for a parametrization of the involute of a circle in order to perform a least-squares fitting to some spiral data points. I started with the usual parametric form of the involute, in ...
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Find $m$ such that $X^2 - X +1 \ | \ (X-1)^m + X^m +1$ [closed]

As the title says, how would I find $m$ such that $X^2 - X +1 \ | \ (X-1)^m + X^m +1$? What I was thinking of is to write $X^2-X+1 = (X-a)(X-b)$, where $a,b$ are the roots of the polynomial and find $...
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Determining if a parametric curve intersects a plane at a right angle

Given a plane described by $$ x + 8y + 12z = 162 $$ and a curve described by $$ \begin{align} x &= t\\ y &= t^2\\ z &= t^3 \end{align} $$ I want to find out if the curve intersects the ...
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Sweeping a shape across a diverging path?

There is a well known kind of parametric surface known as sweeping surfaces, where one curve is swept across another, forming a surface in 3d. i.e. one obtains a formulation of the form: $f(u,v) = R(u)...
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1 answer
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When do two quadratic equations have exactly one common solution?

I have stumbled upon the following two exercises: For $a,b \in \mathbb{Z}$, consider $A=\{x\in\mathbb{R} | x^2+2ax+b=0\}$ and $B=\{x\in\mathbb{R} | x^2+2bx+a = 0\}$. Let $k$ be the number of elements ...
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Is there a parametric equation that describes the D-shaped? [closed]

How can a curve that looks like a D be described by a parametric equation?.It can be defined as in the figure below and also a parametric equation. I don't know if Fourier series can be used to ...
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Where to find velocity and radius of vectors for animation of Fourier parametric shape

I have a parametric equation where the parameters are sums of sines and cosines such as $\left(3+4\cos2t+2\sin2t,7+0.8\cos2t-0.5\sin2t\right)$. I want to animate this with some vectors, rotating at a ...
1 vote
1 answer
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Parametric Bootstrap confidence interval does not contain estimated parameter - Python code [closed]

I have some artificial data with only one feature in the covariates and an integer response on which I perform Poisson regression (see below for a plot). I would like to obtain 95% confidence ...
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Points of intersection of curves given in parametric equations

This is the question, The points of intersection of the curves whose parametric equations are $x=t^2+1, y=2t$ and $x=2s, y=2/s$ is given by: (a) $(1, –3)$, (b) $(2, 2)$, (c) $(–2, 4)$, (d) $(1, 2) ...
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Parameterizing the intersection of two surfaces

Parameterize the equations \begin{align*} z&=a \cos{\left (\cfrac{y^2}{a^2} \right)}\cos{\left(\cfrac{x}{a} \right)}+a \end{align*}and \begin{align*} z&=-4127x+3172y \end{align*}where $a=7500$....
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Analytic form of an ROC curve

I'm studying the problem of combining two sensors for anomaly detection. I want to analyze its performance by the ROC curve. Now I have obtained a parametric equation about the ROC curve: $$(x,y) = (...
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Radius of Curvature for Cam Profile

In the design of a radial design cam-follower mechanism, the radius of curvature of the cam profile is often used to identify areas of high contact stress. The pitch circle of a cam is defined by ...
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finding a parametric equation of a curve formed by deforming an ellipse

I have a deformed curve from ellipse with the following form: $$\left( \frac{x}{a} \right)^2+\left( \frac{y}{b} \right)^2-\alpha\left(\frac{x}{a}\frac{y}{b}\right)^2=1,$$ where for $\alpha=0$ we get ...
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General Ellipse (or Conic Section) to Parametric Equation / Plottable Form

Trying to plot an ellipse or general conic section. From this GaTech open textbook on linear algebra, I can find the equation of the rotated and off-origin centered ellipse (last example in the link) ...
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Solidifying a 3D parametric curve

I have an arbitrary parametric 3D curve which I need to solidify, that is, give it a circular thickness. I have a circle lying in the XY-plane, and for every point (x0, y0) of my curve, the circle ...
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2 votes
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How do we express the equation corresponding to the intersection of the planes $x + y + z = 1$ and $x + 2y + 2z = 0$?

If a system with three unknowns and two equations are such that $$ \begin{align} x+y+z=1&\\ x+2y+2z=0 \end{align} $$ In the answer it says that this system can be represented as $$ \begin{pmatrix} ...
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How do you formally determine the sign of a root?

