Curve of equidistant points on a plane

[borginz]

Dear Forum,
I am hoping for help with the following question:
I am designing an engineering measurement to measure distance by a laser beam to a plane.
Taking all the points on that plane, which have the same distance from laser eye (equidistant points on the plane from a point in space) - what curve will those points make on the plane?
If the plane was perpendicular to the laser beam, the curve of these equidistant points would obviously form a circle. But what if the plane is at a generic angle to the laser beam? Someone told me it's probably a parabola, which seems reasonable, but I am hoping for a mathematical explanation.
Thank you in advance!

 

[Deleted member 4993]

Dear Forum,
I am hoping for help with the following question:
I am designing an engineering measurement to measure distance by a laser beam to a plane.
Taking all the points on that plane, which have the same distance from laser eye (equidistant points on the plane from a point in space) - what curve will those points make on the plane?
If the plane was perpendicular to the laser beam, the curve of these equidistant points would obviously form a circle. But what if the plane is at a generic angle to the laser beam? Someone told me it's probably a parabola, which seems reasonable, but I am hoping for a mathematical explanation.
Thank you in advance!

It will be an ellipse. The plane of the screen will cut the cone of light rays created by the laser beam. The size and shape of the ellipse (major and minor axes) will depend on the distance of the source to screen and angle.

 

[borginz]

It will be an ellipse. The plane of the screen will cut the cone of light rays created by the laser beam. The size and shape of the ellipse (major and minor axes) will depend on the distance of the source to screen and angle.

Thank you for the quick answer! However, the points on the ellipse will not be at the same distance to the source of the laser beam. I agree that the shape resulting from the screen cutting the cone of light is an ellipse, but what is the (theoretical - not phisically visible) curve of the points that are at the same distance from the laser source?

 

[stapel]

Thank you for the quick answer! However, the points on the ellipse will not be at the same distance to the source of the laser beam. I agree that the shape resulting from the screen cutting the cone of light is an ellipse, but what is the (theoretical - not phisically visible) curve of the points that are at the same distance from the laser source?

In what sense are you saying that, yes, the set of points equidistant physically will form an ellipse but that, no, the mathematical equation of that set of points will not form an ellipse?

 

[Ishuda]

Dear Forum,
I am hoping for help with the following question:
I am designing an engineering measurement to measure distance by a laser beam to a plane.
Taking all the points on that plane, which have the same distance from laser eye (equidistant points on the plane from a point in space) - what curve will those points make on the plane?
If the plane was perpendicular to the laser beam, the curve of these equidistant points would obviously form a circle. But what if the plane is at a generic angle to the laser beam? Someone told me it's probably a parabola, which seems reasonable, but I am hoping for a mathematical explanation.
Thank you in advance!

Please define your generic angle. Is it just a rotation about a line in the plane (and if so which line) or possibly something else?

BTW: Just off hand with no proof, I believe you can set up the original co-ordinate as the perpendicular plane defined as
z = 0
with the location of the laser as
(x_L, y_L, z_L) = (0,0,z₀)
where z₀ is non zero and may be take as positive.

 

[borginz]

In what sense are you saying that, yes, the set of points equidistant physically will form an ellipse but that, no, the mathematical equation of that set of points will not form an ellipse?

What I'm saying is that ellipse is the intersection curve between the "laser cone" and the screen - agreed.
But what I'm looking for is the curve of equidistant points, which is not the same as the intersection. What do you think?

 

[Deleted member 4993]

What I'm saying is that ellipse is the intersection curve between the "laser cone" and the screen - agreed.
But what I'm looking for is the curve of equidistant points, which is not the same as the intersection. What do you think?

The only curve that is equidistant from a point is circle.