Question 824160: Hi guys,
I have an equation which i needed to convert to parametric form. Have been stuck for awhile and couldn't find any articles that can help. So please do advice.
Given A to G are constant variables:
sqrt( (X-B)^2 + (Y-C)^2 + (Z-D)^2 ) - sqrt( (X-E)^2 + (Y-F)^2 + (Z-G)^2 ) = A
So I'm suppose to find the parametric equation of X, Y and Z in terms of t.
Thanks a ton!
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! Note that the two square root equations are equations for two sphere's with centers at (B, C, D) and (E, F, G) and the square roots represent the radius of their respective spheres. That is,
let t' = sqrt( (X-B)^2 + (Y-C)^2 + (Z-D)^2 ) and
t'' = sqrt( (X-E)^2 + (Y-F)^2 + (Z-G)^2 )
sphere one
X = B + t' * cos(theta) cos(phi)
Y = C + t' * cos(theta) sin(phi)
Z = D + t' * sin(theta)
and 0 <= theta < 2 pi, and -pi/2 <= phi <= pi/2
sphere two
X = E + t'' * cos(theta) cos(phi)
Y = F + t'' * cos(theta) sin(phi)
Z = G + t'' * sin(theta)
and 0 <= theta < 2 pi, and -pi/2 <= phi <= pi/2
therefore we have
A = 0 if t' = t''
A > 0 if t' > t''
A < 0 if t' < t''
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