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x(b) ... a(x,b) ... to give a(x)
I have following two equations (r and h are constants):
x=x(b)=r (cos(b)+ tan(b)sin(b)-1)+ tan(b)h          (eq. 1)
Â
a =a(x,b)= (1/r)*sqrt((h + rsin(b))^2 + (x + r - rcos(b))^2) - b-h/r        (eg. 2)  Â
to make it readable, if we assume S=Sin(b), C=Cos(b), and T=tan(b), then we get:
x=x(b)=r (C+ TS -1)+ Th    <-- these forms may be useful may be not
a =a(x,b)= (1/r)*sqrt((h + rS)^2 + (x + r - rC)^2) - b-h/r;
I want to have the following functions if possible:
a(x)=....? Â Â Â Â Â Â (eq. 3)
b(x)=....? Â Â Â Â Â (eq. 4) Â <-- this one is not that important. Â Â Â
Question: Â Is it possible to find a(x) and b(x) having (eq. 1) and (eq. 4) above?
for background discussions please see: https://www.experts-exchange.com/questions/27237703/Animation-help-2.html
Thank you
x=x(b)=r (cos(b)+ tan(b)sin(b)-1)+ tan(b)h          (eq. 1)
Â
a =a(x,b)= (1/r)*sqrt((h + rsin(b))^2 + (x + r - rcos(b))^2) - b-h/r        (eg. 2)  Â
to make it readable, if we assume S=Sin(b), C=Cos(b), and T=tan(b), then we get:
x=x(b)=r (C+ TS -1)+ Th    <-- these forms may be useful may be not
a =a(x,b)= (1/r)*sqrt((h + rS)^2 + (x + r - rC)^2) - b-h/r;
I want to have the following functions if possible:
a(x)=....? Â Â Â Â Â Â (eq. 3)
b(x)=....? Â Â Â Â Â (eq. 4) Â <-- this one is not that important. Â Â Â
Question: Â Is it possible to find a(x) and b(x) having (eq. 1) and (eq. 4) above?
for background discussions please see: https://www.experts-exchange.com/questions/27237703/Animation-help-2.html
Thank you
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What is b(x)? I don't see b shown as a function anywhere.
Also, if a is defined as taking two parameters then how do you drop one? How do you get from a(x,b) to just a(x)? The only way I could think of where that would make sense would be if b cancelled out of the equation, but it's not going to.
We need a lot more info before we can answer this question.
Also, if a is defined as taking two parameters then how do you drop one? How do you get from a(x,b) to just a(x)? The only way I could think of where that would make sense would be if b cancelled out of the equation, but it's not going to.
We need a lot more info before we can answer this question.
re:> What is b(x)? I don't see b shown as a function anywhere
lets ignore b(x). Â But, if we had it in a much simpler form say x=rcos(b) then
b=ArcCos(x/r) Â Â and now we have b(x)=ArcCos(x/r)
re:> Also, if a is defined as taking two parameters then how do you drop one? How do you get from a(x,b) ...
This is why I am posting the question here hoping someone familiar with chain function possibly could come up with the answer.
thx
lets ignore b(x). Â But, if we had it in a much simpler form say x=rcos(b) then
b=ArcCos(x/r) Â Â and now we have b(x)=ArcCos(x/r)
re:> Also, if a is defined as taking two parameters then how do you drop one? How do you get from a(x,b) ...
This is why I am posting the question here hoping someone familiar with chain function possibly could come up with the answer.
thx
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Okay I get it. So b(x) actually is important because you replace b in the a(x,b) equation with b(x). I've never seen equations expressed as functions, but it makes sense.
I verified pkwan's math and it looks good. He used the trig identities S^2 + C^2 = 1 and S/C = T to derive eq 6 and the quadratic formula to derive eqs 7 and 8. The rest is just pretty simple algebra. Very well done.
Don't forget that when you introduce an ArcCos, you need to constrain the domain.
