Intersection between Plane and Sphere in 3d space

Qwertyuiop[]

Qwertyuiop[]

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Hi, I have this question. We are given a point Q and a plane. The distance between the plane and point Q is 1. A sphere is centered at point Q with radius 2. What is produced when sphere and plane intersect. I got "the empty set" because i drew a diagram exactly like in the question. We know the size of the sphere but don't know how big is the plane. I will attach a quick sketch of my diagram. So the empty set because they don't intersect. We get different answers if the size of the plane is not specified. If the plane was bigger then the intersection would be a circle. I did a google search on the question and they say we can either get a point or a circle when a sphere and a plane intersect so other options like a) an ellipse and c) parabola are not possible ?
 

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Qwertyuiop[] said:
Hi, I have this question. We are given a point Q and a plane. The distance between the plane and point Q is 1. A sphere is centered at point Q with radius 2. What is produced when sphere and plane intersect. I got "the empty set" because i drew a diagram exactly like in the question. We know the size of the sphere but don't know how big is the plane. I will attach a quick sketch of my diagram. So the empty set because they don't intersect. We get different answers if the size of the plane is not specified. If the plane was bigger then the intersection would be a circle. I did a google search on the question and they say we can either get a point or a circle when a sphere and a plane intersect so other options like a) an ellipse and c) parabola are not possible ?
Click to expand...
I'm afraid you haven't drawn your "diagram exactly like in the question". Why have you drawn a line segment of length approximately 1 unit (or so) to represent the plane?
A "plane" would normally be a surface of infinite 2 dimensions, so, if you extend your straight line (at least outside the circumference of your circle) you should then see that "a circle" is the correct answer (as given).
 
The Highlander said:
I'm afraid you haven't drawn your "diagram exactly like in the question". Why have you drawn a line segment of length approximately 1 unit (or so)? A "plane" would normally be a surface of infinite 2 dimensions, so, if you extend your straight line (at least outside the circumference of your circle) you should then see that a circle is the correct answer (as given).
Click to expand...
Hmm so I have to assume the plane is infinite ?
 
Qwertyuiop[] said:
Hi, I have this question. We are given a point Q and a plane. The distance between the plane and point Q is 1. A sphere is centered at point Q with radius 2. What is produced when sphere and plane intersect. I got "the empty set" because i drew a diagram exactly like in the question.
Click to expand...
There are three ideas of distance going on here.
1) the distance from a point to a plane is the shortest perpendicular line segment from the point to the plane.
2) the distance from a point to a sphere.
3) the distance from a point to the centre of the sphere.
The point [imath]Q[/imath] is one unit from the plane [imath]\pi[/imath].
A circle [imath]C[/imath] of radius [imath]2[/imath] is centered at [imath]Q[/imath]
Thus because the distance of the centre of [imath]C[/imath] to the plane [imath]\pi[/imath] is less than the radius of the sphere they will intersect.
That intersection is a circle.

[imath][/imath][imath][/imath][imath][/imath]
 
Qwertyuiop[] said:
What about the other options ? Can you get an ellipse or parabola ?
Click to expand...
No, not while the distance from Q to the plane is perpendicular to the plane (see pka's Point 1, above).
Could you imagine making a perfectly straight cut through something like an orange* that would expose anything other than a circular surface of the fruit's flesh?
*further imagine the orange to be perfectly spherical. ?
 
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