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Let's assume $P_1=(x_1, y_1)$ and $P_2=(x_2, y_2)$ and $P_3=(x_3, y_3)$.

How to find the closest distance between $P_3$ and the line segment between $P_1$ and $P_2$?

I tried using the formula $\frac{area(P_1, P_2, P_3)}{distance(P_1, P_2)}$: $$\operatorname{distance}(P_1, P_2, P_3) = \frac{|(x_2-x_1)(y_1-y_3)-(x_1-x_3)(y_2-y_1)|}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$$

Sadly it seems like it only works for infinite lines, not for line segments like in my case.

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    $\begingroup$ How do you define the distance from a point to a segment? $\endgroup$
    – user
    Apr 29 at 12:06
  • $\begingroup$ @user I want the closest* distance between a point and a segment $\endgroup$ Apr 29 at 12:11
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there is three area to take in count:

enter image description here

For $P1$ the distance is $|P1-B|$
For $P3$ the distance is $|P3-A|$

to summarise, it juste the distance between two point when the point P is not between the two perpendicular line passing trough A and B.

the hardest part is P2 :

since it between the two point, the distance to the segment is the distance to the point $C$ on the segment $AB$. We can know where the point C is located with a parameter $h$ (the doted line) because of how a segment is parameterised.

$$(A,B) = (hx_B + (1-h)x_A,hy_B + (1-h)y_A)$$

if C is on B then $h=1$ and $h=0$ if C is on A, in the exemple h might be equal to 0,75 or 0,8.

To find h we need to do :

$$h = \frac{<P_2-A, B-A>}{|B-A|^2}$$

this is a projection of the length of the vector $\overrightarrow{P_2A}$ on to $\overrightarrow{BA}$ we use a dot product for that, then devide it to the length of $\overrightarrow{BA}$ then normalise it ench the power of 2.

then to find the distance $|P2-C|$ we do :

$$|P_2 - (A +h*(B-A))|$$

to finish we can compute the 3 expression at the same time wich lead us to :

$$h = min(1,max(0,\frac{<P_2-A, B-A>}{|B-A|^2}))$$ $$d = |P - A - h*(B-A)|$$

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The line segment can be parameterized as $$(x,y) = (tx_2+(1-t)x_3,ty_2+(1-t)y_3) $$ for $0\le t\le 1$. The distance (squared) from $P_1$ to any point $(x,y)$ on this line is then $$(tx_2 +(1-t)x_3 -x_1)^2 +(ty_2 +(1-t)y_3 -y_1 )^2 $$ This is a positive quadratic in $t$. To minimize this, first check the $t$ value ($t_0$) for the minimum, by differentiating and setting it equal to zero.

$\bullet $ If $t_0 \in [0,1]$, choose $t=t_0$.

$\bullet$ If $t_0 \gt 1$, choose $t=1$.

$\bullet$ If $t_0\lt 1$, choose $t=0$.

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Let $d_{ij}$ and $d_{i(jk)}$ denote the distance between points $i,j$ and the distance between the point $i$ and the line $(jk)$ respectively.

Then the distance you are looking for is $$ \big[d_{3(12)}\big]_{d_{12}^2\ge\left|d_{31}^2-d_{32}^2\right|}+\big[\min(d_{31},d_{32})\big]_{d_{12}^2<\left|d_{31}^2-d_{32}^2\right|}. $$

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  • $\begingroup$ Don't $d_{3(12)}$ gives me the same issue as it uses the same formula I tried before? $\endgroup$ Apr 29 at 12:25
  • $\begingroup$ What I meant is that in order to find $d_{3(12)}$, we need to use the formula $\frac{area(P_1, P_2, P_3)}{distance(P_1, P_2)}$, right? $\endgroup$ Apr 29 at 12:28
  • $\begingroup$ @NathanaelDemacon Right. $\endgroup$
    – user
    Apr 30 at 6:45

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