I can't figure out how to calculate next point on stretch (straight line). I mean, I have points $P_1$ and $P_2$ in one straight line. I need to find the next point ($P_3$) in this line where $P_3$ is in distance $h$ from point $P_2$ ($P_3$ is not between $P_1$ and $P_2$). Also distance from $P_1$ to $P_2$ is known as $r$ So far I was trying to use these expressions:

$h = \sqrt{(x_3-x_2)^2 +(y_3-y_2)^2}$ - distance from $P_3$ to$P_2$

$(h+r) = \sqrt{(x_3-x_1)^2 +(y_3-y_1)^2}$ - distance from $P_3$ to $P_1$

at the beginning I'm solving for $x_3$ in both expressions:

$x_3 = \sqrt{(r+h)^2-(y_3-y_1)^2} +x_1 \hspace{10cm}(1)$


$x_3 = \sqrt{h^2-(y_3-y_2)^2} + x_2\hspace{11cm}(2)$

Left side in both equations is $x_3$, so:

$\sqrt{(r+h)^2-(y_3-y_1)^2} +x_1 = \sqrt{h^2-(y_3-y_2)^2}+ x_2$

and that's where I need your help - I tried to solve for $y_3$:

$y3 = (r^2+2\cdot r\cdot h+x_1^2-y_1^2-x_2^2+y_2^2)/(2(y_2-y_1))$

but calculated result is incorrect. Also tried to use wolfram, but result is... https://www.wolframalpha.com/input/?i=sqrt((r%2Bh)%5E2-(y_3-y_1)%5E2)+%2Bx_1+%3D+sqrt(h%5E2-(y_3-y_2)%5E2)%2B+x_2+solve+for+y_3

Did I made a mistake somewhere? Or I should use other methods to solve it?

There is test data:

$P_2 (24;8)$
$h = 50$
$r = 25$

  • $\begingroup$ I find it very difficult to follow your post. If you have $(x_1, y_1), \;(x_2, y_2)$, you can determine the equation of the line passing through all three points. That, along with the distance equations you have, should define the coordinates of $(x_3, y_3)$ $\endgroup$
    – amWhy
    Commented Oct 22, 2016 at 20:33

2 Answers 2


The line can be parameterized by $\mathbf{x}(t) = (x_1 + t(x_2 - x_1),y_1 + t(y_2 - y_1))$ for all $t\in\mathbb{R}$. Since point $P_3 = (x_3,y_3)$ is on this line, there exists a $t$ such that $\mathbf{x}(t) = (x_3,y_3)$. Assuming the point $P_3$ comes after the point $P_2$ on the line, we can solve for $t$ by observing that the distance from $P_1$ to $P_3$ is

$$ r + h = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} = |t|\sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2} $$


$$ t = \frac{r + h}{\sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}} $$

Plugging this value of $t$ into $\mathbf{x}(t)$ will give you the coordinates of $P_3$.


The answer you've written down for y3 is incorrect. You've forgotten the cross terms. The wolfram alpha answer is accurate. However, as pointed out by K.Miller, you need not complicate this so much.


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