3D Perpendicular Point on Line From 3D point

Question

This question has been asked before in reference to 2D. This question extends it to 3D. How do I find the perpendicular intersected point on a line from a point in 3D space?. If my line is defined by points (x1,y1,z1) & (x2,y2,z2) and I have a point (x3,y3,z3) in space.

How do I find the perpendicular intersection of point (x4,y4,z4) on the line from (x3,y3,z3)?

Answer 1

For starters, you pretty much need some implementation of a Vector3 class, whether you write your own, find a standalone implementation on the internet somewhere, or use a library that contains one like XNA or Sharp3D.Math.

Typically lines in 3d space are not represented by two points, but by parametric equations and operated on by vectors and not scalars. Your parametric equation would be of the form:

x = x1 + t(x2-x1), y = y1 + t(y2-y1), z = z1 + t(z2-z1)

The vector u is defined by the coefficients of t. <x2-x1, y2-y1, z2-z1>.

The vector PQ is defined by your chosen point Q minus a point P on the line. Any point on the line can be chosen, so it would be simplest to just use the line t = 0, which simplifies to x1, y1, and z1. <x3-x1, y3-y1, z3-z1>

The definition of the shortest distance between a point and a line in 3-space is as follows:

D = ||PQ x u|| / ||u||

Where x is the cross product operator, and || ... || gets the magnitude of the contained vector. Depending on which library you choose, your code may vary, but it should be very similar:

Vector3 u = new Vector3(x2 - x1, y2 - y1, z2 - z1);
Vector3 pq = new Vector3(x3 - x1, y3 - y1, z3 - z1);

float distance = Vector3.Cross(pq, u).Length / u.Length;

Edit: I just realized you wanted the actual point of intersection, and not the distance. The formula to find the actual point is a bit different. You need to use inner product space to get the component of u perpendicular to PQ. To do that, you need to find the component of u in the direction of PQ:

((PQ · u) / ||u||^2) * u

This gets us the w1 component, but we want w2, which is the component between Q and the line:

PQ = w1 + w2

w2 = PQ - w1

From there, we take w2 and add it to the point Q to get the point on the line nearest Q. In code this would be:

Vector3 p1 = new Vector3(x1, y1, z1);
Vector3 p2 = new Vector3(x2, y2, z2);
Vector3 q = new Vector3(x3, y3, z3);

Vector3 u = p2 - p1;
Vector3 pq = q - p1;
Vector3 w2 = pq - Vector3.Multiply(u, Vector3.Dot(pq, u) / u.LengthSquared);

Vector3 point = q - w2;

Where point.X is x4, point.Y is y4, and point.Z is z4.

Answer 2

You're asking, in practice, the calculation of a distance between the point and line, so the Length of the vector from the (x3,y3,z3) point to the line which is orthogonal to the vector. If we immagine the line like a vector, that means that dot-product of both vectors is equal to 0. (this is just to give you a hint) .