While reading about parametrization, I came across the example of the cuspidal cubic which is defined by $y^2=x^3$. It's stated that an equivalent function is $\vec{f}(t) = \begin{bmatrix} t^2 \\ t^3 \...
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Verifying that these 2 parametric equations describe the same plane

For $\lambda,\mu \in \mathbb R$ we define: $\begin{align} \begin{pmatrix} 1\\ 1\\ 1\\ \end{pmatrix}+\lambda\begin{pmatrix} 1\\ 0\\ -4\\ \end{pmatrix}+\mu\begin{pmatrix} -1\\ -5\\ 0\\ \end{pmatrix} \...
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Soft: Understanding the difference between machine/statistical learning and parameter estimation

Say we have a known model $M$ with unknown parameters and more specifically, $M$ is a parametric model. Parameter estimation on $M$ is applying an appropriate method for estimating the parameters. My ...
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Convert to parametric equation from implicit equation?

Given an implicit equation such as $x^2+y^2=1$ , I know it corresponds to the parametric equations $ \begin{cases} x=\cos t\\ y=\sin t \end{cases}$. But I don't know how to get from the implicit ...
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Is it possible to convert Cartesian equation to polar or parametric equation?

I was reading Wikipedia to find interesting curves to be plotted using a polar coordinate system. And I've found an interesting shape, which is Squircle: Wikipedia shows this equation: $$ {\vert \...
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Parabola passing through TWO points with known tangents directions

Given the points $ P_1 (0, 5), P_2 (0, 0), P_3 (3, 0) $, a parabola passes through $P_1$ and is tangent to the segment $P_1 P_2$ and passes through $P_3$ and is tangent to the segment $P_2 P_3$. ...
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Doubt on a limit with parameter

I have to study the value of the following limit: $$\lim_{n \to +\infty} n^{\alpha}(\sqrt[n]{n+1} - 1)$$ for $\alpha \in \mathbb{R}$. So I made the substitution $\displaystyle x = \frac{1}{n}$ and got ...
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Parametric curves including complex numbers

$$z = te^{it}$$ This is a parametric curve, but I am not sure on how to exactly plot it. Can I confirm that it is a anticlockwise spiral from the origin? Or am I incorrect?
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Tangent to two circles with parametric coordinates (2+2cos θ , 2 sin θ)?

I was wondering if someone could help solve this question, Show that the point with coordinate$ ( 2 + 2cos (θ) , 2 sin (θ))$ lie on the circle $x^2 + y^2 = 4x$ and obtain the equation of the tangent ...
1 vote
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Parametric equation appears to generate circle and violate $x(t)^2+y(t)^2={R+}$

The context of the equations is from plotting the real against the imaginary impedance of an RC circuit however the problem is entirely mathematical. Given the two equations: $x(w) = \frac{R}{C^2 R^2 ...
1 vote
2 answers
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Sine wave on a trefoil knot

I'm trying to impose a sine wave onto two different curves, sort of similar to this old question about an Archimedes spiral. Unfortunately I can't figure out how to incorporate the new sine wave into ...
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if i have random sample from standard normal then prove indepenednt of $\bar Z$ and $\sum_{i=1}^n (Z_i - \bar Z)^2$

If $ Z_1 , Z_2 , .. , Z_n $ is a random sample from a standard normal distibution , then: $\bar Z$ and $\sum_{i=1}^n (Z_i - \bar Z)^2$ are independent
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Finding polynomial order of parametric curve with trig functions

I was given a parametric curve defined by: \begin{align} x(t) &= \cos(t) \\ y(t) &= \cos(2^kt) \end{align} And asked: For any natural number k, eliminating the parameter gives a polynomial ...
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Equation for an asymptotically flat spiral

I would like to find an expression of a spiral which gets straight at infinity, as in the image below: so that I can make a parametric plot.
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Find coordinates of a point for a derivative of a parametric curve

Find the coordinates of the points at which the given parametric curve has a) a horizontal tangent and b) a vertical tangent. The parametric curve is $$\mathscr{C}=\begin{cases}x=t^2+1\\y=2t-4\end{...
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1 vote
1 answer
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Seemingly unsolvable integral for length of parametric curve

I have come across a question for the length of the loop of a given parametric curve defined by the equations $x(t)= 4t-t^3$ ; $y(t)=-2+2t^2$ The loop is defined about the interval from t=-2 to t=2 or ...
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Showing that $(x,y)=\left(\frac{c\sin t+d}{a+b\cos t},\frac{e\cos t+f}{a+b\cos t}\right)$ parameterizes an ellipse

I want to show that $$\mathbf{P}(t) = (X, Y) = \left( \dfrac{ c \sin t + d}{a + b \cos t} , \dfrac{ e \cos t + f }{a + b \cos t } \right)$$ is actually an ellipse, given that $ | b | \lt | a | $ How ...
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I need help finding the limits of integration for this parametric equation.

