Parametric Equation for Plane-Sphere Intersection: How To Guide

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Homework Help Overview

The discussion revolves around finding the parametric equation for the intersection of a plane and a sphere, specifically the sphere defined by x²+y²+z²=1 and the plane given by x+y+z=0. The original poster seeks guidance on how to express this intersection in parametric form for use in a line integral.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods for parameterizing the sphere and substituting into the plane equation. Some express challenges in eliminating parameters and achieving a suitable form for the intersection. Others suggest using familiar forms and equations to guide the parameterization process.

Discussion Status

The discussion is ongoing, with participants sharing different perspectives on parameterization techniques. Some have provided partial insights into the process, while others express uncertainty about specific steps or assumptions. There is no clear consensus yet, but the dialogue appears to be productive.

Contextual Notes

Participants mention the need for a specific form of the intersection curve for application in a line integral, indicating that the context of the problem may impose certain constraints on the parameterization approach.

dagar
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I'm trying to find the parametric equation for the curve created by a plane intersecting a sphere. Typical sphere x^2+y^2+z^2=1, and plane x+y+z=0. I need the intersection in parametric form so I can use it as the curve in a line integral. I just need to know how to do this, not someone to do it for me.

Thanks
 
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hehe your in waterloo and you can't do this quesiton? hehe j/k Hows waterloo?

3D surface(sphere) requires to two parametrizations, then substitute into the plane equation and your resulting equation shohuld be rather familiar...
or if you'd like use the plane eq'n form N.P+D=0;
 
This is what I was trying but I couldn't get one of the parameters to disappear. I'm probably missing something obvious, which is quite often the case.

Even Waterloo has students that skip too many classes...
 
Last edited:
It is the curve x^2+y^2+xy=1/2, how you choose to use that would depend on the integral.
 
Sorry I should have clarified, finding the intersection is pretty trivial, I just never know how to parameterize the thing so I can use it as the curve in a line integral.
 
parametrization of a sphere is
x=s; y=t; z= ? ...substitue into the plane...get circle equation and arrange to something like (s-A)^2+(t-B)^2= R^2 (this should only been dependent on s,t...and other varaible should have a constant associate with R, or plane variables)
set s=new parametrization...u
and solve t interms of u.

...now you have x(s(u)), y(t(u)),z(s(u),t(u))..
 

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