Re: Intersection of two surfaces in 3D

• To: mathgroup at smc.vnet.net
• Subject: [mg52868] Re: [mg52822] Intersection of two surfaces in 3D
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 15 Dec 2004 04:26:39 -0500 (EST)
• References: <200412141059.FAA24571@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Narasimham wrote:
> There are threads currently on sci.math  on this topic. How do we find
> space intersection curve of two  parameterized surfaces? One needs to
> solve for two unknown functions f1(t1,t2)=0 and f2(s1,s2)=0 to print
> out/output coordinates of intersection. I do believe it is within the
> capability of Mathematica, at least when surfaces are algebraically
> generatable. An example/approach considered is:
>
> Clear[x,y,z,t1,t2,s1,s2];
> x1=4*t2* Cos[t1]; y1=4Sin[t1]; z1=3t2;
> x2=s2 Sin[s1];y2=s2 Cos[s1];z2=(s2^2/4);
> pp1=ParametricPlot3D[{x1,y1,z1},{t1,0,2 Pi},{t2,0,1}];
> pp2=ParametricPlot3D[{x2,y2,z2},{s1,0,2 Pi},{s2,0,4}];
> Show[pp1,pp2];
> S1={x-x1,y-y1,z-z1}; S2={x-x2,y-y2,z-z2};
> NSolve[Join[S1,S2],{x,y,z},{t1,t2,s1,s2}];
>

I'm not sure exactly what you want but I'll show a couple of
computations that might be of use in the algebraic case (that is, when
everything can be cast in the form of rational functions in some set of
variables).

We start with the system below. I took the liberty of algebraicizing the
trigs and also of appending 1's for the parameters of the first surface
and 2's for parameters for the second surface. This has the perhaps
unfortunate effect of changing the locations of some of your parameters.

polys = {x-4*s1*ct1,  y-4*st1, z-3*s1,
x-s2*st2, y-s2*ct2,z-s2^2/4,
ct1^2+st1^1-1, ct2^2+st2^1-1};

To find a "nice" implicit form of the intersection we compute a Gorebner
basis in the space variables {x,y,z}, eliminating all parameters.

InputForm[gbvars = GroebnerBasis[polys, {x,y,z}, {s1,s2,ct1,st1,ct2,st2}]]

Out[10]//InputForm=
{9*y^4 - 72*y^2*z + 144*z^2 - 64*z^3 + 16*y*z^3, 9*x^2 -
16*z^2 + 4*y*z^2}

If instead you would like parameters for the intersection curve you can
eliminate the space variables and the parameters for one surface.

InputForm[gbparams = GroebnerBasis[polys, {s1,ct1,st1},
{s2,ct2,st2,x,y,z}]]

Out[12]//InputForm=
{-1 + ct1^2 + st1, 9*s1^2 - 12*s1^3 + 12*s1^3*st1 -
24*s1*st1^2 + 16*st1^4}

There are other techniques that apply to nonalgebraic parametrizations
(and to algebraic). For example one might attempt to track the
intersection curve via an ODE (or better still, DAE) system. It's a bit
of work to do by hand. You might want to look at a MathSource package by
Steve Wilkinson:

http://library.wolfram.com/infocenter/Articles/3260/

Daniel Lichtblau
Wolfram Research

```

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