Intersecting Line & Unit Sphere: Find Point of Intersection

In summary, the conversation involves discussing the approach to finding the point(s) of intersection between a given line and the unit sphere. The suggested approach involves using the equation of a sphere and solving for t, while another approach involves finding the distance between the line and the origin and determining if it is less than 1. The conversation also briefly mentions using scalar projection.
  • #1
goonking
434
3

Homework Statement


Does the line through the points (−1, −1, −2) and (1, 2, 1) intersect the unit sphere? If so, find the point(s) of intersection.

Homework Equations

The Attempt at a Solution


do i also use r = r0 + vt but instead , use equation of sphere this time?

so it would be:

v=<2,3,3>

then using (-1,-1,-2) as the point

x = -1 + 2t y = -1 + 3t z=-2 + 3t

plugging these into x2 + y2 + z2 = 1

then solve for t

but then the math gets way too long and just doesn't seem to be the correct approach.
 
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  • #2
You could try finding how close the line gets to the origin. If it's less than 1, it must intersect the unit sphere.
 
  • #3
vela said:
You could try finding how close the line gets to the origin. If it's less than 1, it must intersect the unit sphere.
do I need to use scalar projection for this?
 
  • #4
What's wrong with your approach? I used it and got the answer quickly.
 
  • #5
goonking said:

Homework Statement


Does the line through the points (−1, −1, −2) and (1, 2, 1) intersect the unit sphere? If so, find the point(s) of intersection.

Homework Equations

The Attempt at a Solution


do i also use r = r0 + vt but instead , use equation of sphere this time?

so it would be:

v=<2,3,3>

then using (-1,-1,-2) as the point

x = -1 + 2t y = -1 + 3t z=-2 + 3t

plugging these into x2 + y2 + z2 = 1

then solve for t

but then the math gets way too long and just doesn't seem to be the correct approach.

The math does not get way too long; it is a bit messy, but sometimes that is how things are.
 
  • #6
Ray Vickson said:
The math does not get way too long; it is a bit messy, but sometimes that is how things are.
whoops, there seems to be an error from the copy paste, it should be x2 + y2 + z2 = 1 and not x2 + y2 + z2 = 1. I will fix it right now.

hmmm, I cannot seem to edit my OP.
 
  • #7
goonking said:
whoops, there seems to be an error from the copy paste, it should be x2 + y2 + z2 = 1 and not x2 + y2 + z2 = 1. I will fix it right now.

hmmm, I cannot seem to edit my OP.
You can edit your post provided you do so within a short period of time. I have fixed your post for you.
 

1. What is the concept of intersecting line and unit sphere?

The concept of intersecting line and unit sphere refers to finding the point where a straight line and a sphere with a radius of 1 unit intersect in three-dimensional space. This is a common problem in geometry and has many real-world applications.

2. How do you find the point of intersection between a line and a unit sphere?

To find the point of intersection, we can use the equations of the line and the sphere to create a system of equations. The solution to this system will give us the coordinates of the point of intersection. Alternatively, we can use vector algebra to find the intersection point by projecting the line onto the sphere's surface.

3. What are the different cases for the point of intersection between a line and a unit sphere?

There are three possible cases for the point of intersection between a line and a unit sphere. The first case is when the line passes through the center of the sphere, resulting in two points of intersection. The second case is when the line is tangent to the sphere, resulting in one point of intersection. The third case is when the line does not intersect the sphere at all.

4. What are the applications of finding the point of intersection between a line and a unit sphere?

The concept of intersecting line and unit sphere has various applications in fields such as computer graphics, physics, engineering, and navigation. It is used to determine the position of objects in three-dimensional space, calculate the distance between two points, and solve geometric problems, among others.

5. Can the concept of intersecting line and unit sphere be extended to higher dimensions?

Yes, the concept of intersecting line and unit sphere can be extended to higher dimensions, such as four or more dimensions. In higher dimensions, the unit sphere is replaced by a hypersphere, and the same principles and methods can be applied to find the point of intersection between a line and a hypersphere.

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