Need Help Solving a problem for work

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November 2nd 2012, 08:09 AM
macleodjb

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Need Help Solving a problem for work

Hi All,

I've been tasked at work for create an excel based product library for my company which will product parts. I'm working on a product that needs to be parametric. Please see the attached image
Attachment 25525
If the lengths of the angled lines were a fixed length i would know how to calculate all of this. But since they lengths are relative to the front radius I'm kinda at a loss on how to generate the x and y points for each line. Any help would be appreciated. I'm sure this is a snap for most of you but I suck at math especially trig.

Thanks in advance

Joe
November 2nd 2012, 11:09 PM
chiro

Re: Need Help Solving a problem for work

Hey macleodjb.

How much calculus do you know?

The reason is that you will know the tangent for the start and ends of the circle since you know the angles and you you have a formula regarding what the derivatives are for a given circle with an equation x^2 + y^2 = r^2 and the dy/dx will correspond to the gradient as a function of the angle.

So you have two gradients so you will essentially get a relation to where the (x,y) points are for the circle part (you get two solutions and discard one for each tangent) and this will give you the relative points for that circle which gives the (x,y) points at the two positions.
November 3rd 2012, 05:02 AM
macleodjb

Re: Need Help Solving a problem for work

Sorry I don't know any calculus. I googled derivatives but the math jargon is just way over my head. Would you mind breaking it down in laymens terms for me. Like Step 1 Angle1/Radius+Whatever.
November 3rd 2012, 09:23 AM
Soroban

Re: Need Help Solving a problem for work

Hello, Joe!

This takes a lot of work.
I have a start on it.
Maybe you or someone else can finish it.
(I assume the lower-left corner is a right angle.)

Code:

| | o P | * * | * * | * * | * * B| * β * (0,b)o - - - - - * | * b | * | * α - - * - - - - - - o - - - - - - - O a (a,0) A

The line $AP$ has slope $\tan\alpha.$
Its equation is: . $y \:=\:\tan\alpha(x-a) \quad\Rightarrow\quad y \:=\:x\tan\alpha - a\tan\alpha$ .[1]

The line $BP$ has slope $\tan\beta.$
Its equation is: . $y \:=\:x\tan\beta + b$ .[2]

Equate [1] and [2]:

. . $\begin{array}{c}x\tan\alpha - a\tan\alpha \:=\:x\tan\beta + b \\ x\tan\alpha - x\tan\beta \:=\: a\tan\alpha + b \\ x(\tan\alpha - \tan\beta) \:=\:a\tan\alpha + b \\ x \:=\:\dfrac{a\tan\alpha + b}{\tan\alpha - \tan\beta} \end{array}$

Substitute into [2]:

. . $y \:=\:\left(\frac{a\tan\alpha + b}{\tan\alpha - \tan\beta}\right)\tan\beta + b \quad\Rightarrow\quad y \:=\:\frac{\tan\alpha(a\tan\beta + b)}{\tan\alpha - \tan\beta}$

We know the coordinates of point $P\left(\frac{a\tan\alpha + b}{\tan\alpha - \tan\beta},\;\frac{\tan\alpha(a\tan\beta + b)}{\tan\alpha - \tan\beta}\right)$

We are given $r$ , the radius of the circle.

Find the equation of the line parallel to $AP$ (to the left of $AP$ )
. . and at a distance $r$ from $AP.$

Find the equation of the line parallel to $BP$ (and below $BP$ )
. . and at a distance $r$ from $BP.$

The intersection of these lines is the center of the circle.

Your turn . . .
November 3rd 2012, 03:44 PM
macleodjb

1 Attachment(s)

Re: Need Help Solving a problem for work

Attachment 25536Well, thus far I really appreciate the help. Sadly, I can't follow it. I figured I was on the right thought path with using slope. For me to understand this you will have to label the formula with things from the image I attached. I understand the idea that you're telling me but not the formula to get there. Yes the lower left corner is 90 degrees. So in order for us to communicate effectively I have added point names to my original image so you can use those in your formula.
November 3rd 2012, 04:13 PM
chiro

Re: Need Help Solving a problem for work

Remember that the derivative of a point for a circle is given by dy/dx = -y/x and tan(theta) = dy/dx
November 4th 2012, 07:31 PM
macleodjb

Re: Need Help Solving a problem for work

I've been trying to figure out what you guys are telling me but I haven't gotten very far. I figured out how to calculate the slope for each angle, but I dont know how to calculate the length of each segment to the point (p). Then i have to offset those lines the distance of the radius to get the center point. I understand it but cannot do it mathematically. Can you re-phrase