I verified pkwan's math and it looks good. He used the trig identities S^2 + C^2 = 1 and S/C = T to derive eq 6 and the quadratic formula to derive eqs 7 and 8. The rest is just pretty simple algebra. Very well done.
Don't forget that when you introduce an ArcCos, you need to constrain the domain.
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When x = 0, a should be 0...
====
>>Â x = 10 (cos(b)+ tan(b)sin(b)-1)+ tan(b)60
  When b = 0, then x is 0 and vice-versa
http://www.wolframalpha.com/input/?i=10+%28cos%28b%29%2B+tan%28b%29sin%28b%29-1%29%2B+tan%28b%2960+for+b%3D0+to+pi%2F2
====
>>Â a(0,b) = (1/10)*sqrt((60 + 10sin(b))^2 + (0 + 10 - 10cos(b))^2) - b-60/10
   When b and x are 0, a is 0
http://www.wolframalpha.com/input/?i=%281%2F10%29*sqrt%28%2860+%2B+10sin%280%29%29%5E2+%2B+%280+%2B+10+-+10cos%280%29%29%5E2%29+-+0-60%2F10
====
But, ...
>>Â a = (1/r)*sqrt((x+r)^2+h^2-r^2
So, for equivalency, if x = 0, then a should be 0, but..
http://www.wolframalpha.com/input/?i=%281%2F10%29*sqrt%28%28x%2B10%29%5E2%2B60%5E2-10%5E2%29+-60%2F10+-+ArcCos%28%5B10%28x%2B10%29+-++60+sqrt%28-10%5E2%2B%28x%2B10%29%5E2%2B60%5E2%29%5D+%2F+%5B%28x%2B10%29%5E2%2B60%5E2%5D%29%2C+for+x+%3D+0+to+45
====
On another note, not related to this question, when looking at the graphs, it appears that the relationship between a and b is suspect, since for large x, then a small change in b should result in a large change in a. See related question: http://rdsrc.us/qNFw9P
====
>>Â x = 10 (cos(b)+ tan(b)sin(b)-1)+ tan(b)60
  When b = 0, then x is 0 and vice-versa
http://www.wolframalpha.com/input/?i=10+%28cos%28b%29%2B+tan%28b%29sin%28b%29-1%29%2B+tan%28b%2960+for+b%3D0+to+pi%2F2
====
>>Â a(0,b) = (1/10)*sqrt((60 + 10sin(b))^2 + (0 + 10 - 10cos(b))^2) - b-60/10
   When b and x are 0, a is 0
http://www.wolframalpha.com/input/?i=%281%2F10%29*sqrt%28%2860+%2B+10sin%280%29%29%5E2+%2B+%280+%2B+10+-+10cos%280%29%29%5E2%29+-+0-60%2F10
====
But, ...
>>Â a = (1/r)*sqrt((x+r)^2+h^2-r^2
So, for equivalency, if x = 0, then a should be 0, but..
http://www.wolframalpha.com/input/?i=%281%2F10%29*sqrt%28%28x%2B10%29%5E2%2B60%5E2-10%5E2%29+-60%2F10+-+ArcCos%28%5B10%28x%2B10%29+-++60+sqrt%28-10%5E2%2B%28x%2B10%29%5E2%2B60%5E2%29%5D+%2F+%5B%28x%2B10%29%5E2%2B60%5E2%5D%29%2C+for+x+%3D+0+to+45
====
On another note, not related to this question, when looking at the graphs, it appears that the relationship between a and b is suspect, since for large x, then a small change in b should result in a large change in a. See related question: http://rdsrc.us/qNFw9P
Hi pkwan,
Thank you for the solution. I included the functions in a java program shown below to test the equations.
