Plotting Vector Functions (Space Curves)
Lisa Oberbroeckling, Fall 2015
Contents
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You should always have the following commands at the top to "start fresh"
clc, clf % clears the command window and figure window clear % clears ALL variables format % resets the format to the default format
Example 1 (2D example)
In 2D, vector functions are no different than the parametric equations you saw in Chapter 10 in Calculus.
To plot them, you must first establish your domain for \( t \). The easiest way to do this is with the command LINSPACE.
t = linspace(0,2*pi);
IMPORTANT: notice the semi-colon at the end of these lines; if there is none, all of those values for \( t \)(and \( x\), \( y\), etc. below) will appear in your command window.
Now we enter the equations for \( x \) and \( y\), using component-wise calculations.
x = 3*cos(t); y = 2*sin(t);
Now we can plot the x and y values. For 2D plots, it's simple:
plot(x,y)
title('2D Vector Function Example 1')
Example 2 (2D example)
The above equations didn't need component-wise calculations. This example does: \( \vec{r}(t) = \left\langle \dfrac{\sin(2t)}{4+t^2}, \dfrac{\cos(2t)}{4+t^2}\right\rangle\)
t = linspace(0,10);
x = sin(2*t)./(4 + t.^2);
y = cos(2*t)./(4 + t.^2);
plot(x,y)
title('2D Vector Function Example 2')
Example 3 (3D example)
This example involves component-wise calculations, and the domain needed to be adjusted to show the entire trefoil knot.
Notice for a 3D space curve, the command is PLOT3 instead of PLOT.
It is always good to label the axes for perspective.
t=linspace(0,4*pi); x=(2+cos(1.5*t)).*cos(t); y=(2+cos(1.5*t)).*sin(t); z=sin(1.5*t); plot3(x,y,z) xlabel('x'), ylabel('y'), zlabel('z') title('3D Vector Function/Space Curve Example')
Example 4 (bad domain example)
The following is an example in thich the domain doesn't have enough points and it creates a jagged curve that should be smooth.
t=linspace(0,4*pi); x=cos(6*t); y=sin(6*t); z=t; plot3(x,y,z) xlabel('x'), ylabel('y'), zlabel('z') title('Bad Domain Example')
Example 3c (better domain)
The problem above was because by default the linspace(a,b) command creates one hundred values between a and b for MATLAB use to connect to plot the graph. By having linspace(a,b,n), you are specifying n values between a and b. So below, we have t be a vector of 500 values between 0 and 4pi with which to create the x,y, and z values to build points to connect into a graph.
t=linspace(0,4*pi,500); x=cos(6*t); y=sin(6*t); z=t; plot3(x,y,z) xlabel('x'), ylabel('y'), zlabel('z') title('Better Domain Example')
Example 5 (adjusting view)
You can adjust the view. Using this command may involve experimentation with the values and rerunning the script.
t=linspace(0,4*pi); x=(2+cos(1.5*t)).*cos(t); y=(2+cos(1.5*t)).*sin(t); z=sin(1.5*t); plot3(x,y,z) view(0,90) xlabel('x'), ylabel('y'), zlabel('z') title('Adjusting View on 3D Graph')
Example 6 (Sphere Command)
The SPHERE command will set up variables x, y, and z to form a unit sphere using the surf or mesh commands. Without going into details, here are examples
[x,y,z]=sphere(100); mesh(x,y,z) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere Command: Unit Sphere') axis equal axis([-5 5 -5 5 -5 5])
How to get one of a different radius (\( r=3 \)) than 1:
[x,y,z]=sphere(100); mesh(3*x,3*y,3*z) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere with different radius') axis equal axis([-5 5 -5 5 -5 5])
How to get a sphere with a different center \( C(2,-1,3) \) than the origin:
[x,y,z]=sphere(100); mesh(x+2,y-1,z+3) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere with center not at the origin') axis equal axis([-5 5 -5 5 -5 5])
QUESTION: How could you get ANY sphere - different center than origin and radius other than 2?
Here they are side-by-side for comparison
subplot(131) mesh(x,y,z) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere Command: Unit Sphere') axis equal axis([-5 5 -5 5 -5 5]) subplot(132) mesh(3*x,3*y,3*z) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere with different radius') axis equal axis([-5 5 -5 5 -5 5]) subplot(133) mesh(x+2,y-1,z+3) xlabel('x'), ylabel('y'), zlabel('z') title('Sphere with center not at the origin') axis equal axis([-5 5 -5 5 -5 5])
Example 7 (Multiple Plots on One Figure)
You must huse the hold on and hold off commands around the additional plot/plot3/mesh, etc. commands for all to show up within the same figure.
clf t = linspace(-pi,3*pi); x = 2*cos(t); y = sin(t); z = t; plot3(x,y,z, 'k') xlabel('x'), ylabel('y'), zlabel('z') title('Multiple Graphs in One Example') hold on % use this to add more plots to current figure t2 = linspace(-1,1); x2=-2*t2; y2=1 + 0*t2; z2=pi/2 + t2; plot3(x2,y2,z2) hold off % make sure you have this at the end