Extending a line between 2 points in 3D space

Question

Let's say I have 2 points in 3D space, one at:

x=2, y=3, z=5

and the second one at:

x=6, y=7, z=10

What is the fastest way, in code, to calculate the coordinates of a third point from extending (for example, doubling) the distance between those two points (relative to point one)?

Answer 1

If you want a point extended as far beyond (x2,y2,z2) as that is beyond (x1,y1,z1):

x3 = x2 + (x2 - x1) (= 10)
y3 = y2 + (y2 - y1) (= 11)
z3 = z2 + (z2 - z1) (= 15)

or:

(x2 * 2 - x1, y2 * 2 -y1, z2 * 2 - z1)

Simple as that.

If you want something other than double the length, you can scale the (x2 - x1)-type terms. For example, if you want it 50% longer than the current line, multiply them by 0.5 (+50%). If you want it three times longer, multiply them by two (+200%).

In terms of code that can perform this extension, something like this, which gives you an endpoint pDest that, along with, p1 forms a line percent times the size of p1-p2:

typedef struct {
    double x;
    double y;
    double z;
} tPoint3d;

void extend (tPoint3d *p1, tPoint3d *p2, double percent, tPoint3d *pDest) {
    percent -= 100.0;                             // what to ADD
    percent /= 100.0;                             // make multiplier
    pDest->x = p2->x + percent * (p2->x - p1->x); // scale each point
    pDest->y = p2->y + percent * (p2->y - p1->y);
    pDest->z = p2->z + percent * (p2->z - p1->z);
}