This may belong to the computing subforum, let me know if this is more true than having it here in the math forum :)(adsbygoogle = window.adsbygoogle || []).push({});

My questions are

1) Suppose there is a plane in 3D space and I have 3 points to define it:

p_{1}= {x_{1}, y_{1}, z_{1}}

p_{2}= {x_{2}, y_{2}, z_{2}}

p_{3}= {x_{3}, y_{3}, z_{3}}

and I want to put a particular pointp_{4}which I already know the x- and z- coordinates.

What will be the most efficient way to compute the y- coordinate ofp_{4}?

I can think of the nasty method which I compute the determinant of a 4x4 matrix to find the coefficients for:

ax + by + cz + z = 0

Then substitute my known x and z, I get the feeling this is very inefficient and there are more elegant solutions than this. Are there any known algorithms to deal with this problem?

2) Now that I have myp_{4}, I want have the same relative position to the 3 pointsp_{1},p_{2},p_{3}if someone move this triangle around:

f(p_{1},p_{2},p_{3}) =p_{4}

for example, ifp_{4}happens to be in the middle of the triangle, the function would be the average of each of thex, y, zcoordinates of the 3 points. But for a point that is off-centered, how will I go about finding what this function should be?

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# I Putting a point inside a triangle in 3D space

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