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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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 ...
xslittlegrass's user avatar
28 votes
6 answers
9k 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 ...
guest's user avatar
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23 votes
2 answers
5k 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?
LCFactorization's user avatar
19 votes
6 answers
4k 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? ...
Black Milk's user avatar
17 votes
2 answers
1k 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 ...
LCFactorization's user avatar
17 votes
1 answer
32k 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 ...
Physics_maths's user avatar
16 votes
4 answers
3k views

How to create these famous surfaces in topology with the desired color effects?

It was said such real objects of famous topological surfaces were plotted by the cooperation between mathematician Richard Palais and artist Luc Benard, and published as the cover page of Science ...
LCFactorization's user avatar
13 votes
3 answers
1k 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{\...
Cheyne's user avatar
  • 312
11 votes
6 answers
7k views

Filling between parametric plots

I am new with Mathematica, but am trying to fill an area between two parametric plots ...
bob7294's user avatar
  • 121
11 votes
1 answer
1k 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 ...
Cham's user avatar
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11 votes
2 answers
738 views

ParametricPlot3D Self Intersection

I would like to know where my ParametricPlot3D intersects itself and to extract all intersections (points, lines, surfaces...) to animate them by changing ...
chkone's user avatar
  • 145
11 votes
1 answer
175 views

How can I plot a region parametrized by another region?

I am trying to learn how to describe and plot a 2D-region for $x_1$ and $x_2$ based on / parametrized by another 2D-region for $v_1$ and $v_2$. In the following example, the region ...
Mauricio Fernández's user avatar
10 votes
3 answers
528 views

Issue Plotting Torus Link

I'm having an issue plotting torus links in Mathematica. What I'm trying to generate is a (2,8) torus link like the one in the picture in Wikipedia: using equations such as those outlined here. ...
Ztan's user avatar
  • 193
10 votes
2 answers
790 views

Parametric plot on multi-dimensional domain

I wish to plot all the points in 3 space that satisfy the following conditions. $$ (x,y,z) = (a^2+b^2, ac+bd, c^2+d^2) $$ for $a,b,c,d$ in some range. I have tried the following. ...
Dan's user avatar
  • 101
10 votes
2 answers
978 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 ...
Francesco's user avatar
  • 101
10 votes
2 answers
722 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 ...
AegisCruiser's user avatar
10 votes
2 answers
918 views

Convert BSplineFunction into two Interpolating Functions

Is there a way to decompose a BSplineFunction output into two functions x[t], y[t] ...
level1807's user avatar
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10 votes
1 answer
653 views

Issuing with ParametricNDSolveValue

I am having a problem with ParametricNDSolveValue[] . let me start with a simple example. Considered following example with constant parameter $a$. ...
Emad's user avatar
  • 135
9 votes
3 answers
3k views

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

I have found the following parametrisation for a Klein bottle: ...
Marion's user avatar
  • 429
9 votes
4 answers
3k 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....
Bez's user avatar
  • 91
9 votes
1 answer
1k 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 ...
Sam's user avatar
  • 93
9 votes
2 answers
717 views

How to draw a 3D point moving along a helical path?

I have an animation for a helix being drawn in 3D but I really want to have a point on there that draws the helix. I've seen similar problems on here for 2D and I tried adapting the suggestions but I ...
Brennan Rodgers's user avatar
9 votes
2 answers
472 views

Finding the point closest to a parameteric curve produced by a numerical method

Question How can I find the closest point to a parametric curve produced by a numerical method. General Context I have a parametric curve (which I will call $g$) produced from a averaging many ...
AzJ's user avatar
  • 697
9 votes
2 answers
317 views

Problem with plotting the derivative of a ParametricFunction

Bug introduced in 10.0 or earlier and continuing through 11.3; Reported as CASE:3691952 I am trying to calculate and plot the sensitivity of a solution to some master equation using ...
Shahar Kasirer's user avatar
9 votes
2 answers
4k views

ParametricPlot3D with three variables

ParametricPlot3D can be used with one or two variables to plot paths and surfaces in 3-D such as ...
E.O.'s user avatar
  • 1,213
9 votes
1 answer
1k views

A problem about the implementation of Bezier curve

I implemented the Bezier curve like built-in BezierCurve as follow: SplineDegree -> d specifies that the underlying ...
xyz's user avatar
  • 605
9 votes
1 answer
549 views

ParametricNDSolve[ ] and NMinimize[ ] in Combination

I have the following code that is used to simulate a spacecraft orbiting Mars, and uses ParametricNDSolve to vary the spacecraft's initial conditions using a parameter value ...
indigoblue's user avatar
8 votes
5 answers
737 views

2D cross-section of ParametricPlot3D

Now I'm trying to get the two-dimensional cross-section from ParametricPlot3D, the details of this question are as follows As an example, here is a parametric equation: x=x(u,v) y=y(u,v) z=z(u,...
zhongwt's user avatar
  • 81
8 votes
3 answers
6k 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 ...
user avatar
8 votes
4 answers
2k views

ParametricPlot3D for only part of a sphere

I want to plot the region of a sphere that satisfies the equation $$ \tan\theta\geq-\frac{rz}{rx\cos\phi + ry\sin\phi} $$ where $rx$, $ry$, $rz$ are known constants. The spherical coordinates $\...
ahle6481's user avatar
  • 565
8 votes
3 answers
1k views

How to combine ParametricPlot and RegionPlot?

