Locus Problems in 3D

[surrealist]

A parabola is defined at the locus of points equally distant from a point (called a focus) and a line (called the directrix).

 

I would like to generalize on this idea.

 

My questions are:

1) In 3-space, what is the locus of points equally distant from a point and a plane? (My guess is a paraboloid.)

 

2) In 3-space, what is the locus of points equally distant from a line and a plane?

 

3) In 3-space, what is the locus of points equally distant from two lines?

 

Can any of you give me help in either computing or visualizing these loci?

[JS]

in 1) set your (0,0,0) in the proyection of the point in the plane. given a line over (0,0,0) using that point we have a parabola , "rotating" the line over the plane we have more parabolas by simetry we have there a paraboloid that satisfies the condition of equidistance...

 

i think that for visualize the others may be useful think about planes that intersect the lines and planes in case 2) & 3)... trying to get a point and a line over the cut-plane having a parabola over it

[DQW]

My questions are:

1) In 3-space' date=' what is the locus of points equally distant from a point and a plane? (My guess is a paraboloid.)[/quote']Correct.

 

2) In 3-space, what is the locus of points equally distant from a line and a plane?

I think it is a pair of congruent, coaxial, oblique cones with vertex at the point of intersection.

 

When the line is parallel to the plane, it is clearly the surface of translation (parallel to the line) of a parabola.

 

3) In 3-space, what is the locus of points equally distant from two lines?

In general, for a pair of skew lines, I guess this surface would be something like a distorted saddle.

 

For a pair of parallel (coplanar) lines, the locus would be the plane normal to the plane containing the lines and midway between them.

 

For a pair of intersecting, coplanar lines, the loci would be the pair of planes normal to the plane that contains them, and hwose intersections with this plane are the angle bisectors of the lines.

[anandvivek88]

Can you tell me How is the answer of 1st question as 'Parabloid' ?

[BigMoosie]

Can you tell me How is the answer of 1st question as 'Parabloid' ?

 

Visualise a cross section that passes through the point and is perpendicular to the plane, for any angle you choose it will be just the same as a porabola (i.e. equidistant to a point and a line) so it goes to reason that it will be a porabola rotated around its axis.