At x=0, I was expecting to get zero for both a and b, but I am getting:
angle a       angel b   x
 -02.81  02.81  00.00
 -02.79  02.80  00.25
 -02.78  02.80  00.50
 -00.01  02.26  23.75
 00.03  02.25  24.00
 00.07  02.25  24.25
 00.11  02.25  24.50
 00.15  02.24  24.75
 00.19  02.24  25.00
 06.26  01.94  59.50
 06.30  01.93  59.75
Most likely this dicrepency is due to my mistkes converting from your math equation to java equations shown below:
I will go over it to locate my mistakes.
Thank you for the solution. I included the functions in a java program shown below to test the equations.
At x=0, I was expecting to get zero for both a and b, but I am getting:
angle a       angel b   x
 -02.81  02.81  00.00
 -02.79  02.80  00.25
 -02.78  02.80  00.50
 -00.01  02.26  23.75
 00.03  02.25  24.00
 00.07  02.25  24.25
 00.11  02.25  24.50
 00.15  02.24  24.75
 00.19  02.24  25.00
 06.26  01.94  59.50
 06.30  01.93  59.75
Most likely this dicrepency is due to my mistkes converting from your math equation to java equations shown below:
I will go over it to locate my mistakes.
import java.text.DecimalFormat;
public class Data1 {
public static void main(String args[]) {
DecimalFormat df = new DecimalFormat("00.00");
double h = 30;
double r = 5;
double a = 0;
double b = 0;
System.out.println("angle a\t angel b x");
for(double x = 0; x < 60; x +=0.25){
//b = ArcCos([r(x+r) + h sqrt(-r^2+(x+r)^2+h^2)] / [(x+r)^2+h^2]) ................... (eq. 7)
b = Math.acos((r*(x+r) - h*Math.sqrt(-Math.pow(r,2)+Math.pow((x+r),2)+
Math.pow(h,2))) / (Math.pow((x+r),2)+Math.pow(h,2)));
// a = (1/r)*sqrt((x+r)^2+h^2-r^2) -h/r - ArcCos([r(x+r) - h sqrt(-r^2+(x+r)^2+h^2)] / [(x+r)^2+h^2]) ... (eq. 11)
a=(1/r)*Math.sqrt(Math.pow((x+r),2)+Math.pow(h,2)-Math.pow(r,2))-(h/r)-
Math.acos((r*(x+r)-h*Math.sqrt(-1*Math.pow(r,2)+Math.pow((x+r),2)+
Math.pow(h,2)))/((Math.pow((x+r),2)+Math.pow(h,2))));
// System.out.println(" " + df.format(a*(180/Math.PI)) + " " + df.format(b*(180/Math.PI)) + " " + df.format(x));
System.out.println(" " + df.format(a) + " " + df.format(b) + " " + df.format(x));
}
}
}
Hi phoffric,
I didn't see your post.
I didn't see your post.
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I should have used:
x=x(b)=r (cos(b)+ tan(b)sin(b)-1)+ tan(b)h          (eq. 1)
a = (r *tan(b) + h/cos(b) - h)/r - b                 (eg. 2) much simpler
x = r (C+T S -1)+T/h                            (eq. 1a)
a = (r T + h/C - h)/r - b                          (eg. 2a)
now will try the solution from pkwan on these two equations,
brb
x=x(b)=r (cos(b)+ tan(b)sin(b)-1)+ tan(b)h          (eq. 1)
a = (r *tan(b) + h/cos(b) - h)/r - b                 (eg. 2) much simpler
x = r (C+T S -1)+T/h                            (eq. 1a)
a = (r T + h/C - h)/r - b                          (eg. 2a)
now will try the solution from pkwan on these two equations,
brb
pkwan,
I wonder if you have some time to take a look at this:
https://www.experts-exchange.com/questions/27242530/have-x-b-to-find-b-x.html
Thank you,
Mike
I wonder if you have some time to take a look at this:
https://www.experts-exchange.com/questions/27242530/have-x-b-to-find-b-x.html
Thank you,
Mike
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