I have two implicitly defined equations. I plot the implicit curve using parametric plot. ...
Alex Ivanov's user avatar
8 votes
2 answers
621 views

return {} when the function is given {}

I have a function taking an array as argument and giving an output. However, sometimes my argument is {} (the empty array), then how can I get an output {} when my argument is {}? Actually, I want the ...
lol's user avatar
  • 677
8 votes
2 answers
3k 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 ...
Zoldor's user avatar
  • 105
8 votes
2 answers
3k 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 ...
user5851's user avatar
8 votes
1 answer
571 views

Calculate Radius of Curvature

I have a list of coordinates that make a near-parabolic graph, which I managed to fit and calculate the curvature. However, the curvature only outputs a equation. Is there any way to calculate the ...
Char's user avatar
  • 519
8 votes
2 answers
2k views

Fitting parametric Spiral Function to Data Points

How would I fit the following parametric function describing a spiral to a set of data points? spiral = t*{Cos[t + x], Sin[t + x]} x is the unknown parameter ...
Gerhard Wonner's user avatar
7 votes
2 answers
501 views

Plotting the image of a curve under a flow

I have some explicit time-independent vector field on the plane, and I would like to study how points evolve under the flow generated by this vector field. The flow is rather complicated and cannot be ...
user45824's user avatar
7 votes
4 answers
339 views

Unexpected behavior of `Exclusions` in `ParametricPlot`

I'm using Mathematica to prepare some teaching materials based on the discussion of the damped, driven pendulum in Chapter 12 of Taylor's Classical Mechanics. This involves using ...
Michael Seifert's user avatar
7 votes
2 answers
3k views

Plotting Limit Cycle

I have Lotka Volterra Equations where the prey equation is modified. The Equations are x'[t]=a*x[t]*(1-x[t]/k)-b*x[t]*y[t] y'[t]=-c*y[t]+d*x[t]*y[t] To plot the ...
Sum-Meister's user avatar
7 votes
3 answers
405 views

How to plot a surface given implicitly by 3 equations in 5 coordinate variables?

I have 3 implicit equations in 5 variables: $f_1(S, y_h, y_d, x_h, J)=0$, $f_2(S, y_h, y_d, x_h, J)=0$, and $f_3(S, y_h, y_d, x_h, J)=0$. These equations determine a 3D surface in $S-y_h-y_d$ ...
yuhao chen's user avatar
7 votes
2 answers
194 views

Is Function broken is this example?

This program f[x_] := g[Function[a, x]]; g[fn_] := Module[{h}, h[a_] := fn[a]; h[0]]; f[999] returns Function[0, 999][0] but ...
Pierre ALBARÈDE's user avatar
7 votes
2 answers
241 views

Parametric plot showing an unwanted boundary at the maximum parameter value

I'm trying to fill a parametric plot, but I get this ugly seam for some values of $a$. Observe the following code: ...
EllipticalInitial's user avatar
7 votes
1 answer
330 views

Finding "nice" solution for under-determined system with constraint?

Consider the following equation: ...
Kagaratsch's user avatar
7 votes
3 answers
722 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. ...
Tom's user avatar
  • 3,416
7 votes
1 answer
1k views

Fitting a Parametric Curve

I apologize if this question is obvious or simple. I'm in high school and have started playing around with Mathematica only recently. For my Calculus Final, I am supposed to use rotation about the $y$...
Zaid Mansuri's user avatar
7 votes
2 answers
1k views

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

I am interested in the following implicit curve with parametric equation: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
LCFactorization's user avatar
7 votes
2 answers
257 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 ...
jfaraday's user avatar
6 votes
5 answers
371 views

Replace elements of a list based on a function

I have a function of k (written in the form of a table/list to get values) for 16 integer values from 1 to 16: f = Table[3*k*k, {k, 1, 256}] I want to replace ...
tabi_k's user avatar
  • 157
6 votes
4 answers
3k views

Visualizing Line Integrals

I have a plane curve $C$ described by parametric equations $x(t)$ and $y(t)$ and a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. The line integral of $f$ along $C$ is the area of the "fence" ...
Austin Mohr's user avatar
6 votes
6 answers
845 views

How to wrap a cylinder?

i'm trying to wrap a cylinder in a torus, the best i've done is the following code: ...
Oscar Hernández's user avatar

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