Distance between a line and a point in three-dimensional space

Question

Let’s say there are three points in a 3D space (dots A, B and P). Let’s also say that there is a straight line through A and B.

What I’m trying to do, actually, is find out if a reta AB intercept one esfera de raio r that has origin in P. For this I just need to check if the distance of the line AB and the point P is less than the radius of the sphere.

Vector A, B, P; //A minha classe Vector, guarda três valores (x, y e z)

if(distance(A, B, P) < r){
   return sphere.color;
}

My problem is this method distance(Vector A, Vector B, Vector C)... How can I calculate the distance from a point to a line (given two points of the same), and I have to write this in JAVA?

The distance between a point and a line is the length of a semi-straight starting at P, and ending at line AB, forming an angle of 90 degrees with the same.

Discover the distance between a point and a straight line, drawing for example, it is relatively easy, just draw a straight line from the point to the other line and measure the size.

In 2D, this would be easy, since the reta D (the distance) forms a rectangular triangle, the hypotenuse being the vetor V representing the position of ponto P, and the adjacent cathode is this same vector designed (which I don’t know the length) on reta AB.

This way I could calculate the reta D, since I have the means to discover all three angles of the triangle (one is a right angle, and the other is the difference of the angles of the reta AB and the vetor V, and the third complete 180 degrees with the others), I can use the only side I have the value (the hypotenuse) and calculate the opposite cathode.

It turns out that I’m working in an environment three-dimensional, where it is not so simple to find an angle between the reta AB and the vetor V to complete the triangle. Also, I’m not a mathematician, but a programmer, So I wanted a method of figuring this out using algebraic expressions (and methods in Java), rather than geometric techniques, so that it worked for any line and point. After all math writing is not easy to convert into actual code.

Finally, I made this post here, because I may be trying to find the distance in an unnecessarily complicated way, or simply wrong.

From now on, thank you.

Answer 1

"Let’s say there are three points in a 3D space (points A, B and P). Let’s also say that there is a line that goes through A and B."

What I’m trying to do, in fact, is to find out if an AB line intersects a sphere of radius r that originates in P. For this I just need to check if the distance from the AB line and the P point is less than the radius of the sphere.

So what you want to find out is, given a straight R = [x,y,z]t + [a,b,c] (take a look here to know how to write the equation. You want to know if ||R - P|| <= r

And answer of this inequality of the "t" where the line and sphere intersect, is of second order. If there are real solutions the intersection exists, if not, there is no.