The equation is $$x= 2\cot(\theta), y=2\sin^2(\theta)$$ $$0\le \theta\le\pi $$ I believe the answer is $4\pi$ but I keep getting $-4\pi$ I know this is wrong because the area is negative and I ...
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Finding the intersection of two cylinders?

How do I find the equation of intersection of these two cylinders? I'm try to draw a line at this intersection in three-dimensional modeling with equations to transition from a vertical line to a ...
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Validation of statistical models

A statistical model can be generally defined as $(E,\{P_\theta\}_{\theta∈\Theta})$ where $E$ is the sample space containing all possible outcomes of a r.v., $\{P_\theta\}_{\theta∈\Theta})$ is a family ...
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Identify geometric figures from parametric equations

Is there a general approach which can transform parametric equation into the set of geometric figures with characteristics (points, arcs, circles, lines, segments, ellipses and other quadratics). Let ...
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Is it possible to convert vector valued function in this form: $r(t)=\langle3,t,t^2-1\rangle$ into non parametric form?

For instance with $r(t)=\langle t+1, t^2-2\rangle$, we can substitute it with x values and y values to get $y=t^2-2=(x-1)^2-2$ form that is easy to graph. How can we do the same for $r(t)=\langle 3,t,...
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Parametric families of area-preserving curves?

I have an interesting PDE problem that I would like to solve. Consider the diagram below. I would be very interested to hear about whether there are any "nice" families of parametric curves ...
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Is there an easy way to combine a piecewise curve into one curve?

The problem related to shape below. I need to find a parametric equation curve that is split into four different equations depending on the range. How can piecewise equations be combined into a single ...
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Nonlinear cartesian equation to parametric equation

how can I parametrize the following equation? Not sure what techniques I should use here since it's my first time. $\begin{cases} M_1 : z=16-3x^2-y^2 \\ M_2 : z=(x-4)^2+3y^2 \end{cases}$
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vector questions finding another line with the two given lines.

I have questions about this problem. The vector equation of the lines L1 and L2 are r=(15,11,6)+t(6,3,2) and r=(0,7,-4)+s(-3,2,-6) respectively. Let A and B be the points on L1 and L2 with parameters ...
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How to find the radius of the circle such that it intersects one ellipse at only one point? [duplicate]

I have a circle centered at (0, 0) with an unknown radius r. I also have a shifted and scaled ellipse (s.t. $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$). I want to find the radius that intersects ...
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Distance and Arc Length (Math GRE)

A circular helix in $xyz$-space has the following parametric equations, where $\theta \in \mathbb{R}$. $x(\theta)= 5\cos(\theta)$ $y(\theta)=5\sin(\theta)$ $z(\theta)=\theta$ Let $L(\theta)$ be the ...
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2 answers
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Is there a way to modify the solid of revolution integral to allow for solids of increasing and decreasing radius?

I am doing a project on tori as they relate to pool floaties and the volume of a normal torus can be calculated by the solids of revolution integral on a circle, Is there a way to modify the integral ...
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Write the parametric equations of the normal line to the given surface at the specified point.

I am struggling with finding the parametric equation for this problem. I just started this chapter so I do apologize if this seems like a dumb question. "Find equations for the tangent plane and ...
3 votes
1 answer
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Continuity of parametric integral $I(\alpha)=\int_0^\infty \frac{\ln{(1-\alpha^2+\alpha^2x^2)}}{x^2-1}dx$

How to prove that parametric integral $I(\alpha)=\int_0^\infty \frac{\ln{(1-\alpha^2+\alpha^2x^2)}}{x^2-1}dx$ is differentiable on $(1,\infty)$? I wrote $I(\alpha)$ as $\int_0^1 \frac{\ln{(1-\alpha^2+\...
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5 votes
4 answers
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Trouble with tedious algebra (Oxford 1992 Admissions Test 2 1992)

(i) Show that the condition that the points $P$ $(a\cos A,b\sin A)$ and $Q$ $(a\cos B,b\sin B )$ should subtend a right angle at O is $$a^2\cos A\cos B+b^2\sin A\sin B=0$$ (ii) Let S be a circle ...

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