I want to sketch the parametric euqations lies on the cone, but the cone is so small and the lines are sparser than it should be.

parametric euqations is x=tcost, y=tsint, z=t

and the euqation of the cone is z^2=x^2+y^2

my code looks like this

t=linspace(-2*pi,2*pi,100);

x=t.*cos(t);

y=t.*sin(t);

z=t;

plot3(x,y,z);

hold on;

u=linspace(-2*pi,2*pi,100);

v=linspace(-2*pi,2*pi,100);

[u,v]=meshgrid(u,v);

x=cos(u).*sin(v);

y=cos(u).*cos(v);

z=cos(u);

surf(x,y,z)

I can't find any mistake in my code.

Answer is that the line should be denser.

What is the problem of my code?

I appreciate any help. Thank you.

Answer 1

[Star Strider]

Scale the size of the cone up by multiplying its variables with some appropriate constant (that I call a ‘Magnification Factor’ here).

Example —

t=linspace(-2*pi,2*pi,100);
x=t.*cos(t);
y=t.*sin(t);
z=t;
plot3(x,y,z, '-r', 'LineWidth',2);
hold on;
u=linspace(-2*pi,2*pi,100);
v=linspace(-2*pi,2*pi,100);
[u,v]=meshgrid(u,v);
x=cos(u).*sin(v);
y=cos(u).*cos(v);
z=cos(u);
mf = 6;                                                 % ‘Magnification Factor’
surf(x*mf,y*mf,z*mf, 'EdgeColor','none')
grid on
axis('equal')

producing:

.

[수정 이]

I really appreciate your help! Thanks sir!

Actually, I found that the major problem was the range of linespace. I broaden it -6*pi to 6*pi and the parametric equation worked well.

However, I found that without 'mf', the cone is still too small. What is the actual function of 'mf'? How can I know that there's no mathematical problems using mf despite the change of the size of cone?

[Star Strider]

As always, my pleasure!

The ‘mf’ variable (that actually should be ) simply scales the cone up to the correct size to match the length of the red line. If you increased ‘t’ from to , the cone size should be increased proportionally.

If the code changes slightly to add scaling factor ‘sf’ (used both for the line and the cone), eliminate ‘mf’ (since it is no longer necessary with ‘sf’), increase the number of elements in ‘t’ to make the line smoother, (and changing the cone code slightly to make the surface smoother), everything scales automatically and the plot looks better:

sf = 6*pi;                                                      % Scale Factor
t=linspace(-sf,sf,250);
x=t.*cos(t);
y=t.*sin(t);
z=t;
plot3(x,y,z, '-r', 'LineWidth',2);
hold on;
u=linspace(-2*pi,2*pi,100);
v=linspace(-2*pi,2*pi,100);
[u,v]=meshgrid(u,v);
x=cos(u).*sin(v);
y=cos(u).*cos(v);
z=cos(u);
surf(x*sf,y*sf,z*sf,'FaceAlpha',0.95)
grid on
shading('interp')
axis('equal')

producing:

Changing ‘sf’ to whatever you want will scale everything else proportionally, without changing anything else in the code.

Increassing the number of elements in ‘t’, ‘u’, and ‘v’ will increase the resolution of the line and the cone, creating a better-looking plot. Also, you can use the view function to rotate the plot to see it from different angles.

[수정 이]

Thank you very much. I really aprreciate your help!

I understood it finally.

Have a nice day :)

[Star Strider]

As always, my pleasure!