MHB Help finding length of perpendiculars in a box of known dimension

AI Thread Summary
The discussion focuses on calculating the distances from a point within a box to its walls along perpendicular lines. The user has known distances from the center of each wall to the point of interest and seeks to determine the closest distances to the walls, labeled as B1 to B4. A mathematical approach is proposed, using a coordinate system centered in the box and a system of equations to relate the known distances (C1 to C4) to the unknown coordinates (x, y). The solution provided allows the user to derive the distances to the walls effectively. This method successfully addresses the user's research problem.
ttk
Messages
2
Reaction score
0
For a research problem, I'd like a way to find the distance of each of 4 lines perpendicular to one of 4 walls connected to a point that is within a box of known dimension. I know the distance from the center of each wall to the point of interest (C1 to C4), but I do not know the angle this line makes relative to the wall (usually it will not be perpendicular). I'm attaching an image of the problem. C1 to C4 are known, as is the dimension of the square box. What I want to know is B1 to B4. My overall goal is to understand the distance of the point of interest (which could be anywhere within the box) from the nearest wall along a perpendicular to that wall (it's closest distance from the wall).

View attachment 5915
 

Attachments

Mathematics news on Phys.org
ttk said:
For a research problem, I'd like a way to find the distance of each of 4 lines perpendicular to one of 4 walls connected to a point that is within a box of known dimension. I know the distance from the center of each wall to the point of interest (C1 to C4), but I do not know the angle this line makes relative to the wall (usually it will not be perpendicular). I'm attaching an image of the problem. C1 to C4 are known, as is the dimension of the square box. What I want to know is B1 to B4. My overall goal is to understand the distance of the point of interest (which could be anywhere within the box) from the nearest wall along a perpendicular to that wall (it's closest distance from the wall).

Hey ttk! Welcome to MHB! ;)

Let's pick the center of the square to be our origin.
And let's pick the coordinates of the point of interest to be $(x,y)$.
Oh, we already had an $x$ for the side of the square. :eek:
Well, let's discard that one, and let's pick $h$ to be half the side of the square, if you don't mind.
I just like $x$ and $y$ to be my unknowns, and use other letters for known values.

With those choices, we have the following system of equations:
\[\begin{cases}
x^2 + (h-y)^2 = C_1^2 \\
(h-x)^2 + y^2 = C_2^2 \\
x^2 + (h+y)^2 = C_3^2 \\
(h+x)^2 + y^2 = C_4^2
\end{cases}
\Rightarrow\begin{cases}
x^2 + y^2 - 2hy = C_1^2 - h^2\\
x^2 + y^2 - 2hx= C_2^2 - h^2 \\
x^2 + y^2 +2hy = C_3^2 - h^2 \\
x^2 + y^2 +2hx = C_4^2 - h^2
\end{cases}
\Rightarrow\begin{cases}
4hx= (C_4^2-h^2) - (C_2^2 - h^2) \\
4hy = (C_3^2 - h^2) - (C_1^2 - h^2) \\
\end{cases}
\Rightarrow\begin{cases}
x= \frac{C_4^2 - C_2^2}{4h} \\
y = \frac{C_3^2 - C_1^2}{4h} \\
\end{cases}
\]

Does that satisfy your needs? (Wondering)
 
You cracked it! Thanks so much. With x,y, I can easily determine the distance to the walls. Can't thank you enough. Take care,

Terry
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top