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Tangent line to a sphere

Posted on 2004-04-15
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Last Modified: 2012-08-14
I have a arbitrary point on a sphere.  The sphere has an arbitrary radius.

I want to find a tanget line to the point on the sphere, or perhaps more correctly put, I want to find the tangent plane and from the tanget plane obtain any arbitrary line within the plane that touches the point on the sphere.
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Question by:bbangerter
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by:
ozo earned 100 total points
ID: 10839495
If the point on the sphere is x1,y1,z1, and the center of the sphere is x0.y0.z0
the tangent plane is (x-x1)(x1-x0)+(y-y1)(y1-y0)+(z-z1)(z1-z0)=0
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by:PointyEars
PointyEars earned 25 total points
ID: 10841152
ozo (as usual!) has been quick, good, and to the point (pun intended), but perhaps you would like to know a bit more about the solution.

A generic vector with origin in (x1,y1,z1) has the following components:
  (x-x1,y-y1,z-z1)

A vector corresponding to the radius of the sphere ending on (x1,y1,z1) has components:
  (x1-x0,y1-y0,z1-z0)

Two vectors are perpendicular when their scalar product is zero.  If you apply this to the two vectors just defined you get ozo's formula:
  (x-x1)(x1-x0)+(y-y1)(y1-y0)+(z-z1)(z1-z0)=0

You only need to remove the parentheses to get the equation of a plane in the usual form:
  a*x + b*y + c*z + d = 0
This plane passes through (x1,y1,z1) and, being perpendicular to the radius ending in the same point, is tangent to the sphere.
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by:bbangerter
ID: 10852047
Thanks ozo and thanks PointyEars for giving me that connection between ozos formula and the standard plane equation.
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