Questions tagged [parametric-functions]

Questions about explicit functions expressed in terms of independent variables, i.e. parameters. Including approaches for rewriting functions using different parameterizations.

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28
votes
6answers
8k views

How to get intersection values from a parametric graph?

I have this graphed: ParametricPlot[{2.4*Cos[t] + 1.6*Cos[3 t/2], 2.4*Sin[t] - 1.6 Sin[3 t/2]}, {t, 0, 4*Pi}] It is a star, and the lines cross each other twice ...
18
votes
6answers
3k views

How do I obtain the enclosed area of this particular parametric plot?

I'm trying to find a way to obtain the enclosed area of this particular plot. Can someone show me how? ...
49
votes
5answers
24k views

Find eigen energies of time-independent Schrödinger equation

I'm trying to get the eigenvalues of a one dimensional time-independent Schrödinger equation, $-\frac{h^2}{2m_0}\frac{d^2\psi}{dx^2}+U(x)~\psi=Ei~\psi$ where U(x) is some potential and Ei is the ...
6
votes
2answers
2k views

Solve stiff system by shooting method

I'm trying to solve a second order differential equation with the shooting method but, it appears to be a stiff system. This is worst for a larger parameter $\mu$. I've tried different methods: ...
3
votes
4answers
755 views

Find Where a Parametric Equation Overlaps [duplicate]

I have a parametric equation: r[t_] = {2.9 Cos[3.2 Pi t], Sin[4 Pi t] + 5 t}; which overlaps at a point in the $\{-x, y\}$ region of a plot. I'm trying to ...
11
votes
1answer
968 views

How to add a fading trail effect in Animate?

I would like to add a smooth fading effect to the end of a curve, while it is animated. Here's a minimal working example which shows a particle moving on a circle (the circle is drawn while the ...
8
votes
3answers
3k views

My 3D plot of a Klein bottle doesn't look right

I have found the following parametrisation for a Klein bottle: ...
3
votes
1answer
370 views

Minkowski sum and product of 2D-regions

Let $A_j$, $j=1,\dots,n$ be subsets of $\mathbb C$ defined by $$ A_j=\{z\,|\,f_j(z)\geq0\} $$ where $f_j:\mathbb C\to\mathbb R$ are given functions. I am interested in the following subsets of $\...
23
votes
2answers
4k views

How to plot a certain surface? What is its parametric equation?

The surface show below is very beautiful; however, I don't know its function either as an implicit function or in parametric form. Anyone have an idea about it and how to draw it with Mathematica?
11
votes
1answer
22k views

Plot a function for different parameters

Suppose I have a function that I want to plot for several values of a parameter all in the same window. For instance the function $f$ defined as ...
12
votes
6answers
6k views

Filling between parametric plots

I am new with Mathematica, but am trying to fill an area between two parametric plots ...
10
votes
2answers
575 views

Continuously varying tube radius

I'm currently trying to draw tubular neighborhoods of torus knots, which Mathematica's Tube function allows me to do quite easily. My question regards the ...
7
votes
3answers
4k views

Spherical parametric plot

How can I plot such a parametrized curve: $r = 2\cos t + \sin t$ $\theta = t$ $\phi = \sin 10t$ where $0<t<\pi/2$. In the documentation only has been discussed parametric plot in Cartesian ...
4
votes
2answers
2k views

Logarithmic scale in an ParametricPlot obtained from ODE boundary conditions

How do I plot an ParametricPlot with the x-axis using an logarithmic scale? Since I need to use an ParametricPlot, I can not use the LogLinPlot[] and I also was not able to find any viable Solution in ...
1
vote
1answer
907 views

How to find the parametric equation of the von Mises yield function?

