This example treats the segment as parameterized vector where the parameter `t` varies from `0` to `1`. It finds the value of `t` that minimizes the distance from the point to the line.

If `t` is between `0.0` and `1.0`, then the point on the segment that is closest to the other point lies on the segment. Otherwise the closest point is one of the segment’s end points. The program finds this closest point and calculates the distance between it and the target point.

The following code shows how the program finds the distance between the point `pt` and the segment `p1 --> p2`.

// Calculate the distance between // point pt and the segment p1 --> p2. private double FindDistanceToSegment( PointF pt, PointF p1, PointF p2, out PointF closest) { float dx = p2.X - p1.X; float dy = p2.Y - p1.Y; if ((dx == 0) && (dy == 0)) { // It's a point not a line segment. closest = p1; dx = pt.X - p1.X; dy = pt.Y - p1.Y; return Math.Sqrt(dx * dx + dy * dy); } // Calculate the t that minimizes the distance. float t = ((pt.X - p1.X) * dx + (pt.Y - p1.Y) * dy) / (dx * dx + dy * dy); // See if this represents one of the segment's // end points or a point in the middle. if (t < 0) { closest = new PointF(p1.X, p1.Y); dx = pt.X - p1.X; dy = pt.Y - p1.Y; } else if (t > 1) { closest = new PointF(p2.X, p2.Y); dx = pt.X - p2.X; dy = pt.Y - p2.Y; } else { closest = new PointF(p1.X + t * dx, p1.Y + t * dy); dx = pt.X - closest.X; dy = pt.Y - closest.Y; } return Math.Sqrt(dx * dx + dy * dy); }

The least obvious part of this code is the following statement.

// Calculate the t that minimizes the distance. float t = ((pt.X - p1.X) * dx + (pt.Y - p1.Y) * dy) / (dx * dx + dy * dy);

So where does this formula come from?

To find the shortest distance between that point and the line segment, you need to know some relatively easy calculus and one clever fact. The clever fact is that if T minimizes an equation, then T^{2} minimizes the equation squared. In this example that means we can minimize the distance squared between the point and the line segment, and then the value `t` that we find will also minimize the non-squared distance.

A point on the line segment has coordinates X = Pt1.X + t*dx, Y = Pt1.Y + t*dy.

The distance squared between that point and the point P is:

[Pt.X - (Pt1.X + t*dx)]^{2}+ [Pt.Y - (Pt1.Y + t*dy)]^{2}

Taking the derivative with respect to t gives:

2*[Pt.X - (Pt1.X + t*dx)]*dx + 2*[Pt.Y - (Pt1.Y + t*dy)]*dy

To find the minimum, we set this equal to 0 and solve for t.

2*[Pt.X - (Pt1.X + t*dx)]*dx + 2*[Pt.Y - (Pt1.Y + t*dy)]*dy = 0

Now if you divide both sides by 2 and then combine the t terms you get:

-t*(dx^{2}+ dy^{2}) + dx*(Pt.X - Pt1.X) + dy*(Pt.Y - Pt1.Y) = 0

Subtracting the t term from both sides of the equation gives:

dx*(Pt.X - Pt1.X) + dy*(Pt.Y - Pt1.Y) = t*(dx^{2}+ dy^{2})

Now you can divide both sides by (dx^{2} + dy^{2}) to get:

t = [dx*(Pt.X - Pt1.X) + dy*(Pt.Y - Pt1.Y)] / (dx^{2}+ dy^{2})

That’s the equation used in the code.

Thank you! This worked great for me. 🙂

Thank you!

Finally a function that works for line segments and not just for infinite lines.

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THX BRO!!!

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I have been getting some strange results when using this method is minus numbers typically less and 1. Does any one know if this method should also work correctly for

a)Negative numbers x / y input

b) when x / y could be less than 1.

Also is there a name for the theory / method this code is derived from. I really want to use this method as its alot faster than the alternatives!

It seems to work for me. Can you give me some example coordinates where it doesn’t seem to work?

I don’t know if this method has a name. I derived it a long time ago. If you use it in your code, you might include a link to the page so you can find it later if you need to.

Thank you!

can you just give the code in my sql

I don’t think MySQL (or any database tool) has any code to do this. You would need to write some code to get the segments’ end points and perform the calculation.

Been years since I’ve had to look at Algebra so probably a silly question, P1 and P2 does it matter which point on the line segment is which (i.e. most left needs to be P1) or does it make no difference?

Thx

This isn’t a silly question at all. The short answer is, no it doesn’t matter.

The value t tells you how far you go from point A to point B. If you switch them around, then t tells you how far you go from B to A, but you get the same point either way.

The reason why this isn’t a silly question is that this method uses vector arithmetic and the parameter t to make sure it doesn’t matter which order you use. If you try to solve equations like y = mx + b, you run into special cases for things like a vertical line (where the slope m is infinite). This technique lets you handle all cases the same way whether the line is vertical, horizontal, or you swap the order of the points.

How can i call this function?

If you download the example, you should be able to see how the code does it.

Thank you!

Good day sir!

I’m a student studying C# for my programming subject. I tried replicating this program to function on a Picturebox object in VS2013 but it doesn’t work. I’m currently comparing the code of the two programs, your original version and my modified version and to my rookie eyes, I couldn’t find a difference. I think I’m missing something to allow this program to work inside the Picturebox.

Did you download the example? Or just copy and paste the code into your program? The example has more code that’s not shown here to hook up the PictureBox to event handlers that let you pick the segment and point.

“float t = ((pt.X – p1.X) * dx + (pt.Y – p1.Y) * dy) /

(dx * dx + dy * dy);”

Can you explain where this formula comes from? I’ve been racking my brain this morning but it’s been too long since I took Linear Algebra or Calc 3. I understand the concept of a parametric equation, just not why the above formula produces the t that minimizes distance. Thanks!

I’ve added an explanation to the bottom of the post above.

Thank you! Very clear explanation. I had missed the fact that you could minimize the square of the function, so I was left with a really ugly equation.