Solving for 2 parametric equations

Hey everyone! I got a question: I'm trying to calculate where a sphere and a cone intersect. I've got 2 parametric functions describing a sphere and a cone, K and S. These are both functions of "s and t" which are the two vars describing each point.

(so S(s,t) and K(s,t) note that S is not equal to s)

What I want to do is solve for K==S. Any idea why matlab does not give me an answer?

The code is added below: it currently plots the 2 functions, if you remove the commented section u'll see the problem.

Thanks in advance!

syms t s theta r x0 y0 z0 c vx vy vz K(theta,s) S(theta,s) real

%%defining the constants
x0=1;
y0=1;
z0=1;
c=0.5;
vx=1;
vy=-2;
vz=3;
r=5;

P=[x0 y0 z0]';
A=[vx vy vz]';
x=A(:).'/norm(A); %normalise the input vector
yz=null(x).'; %find the null spaces of normalised A
xyz=[x;yz]  %The rows of this matrix are the axes of a normalised 
u=xyz(2,:)'
v=xyz(3,:)' %u and v are a ortogonal normal basis for normalised A
%t=(tan(theta+pi/2));
%K(t,s)=P+A*t+t*tan(C/2)*(u*cos(s)+v*sin(s))
K(theta,s)=P+x'*(tan(theta+pi/2))+(tan(theta+pi/2))*tan(c/2)*(u*cos(s)+v*sin(s))
%K described a cone with directional vector A, vertex(centre)=P and opening
%parameter=C
S(theta,s)=r*[sin(theta) 0 0;0 sin(theta) 0;0 0 cos(theta)]*[cos(s);sin(s);1]
%S described a sphere with centre @0,0,0 and radius r.

%I want to know the function that described the intersection of these 2
%objects. So for K=S.

%%solve(K==S) % does not work. Why?

%%plotting
KK=formula(K) %rewrite for indexing.
SS=formula(S)
%GG=formula(G)

figure
ezsurf(KK(1), KK(2), KK(3), [0,2*pi,0,pi])   
hold on

spheresurf=ezsurf(SS(1), SS(2), SS(3), [0,2*pi,0,pi])
set(spheresurf,'facealpha',0)
set(spheresurf,'edgecolor',[.2 .4 .9])

%inters=ezsurf(GG(1),GG(2),GG(3),[0,2*pi,0,pi])
%set(inters,'edgecolor',[.6 .9 .2])

%%there are clearly 2 circle like figure where the cone and the sphere
%%intersect. Why doesn't the solve function work?

[luc]

Right. I've been simplifying my formulas to see when the solve function will work. But even this simple example of finding the positions where the two circle intersect gives me 0 solutions.

syms x z
figure
ezplot(x^2+(z+1)^2==5^2)
hold on
ezplot(x^2+(z)^2==5^2)
solve(x^2+(z+1)^2==x^2+(z)^2)

could someone please tell me what I am doing wrong?

[Torsten]

You will have to tell which variable to solve for.

I guess that

solve(x^2+(z+1)^2==x^2+(z)^2,z)

will work.

Best wishes

Torsten.

[luc]

Thanks Torsten,

That works, but for my main problem this addition does not work. Any idea why that is?

[Torsten]

At least one problem is that a pair (theta,s) - applied to both surfaces - do not yield the same point in 3d-space.

I think this is the minimum requirement to make solve work for your problem.

Further: What do you expect as output from solve ? A parametrized 1d-curve ?

Some curves are that complicated that they can only be represented as the intersection of two surfaces - they don't have a representation on their own.

Best wishes

Torsten.

[luc]

Okay,

Thanks for the advice.

I think the output would be a semi-sphere in 3D. So a line in 3D.

If you look at this picture u can see that the outcome would indeed be something of that ilk.

I was thinking of rewriting my equations to match a cone with z vector, 0 origin and a sphere with a different centre to make it easier.

Any thoughts?

[Torsten]

Did you try to use "solve" if the result is 1d, 2d ... instead of 0d ?

I'd test it: Try to find the intersection of the sphere x^2+y^2+z^2=1 with the plane x+z=1.

And as I already said: Don't use a parametrization of your sphere/cone. Or a parametrization such that one pair (theta,s) - applied to both equations - yields the same point in 3d-space.

Best wishes

Torsten.