The von Mises yield function is given by: $ \Phi(\sigma_1,\sigma_2)=\sqrt{\sigma_{1}^{2} +\sigma_{2}^{2}-\sigma_{1} \sigma_{2}} - \sigma_y $ were $\sigma_1$ and $\sigma_2$ are the principal ...
9
votes
4answers
2k views

How do you find the self-intersection points of a single parametric trigonometric function? [duplicate]

How do you use Solve or FindRoot or other Mathematica functions to find the points at which the graph of ${\bf a}(t) = 3 \cos (3....
8
votes
2answers
2k views

Find the parameter values satisfying conditions for all roots of an equation

Hi I'm very new at Mathematica. My real problems concerns a much more difficult equation with many parameters, but I want to check that the method is good, so I made up a test equation. My test ...
5
votes
3answers
1k views

Solve for the intersection point of 3 surfaces

Basically, I have three 2-torus (tilted) in 3-dimensional Euclidean space. They are expressed by parametric equations: ...
5
votes
4answers
1k views

Finding intersections of two parametric curves

I am dealing with the root finding of two parametric data curves (data below). I have already checked Parametric Interpolation of 2D data, which I found useful for plotting, though it is not adequate ...
2
votes
2answers
1k views

Create a moving plot along a parametric plot using Manipulate?

I have been looking through Manipulate examples and haven't been able to create a working plot yet. I have a parametric plot defined: ...
5
votes
1answer
643 views

Using ParametricNDSolve and NMinimize to find accurate interplanetary trajectory arrival periapse radius

I have an interplanetary trajectory simulation that calculates the required velocity (using a Lambert solver) and escape angle for a spacecraft to travel from Earth to Mars. This model does not use a ...
4
votes
1answer
3k views

How do I plot a parametric family of curves given a list of parameter values?

I tried plotting the function shown below in Mathematica; however the plot showed nothing. Could you help me explain it? Note that here D' = 1- x. ...
3
votes
1answer
324 views

How to `ParametricPlot` a system of ODEs for a set of initial conditions?

I want to ParametricPlot the solution to a differential equation for which an initial value ranges from 1 to 10. The problem (I believe) is that I don't know how ...
2
votes
2answers
326 views

How to use ParametricPlot and fill closed areas with blue and red?

The current code is : ...
2
votes
3answers
154 views

Solve / brute-force for parameters that drive a function to zero at specified points

Edit: Made question a lot easier to read. Simple example. - Consider $p(x,y,z,t) = e^t(ax^2 + by^2 + cz^2)$ and I seek the values of $a,b,c$ which make $p\approx0$ at the point $(x=r\cos\theta,y=r\...
2
votes
1answer
772 views

How to plot a rotated ellipse using `ParametricPlot`?

I'm plotting the von Mises yield surface using CountourPlot and ParametricPlot. With ...
0
votes
1answer
190 views

PlotStyle based on If condition

So in the If[evecs==Re[evecs], I want to then change the colour in PlotStyle of the ...
0
votes
0answers
246 views

Eliminating variables from system of equation using Eliminate or Solve

I have a system of equations for algebraic curve given by the zero locus of some polynomial encoded in the system of equations (I want to eliminate variable z and get algebraic curve in terms of x and ...
9
votes
2answers
3k views

ParametricPlot3D with three variables

ParametricPlot3D can be used with one or two variables to plot paths and surfaces in 3-D such as ...
6
votes
2answers
399 views

Problem combining ParametricNDSolve with NonlinearModelFit

Final Edit: I have found a solution! Specifying Method->"LevenbergMarquardt", Gradient->"FiniteDifference" gives the behavior I am looking for. Huzzah! It appears that the crashing behavior was coming ...
8
votes
3answers
374 views

Finding the Period of a Limit Cycle

I'm interested in the periods of limit cycles of the Wilson-Cowan equations which have the form $$x'(t) = -x + S(ax(t) - by(t) +e)$$ $$y'(t) = -y + S(cx(t) - dy(t) + f)$$ where $$S(x) = 1 + \frac{...
16
votes
2answers
840 views

Symbolic area calculation for a parametric self-intersecting closed curve

The parametric equation of the curve is: $$\begin{cases} x &= -9 \sin (2 t)-5 \sin (3 t) \\[6pt] y & = 9 \cos (2 t)-5 \cos (3 t) \end{cases}\quad t\in[0,2\pi]$$ which can be easily ...
10
votes
2answers
877 views

Fitting experimental data by using ParametricNDSolveValue and NonlinearModelFit

I am a newcomer to Mathematica. Basically I just want to fit the data (enzyme kinetic data) shown below to a system of odes' by using NonlinearModelFit. The data, an 101x2 array, contains time in the ...
5
votes
1answer
287 views

Switching parameter lines on/off in a 3D parametric plot

In a classroom demonstration for ParametricPlot3D [{f[u, v], g[u, v], h[u, v]}, {u, ... }, {v, ... }] I want to show u lines only without v, v lines only ...
4
votes
2answers
2k views

Is Mathematica capable of converting parametric equations into a function? If so, how?

