plotting a nonlinear curve in matlab

2015 8 11
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更新时间：2015 8 12
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I would like MATLAB to draw the plot for the curve described by the following equation, where for instance (a,b,d) = (-10,-10,-10) is a point in 3D cartesian space and c = 2. x and y are the variables. Are there any thoughts?
x^2(1+1/c.^2) -2ax + y^2(1+1/c.^2) - 2by+(a^2+b^2+d^2-r^2)-2d/c(sqrt(x^2+y^2)) = 0
ps. the equation is actually the intersection of a sphere and cone and can be seen in http://mathworld.wolfram.com/Cone-SphereIntersection.html

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Walter Roberson
Walter Roberson 2015-8-12

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If you have the Symbolic Toolbox, you can use solve(). But the solution involves a quartic so the solution will probably be in terms of RootOf(). You need two of the roots. Be careful, two of the roots might be spurious (and might be real-valued) so back-substitute and test before you accept a root at any given location.
For any given x you can use roots() to find the 4 y numerically:
roots([c^4+2*c^2+1, (-4*c^4-4*c^2)*b, (2*c^4+2*c^2)*a^2+(-4*c^4*x-4*c^2*x)*a+(6*c^4+2*c^2)*b^2+(2*d^2-2*r^2+2*x^2)*c^4+(-2*d^2-2*r^2+4*x^2)*c^2+2*x^2, -4*a^2*b*c^4+8*a*b*c^4*x-4*b^3*c^4+((-4*d^2+4*r^2-4*x^2)*c^4-4*c^2*x^2)*b, a^4*c^4-4*a^3*c^4*x+(2*b^2*c^4+(2*d^2-2*r^2+6*x^2)*c^4+2*c^2*x^2)*a^2+(-4*x*c^4*b^2+(-4*d^2*x+4*r^2*x-4*x^3)*c^4-4*x^3*c^2)*a+b^4*c^4+((2*d^2-2*r^2+2*x^2)*c^4+2*c^2*x^2)*b^2+(d^4+(-2*r^2+2*x^2)*d^2+r^4-2*x^2*r^2+x^4)*c^4+(-2*d^2*x^2-2*r^2*x^2+2*x^4)*c^2+x^4])
Be sure to cross-check the values.
For the a, b, c, d that you provide, the smallest positive r for which there are real-valued x and y that are solutions is r = 4*sqrt(5)+2*sqrt(5)*sqrt(2)

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MatG
MatG 2015-8-12
编辑：Walter Roberson 2015-8-12
Thanks for the reply Walter.
The code I wrote:
a= -10; b=-10; d=-10; c = 2; r= 50; syms x y z [Sx,Sy,Sz] = solve( [(x-a)^2 + (y-b)^2+(z-d)^2 == (r^2), x^2+y^2 - (c^2)*z^2 == 0],[x,y,z]);
The output is:
??? Error using ==> char Conversion to char from logical is not possible.
Error in ==> solve>getEqns at 165 vc = char(v);
Error in ==> solve at 67 [eqns,vars] = getEqns(varargin{:});
Error in ==> Quadric at 9 [Sx,Sy,Sz] = solve( [(x-a)^2 + (y-b)^2+(z-d)^2 == (r^2), x^2+y^2 - (c^2)*z^2 == 0],[x,y,z]);
Can any one help with this?
Walter Roberson
Walter Roberson 2015-8-12
I do not know at the moment why char is being mentioned. The code you posted is not the same as the code the error message was generated for: the code you posted is all on one line but the error message was for line 9. This does make a difference because you have no ";" between the "syms" and the assignment following in the code you posted.
If your MATLAB version is before (I think it is) R2010b, you might have to rewrite your comparisons A==B as (A)-(B) and allow it to assume the implicit "== 0".

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