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.
717
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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 ...
28
votes
6
answers
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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 ...
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?
19
votes
6
answers
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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?
...
17
votes
2
answers
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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 ...
17
votes
1
answer
32k
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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
...
16
votes
4
answers
3k
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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 ...
13
votes
3
answers
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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 $$S(x) = 1 + \frac{\...
11
votes
6
answers
7k
views
Filling between parametric plots
I am new with Mathematica, but am trying to fill an area between two parametric plots
...
11
votes
1
answer
1k
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How to add a fading trail effect in Animate?
I would like to add a smooth fading effect to the end of a curve, while it is animated. Here's a minimal working example which shows a particle moving on a circle (the circle is drawn while the ...
11
votes
2
answers
738
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ParametricPlot3D Self Intersection
I would like to know where my ParametricPlot3D
intersects itself and to extract all intersections (points, lines, surfaces...) to animate them by changing ...
11
votes
1
answer
175
views
How can I plot a region parametrized by another region?
I am trying to learn how to describe and plot a 2D-region for $x_1$ and $x_2$ based on / parametrized by another 2D-region for $v_1$ and $v_2$.
In the following example, the region ...
10
votes
3
answers
528
views
Issue Plotting Torus Link
I'm having an issue plotting torus links in Mathematica. What I'm trying to generate is a (2,8) torus link like the one in the picture in Wikipedia:
using equations such as those outlined here.
...
10
votes
2
answers
790
views
Parametric plot on multi-dimensional domain
I wish to plot all the points in 3 space that satisfy the following conditions.
$$ (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.
...
10
votes
2
answers
978
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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 ...
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 ...
10
votes
2
answers
918
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Convert BSplineFunction into two Interpolating Functions
Is there a way to decompose a BSplineFunction output into two functions x[t], y[t] ...
10
votes
1
answer
653
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Issuing with ParametricNDSolveValue
I am having a problem with ParametricNDSolveValue[] . let me start with a simple example. Considered following example with constant parameter $a$.
...
9
votes
3
answers
3k
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My 3D plot of a Klein bottle doesn't look right
I have found the following parametrisation for a Klein bottle:
...
9
votes
4
answers
3k
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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....
9
votes
1
answer
1k
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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 ...
9
votes
2
answers
717
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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 ...
9
votes
2
answers
472
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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 ...
9
votes
2
answers
317
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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 ...
9
votes
2
answers
4k
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ParametricPlot3D with three variables
ParametricPlot3D can be used with one or two variables to plot paths and surfaces in 3-D such as
...
9
votes
1
answer
1k
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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 ...
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 ...
8
votes
5
answers
737
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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,...
8
votes
3
answers
6k
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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 ...
8
votes
4
answers
2k
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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 $\...
8
votes
3
answers
1k
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How to combine ParametricPlot and RegionPlot?
I have two implicitly defined equations. I plot the implicit curve using parametric plot.
...
8
votes
2
answers
621
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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 ...
8
votes
2
answers
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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 ...
8
votes
2
answers
3k
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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 ...
8
votes
1
answer
571
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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 ...
8
votes
2
answers
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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 ...
7
votes
2
answers
501
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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 ...
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 ...
7
votes
2
answers
3k
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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 ...
7
votes
3
answers
405
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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$ ...
7
votes
2
answers
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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 ...
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:
...
7
votes
1
answer
330
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Finding "nice" solution for under-determined system with constraint?
Consider the following equation:
...
7
votes
3
answers
722
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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.
...
7
votes
1
answer
1k
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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$...
7
votes
2
answers
1k
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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
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 ...
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 ...
6
votes
4
answers
3k
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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" ...
6
votes
6
answers
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How to wrap a cylinder?
i'm trying to wrap a cylinder in a torus, the best i've done is the following code:
...