I've been trying to find the formula for the offset/parallel to a sine wave. Not just the parametric equation, but the y = f(x) form. Here's what I've done so far: Read up on the parametric form and ...
3
votes
2answers
127 views

Two different solutions for the same ContourPlot while using AccuracyGoal and WorkingPrecission

I was making a contour plot, of some differential equation however I get two different solutions If I use "AccuracyGoal" and "Working Precision". The question will appear long but ...
3
votes
2answers
321 views

Plot a parametric function from $\mathbb{R}^2$ to $\mathbb{R}^2$

I want to plot the following function defined on $[-1,1]\times[-1,1]\setminus(0,0)$: $$ F(x,y,t)= \begin{cases} (1-t)(x,y)+t(1,\frac{y}{x})& \text{if}\ -x\leq y\leq x \\ (1-t)(x,y)+t(\frac{x}{y}...
9
votes
1answer
970 views

Oval or Bunimovich stadium

The dynamic billiard demonstration, that can be found on wolfram demonstrations (here), is only designed for an ellipsoid-shape boundary(snapshot below). The idea is to have a dynamic billiard with ...
7
votes
2answers
658 views

How to fill the closed region by ParametricPlot with solid color?

I am interested in the following implicit curve with parametric equation: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
7
votes
1answer
542 views

Finding a parametric curve fitting a two-dimensional dataset

I have a 2-dimensional data set where I want to find the function that fits best the points. My issue is that I am looking for a function of the for $f(x,y)=c$ with $c$ a constant. I could not find in ...
7
votes
2answers
190 views

Model calibration with phase space data

Forgive me if this is answered elsewhere, I am new to Mathematica and this forum and have tried a number of search phrases to no avail. My question is very general but for example, suppose I throw a ...
7
votes
3answers
529 views

Finding the conditions for degenerate solutions to a cubic equation

I have a cubic equation (corresponding to the band structure of a physical system) given by w in the code below. ...
3
votes
1answer
455 views

Using ParametricNDSolveValue with initial conditions as parameters [duplicate]

Recently I've came across the command ParametricNDSolveValue, which allows the user to solve DEs with one or more parameters. I've tried setting ...
3
votes
1answer
944 views

Limitations of ParametricNDSolve family w.r.t objective functions

Observation: I can see even for very simple modification in case of an scalar objective involving an definite integral in time ParametricNDSolve fails. Here is an ...
3
votes
1answer
611 views

How to plot a function whose constant parameters have associated uncertainties in Mathematica 8.0?

Let's assume I want to plot the function $f(x)=a + bx+ cx^{2}$, where experimentally known values of parameters $a, b$ and, $c$ are as follows: $a=1.01 \pm 0.18$ $b=0.92 \pm 0.11$ $c=2.2 \pm 0.2$ ...
2
votes
2answers
4k views

Eliminating the Parameter: Transform parametric equation to Cartesian equation and draw arrows along parametric growth

Hey guys I really could use some help on this calc 3 problem. I'm stuck on how to write the code for this problem: a) Eliminate the parameter to find a Cartesian equation for a parametric curve. b) ...
1
vote
1answer
1k views

Plotting knot torus

I'd like to plot together a torus and a knot on it, that is, closed curve wrapping around the torus, $m$ times around a circle and $n$ times around the other circle. It will be used to explain ...
1
vote
1answer
285 views

Estimating parameters on system of differential equations

I am looking for some help estimating the parameters of a series of differential equations to assist fitting the experimental data I have collected. The Problem: The equations are based off of the ...
0
votes
2answers
316 views

Interpolating a ParametricFunction

Original domain of my independent variable t is 0 to 10 nanometers, but in Mathematica (as in most numerical software), it's good idea to solve differential ...
0
votes
1answer
691 views

Use ParametricNDSolve with Plotting

I am attempting to solve a differential equation of the form $$ \frac{dx}{dt}=ax+by \\ \frac{dy}{dt}=cx $$ with parameters $0<a<1$ and $a,b,c \in \mathbb{R}$. I am able to use NDParametric ...