Is the point (-3,-4,2) visible from the point (4,5,0) if there is an opaque ball of radius 1 cen

step 1 So we have an interesting question here. If we are standing at the point 4, 5, 0, is the point n

Is the point (-3,-4,2) visible from the point (4,5,0) if...

Key Concepts

Line-of-Sight in 3D Space

This concept refers to the direct, unobstructed visual path between two points in a three-dimensional environment. Determining line-of-sight involves checking if any obstacles are in the way of the straight line joining the points, which is fundamental in both theoretical geometry and practical applications like computer graphics and robotics.

Ray-Sphere Intersection

This is a mathematical method used to assess whether a ray or line segment intersects a sphere. By parameterizing the line connecting two points and using the sphere’s equation, one can solve for the intersection points. If the line segment passes through the sphere's volume, the sphere obstructs the view between the two points.

Geometric Obstruction

This concept pertains to any object that blocks the view between two points, preventing a clear line-of-sight. In visibility problems, determining if an opaque object, such as a sphere, intersects the path between points is key to deciding whether one point is visible from the other.

Transcript

00:01 So we have an interesting question here.

00:04 If we are standing at the point 4, 5, 0, is the point negative 3, negative 4, 2 visible if there's an opaque ball setting at the origin here? let's see, our line of sight from 4, 5, 0 to minus 3, minus 4, 2 can be parametrized as such.

00:24 We have our 4, 5, 0 times 1 minus t plus negative 3, negative 4, 2 times t, which ends up being, let's see, 4 minus 4t minus 3t, 5 minus 5t minus 4t, and just 2t here is 4 minus 7t, 5 minus 9t, 2t.

01:01 So this line segment, this is a parameterization for the line segment from 4, 5, 0 to minus 3, minus 4, 2, and we want to see if that is obstructed by our ball radius 1.

01:14 Well, it is obstructed if and only if there's a point on here that is inside the ball, so to speak.

01:24 If we imagine doing this in two dimensions, here's some point, here's some other point, here's a ball.

01:31 If the straight line path between them is outside the ball, here you can see it's barely outside, then it's perfectly fine, it's visible.

01:42 Whereas if the straight line path, say in this case, would cross the ball, it's no longer visible.

01:48 That occludes our vision.

01:51 So we'd like to see, is there a point on this that has radiance that is inside the ball, which is to say that it is within one distance, one of the origin.

02:03 Well, the distance here is going to be the square root of 4 minus 7t squared, plus 5 minus 9t squared, plus 2t squared.

02:16 We want to see, is this ever equal to 1? in fact, we can ignore the square root here, because that makes math substantially harder, and it doesn't actually change the 1.

02:29 We can see if the square distance is ever equal to 1.

02:32 And then let's just expand this out...

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