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Math people, I could use your help

dank69

dank69

Lifer
akdnBbO.png


Can anyone help me with a formula for determining the length of lines CD and EF? I know the radius of the big circle (r), the locations of points A & B, and the angle of lines AC and BD is always 45 degrees.

Thanks for any help.
 
Given the numbers I could draw it up in cad in a few seconds. Is this a practical application, or does it require actual formulae?
 
1) You know the x-distance from FE to B (call it xB). You know the y-distance from CD to B (call it yB). And you know the angle between CD and BD (45°). Thus, you can calculate the x-distance from FE to D (Call it xD):
xD = xB + yB / tan(45°)
Just be sure to calculate tangent in degrees.

2) Repeat on side A. x-distance from FE to C is:
xC = xA + yA / tan(45°)
Just be sure to calculate tangent in degrees.

3) Total distance xCD= xD + xC = xB + xA + (yB + tA) / tan(45°)

4) You have a circle radius and a chord length. Look up a circle chord calculator to get yFE.
 
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Tried to draft it but can't.
not enough information
you also need, say, where point F is in relation to A or B.
Edit:
See post above. One more piece is required
 
dullard said:
1) You know the x-distance from FE to B (call it xB). You know the y-distance from CD to B (call it yB). And you know the angle between CD and BD (45°). Thus, you can calculate the x-distance from FE to D (Call it xD):
xD = xB + yB / tan(45°)
Just be sure to calculate tangent in degrees.

2) Repeat on side A. x-distance from FE to C is:
xC = xA + yA / tan(45°)
Just be sure to calculate tangent in degrees.

3) Total distance CD= xD + xC = xB + xA + (yB + tA) / tan(45°)

4) You have a circle radius and a chord length. Look up a circle chord calculator to get yFE.
Click to expand...

The problem is I do not know the y distance from CD to B, because I do not know where CD is. I drafted up this example using one of the known configurations, but I need a formula to calculate the lengths of those two lines for any value r. :\
 
dank69 said:
The problem is I do not know the y distance from CD to B, because I do not know where CD is.
Click to expand...
I misunderstood your post. I assumed you knew the full coordinates with respect to CD (now I must assume you know the full coordinates with respect to the center of the circle). So what you are saying, is that you want to vary FE and from that, know the distance CD for any radius R?

Is CD parallel to AB?
 
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dullard said:
I misunderstood your post. I assumed you knew the full coordinates with respect to CD (now I must assume you know the full coordinates with respect to the center of the circle). So what you are saying, is that you want to vary FE and from that, know the distance CD for any radius R?

Is CD parallel to AB?
Click to expand...
Yes CD will always be parallel to AB since B is a mirror of A though the center of the circle. EF will naturally vary as r changes.
 
I can figure out the point where AC intersects FE if they were extended. In trigonometry terms, that would mean I have the values a, b and angle A for a triangle with a = r and angle A = 45. If I can use those to get c I would be golden.
 
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dank69 said:
Bump for morning crowd.
Click to expand...

While I appreciate a new thread (there have been many boring, boring threads as of late - I miss refreshing to an entire new page of threads!) my brain does NOT function this early. I'm barely through my first cup of coffee. The West coasters aren't awake yet either.... it will start to pick up around 1030EST. 😀

This thread has reminded me how much I don't miss school. 😎

Edit: solution above. that was easy..... 😛
 
lxskllr said:
I gave it to you...
Click to expand...
I see the textbook and all the formulas but I still don't know the formula I need. I don't even see a secant line on that page, only tangent lines which I don't need. Maybe you could dumb it down for me?

I don't have the angle that the formula for O seems to need.
 
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Unrelated question: I can't edit post 9 and earlier. Has there always been a timeout limit for editing your posts?

Anyway, the word though should have been through in post 8.
 
If you give me the radius, I can give you the answer via software. I'm getting ready to run out, and don't have time to think tnrough the math. I don't see a ready solution for determining the midordinate from my book example.
 
That's my problem. I am making a parameterized component category for many components, each with a different radius. I need a formula. 🙁
 
dank69 said:
I can figure out the point where AC intersects FE if they were extended. In trigonometry terms, that would mean I have the values a, b and angle A for a triangle with a = r and angle A = 45. If I can use those to get c I would be golden.
Click to expand...
I was on the right track with this. Since I now have two sides and an angle, I can use the law of sines to get the other angles:
TXMNxZC.png


a = radius of circle
angle A = 45 degrees
law of sines:

sin(A)/a = sin(B)/b

angle B = arcsin(b(sin(45)/radius))
However, angle B will always be obtuse if the line intersects the circle, so angle B will actually be:
angle B = 180 - arcsin(b(sin(45)/radius))

angle C = 180 - B - A
angle C = 135 - B
angle C = 135 - (180 - arcsin(b(sin(45)/radius)))

With angle C I can now calculate the chord CD :
CD = 2radius(sin(C))
CD = 2radius(sin(135 - (180 - arcsin(b(sin(45)/radius)))))

And the versin EF:
EF = radius - cos(C)(radius)
EF = radius - cos(135 - (180 - arcsin(b(sin(45)/radius))))(radius)

Of course, b is its own formula:
b = radius + x + y

y = distance from point A to the perpendicular intersection with line EF which is half the length of line AB. Let's call that distance z:
OD9FlwX.png


y = z/2

Let's call the length of the line connecting points G and H length w.

x = 3.1 + tan(20)((z - w)/2 - 4.83) + 4.83 / cos(20)

Plug those formulas in the formula for b:

b = radius + 3.1 + tan(20)((z - w)/2 - 4.83) + 4.83 / cos(20) + z/2

Plug formula for b into formulas for CD and EF:
CD = 2radius(sin(135 - (180 - arcsin((radius + 3.1 + tan(20)((z - w)/2 - 4.83) + 4.83 / cos(20) + z/2)(sin(45)/radius)))))

EF = radius - cos(135 - (180 - arcsin((radius + 3.1 + tan(20)((z - w)/2 - 4.83) + 4.83 / cos(20) + z/2)(sin(45)/radius))))(radius)
 
Do you know the coords of the center of the circle? If you do, and you know the coords of A and the angle of the lines is always 45 degrees, then you know the slope is 1, so you can use the point slope formula to traverse the line until the distance from the point you're at on the line is equal to the radius and that gives you the coords of C. Do the same with B to get D. Then you just get the distance between the points to find E. To get F-E you'd take the y coord of the circle's center + R minus the y coord of C or D (they are the same).
 
ioni said:
Do you know the coords of the center of the circle? If you do, and you know the coords of A and the angle of the lines is always 45 degrees, then you know the slope is 1, so you can use the point slope formula to traverse the line until the distance from the point you're at on the line is equal to the radius and that gives you the coords of C. Do the same with B to get D. Then you just get the distance between the points to find E. To get F-E you'd take the y coord of the circle's center + R minus the y coord of C or D (they are the same).
Click to expand...
Yes, the center of circle is always 0,0 and I always know the x,y coords of A and B relative to 0,0. I do not know how to apply the point slope formula though.
 
For distance E-F it would be quicker to use the formula for the midordinate from my link...

R(1 - cos delta/2)

delta/2 in your example is 45°
 
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