Intersection between two (perpendicular) lines

Question

Let's assume I have a (infinite) line defined as y = z * x how can I find the closest point in this line to any given coordinate? Technically I seek the intersection between the initial line and its perpendicular passing on the given coordinate.

I found libraries which check for possible intersections between segments, but I would like to find a simpler formula since I know the intersection exists. — Nuthinking, Jul 24 '18 at 16:18
`y =z * x` is not a line, but a ruled surface. Setting a plane (e.g. z= k) gives a line and the problem becomes a 2D case, very documented. — Ripi2, Jul 24 '18 at 18:09

score 1 · Accepted Answer · answered Jul 24 '18 at 16:53

1

Let (x, zx) be a point on the given line, and (u, v) the outside point.

The squared distance is

(x - u)² + (zx - v)² = (z² + 1) x² - 2 (u + zv) x + u² + v²

and the minimum of this quadratic expression is achieved by

x = (u + zv) / (z² + 1)

giving you the orthogonal projection of the point onto the line.

answered Jul 24 '18 at 16:53

Something doesn't work on my implementation: https://codepen.io/nuthinking/pen/8b9d12eca5c66011b5645d595c49ae82?editors=0011 – Nuthinking Jul 24 '18 at 17:59
@Nuthinking When I changed your code into y(x) dependence instead of weird mix, it became to work as needed – MBo Jul 24 '18 at 19:00
@MBo do you have a fork? – Nuthinking Jul 24 '18 at 20:11

score 0 · Answer 2 · answered Jul 24 '18 at 17:07

0

I did a line intersection sample a few weeks a go. You could try that:

https://github.com/feldhaus/math-geometry-playground/blob/master/line-intersection/index.html

answered Jul 24 '18 at 17:07

Maicon Feldhaus

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looks like a 2 segments intersection. – Nuthinking Jul 24 '18 at 17:11

Intersection between two (perpendicular) lines

2 Answers2