 Angle  Y=mX  Xvalue  Yvalue    

θ  Slope (m)  r*Cosθ  r*Sinθ    

(Deg)  Tanθ  Cosθ  Sinθ  Tan^{2}θ  Cos^{2}θ  Sin^{2}θ 

0°  0  1  0  0  1  0  30°  1/√3  √3 / 2  1/2  1/3  3/4  1/4  45°  1  √2 / 2  √2 / 2  1  1/2  1/2  60°  √3  1/2  √3 / 2  3  1/4  3/4  90°  ∞  0  1  ∞  0  1                0°  0  1  0  0  1  0  26.565°  1 / 2  2/√5  1/√5  1/4  4/5  1/5  45°  1  √2 / 2  √2 / 2  1  1/2  1/2  63.435°  2  1/√5  2/√5  4  1/5  4/5  90°  ∞  0  1  ∞  0  1         Examples of convenient trigonometric ratios.



     
     
 Ellipse Equation 
x^{2} a^{2}  +  m^{2}x^{2} b^{2}  =  1 
 Recall we're looking for intersection of line and ellipse. Thus, substitute line equation (y=mx, slope times x) into term containing "y". 
x^{2}* b^{2}  +  m^{2}x^{2} * a^{2}  =  a^{2}b^{2} 
    
Solve for aa^{2 }b^{2}  +  a^{2}m^{2}x^{2}  =  x^{2}b^{2}  a^{2 }* (b^{2}  +  m^{2}x^{2})  =  x^{2}b^{2}  a^{2 }* (b^{2}    m^{2}x^{2})  =  x^{2}b^{2}   Solve for ba^{2 }b^{2}  +  b^{2}x^{2}  =  m^{2}x^{2}a^{2}  b^{2 }* (a^{2}  +  x^{2})  =  m^{2}x^{2}a^{2}  b^{2 }* (a^{2}    x^{2})  =  m^{2}x^{2}a^{2}      
a^{2 }  =  x^{2}b^{2} (b^{2}  m^{2}x^{2})   
 b^{2 }  =  m^{2}x^{2}a^{2} (a^{2}  x^{2})   
    
a^{ }  =  xb √(b^{2}  m^{2}x^{2})   
 b^{ }  =  mxa √(a^{2}  x^{2})   
    
Assume 1st vector, (r_{1}, θ_{1}):a^{ }  =  r_{1}cos(θ_{1})b √(b^{2}  tan^{2}(θ_{1})r_{1}^{2}cos^{2}(θ_{1}))     Assume 2nd vector, (r_{2}, θ_{2}):b^{ }  =  tan(θ_{2})r_{2}cos(θ_{2})a √(a^{2}  r_{2}^{2}cos^{2}(θ_{2}))     Make following substitutions: m = tan(θ) x = rcos(θ)

a^{ }  =  b * r_{1} * cos(θ_{1}) √(b^{2}  r_{1}^{2}sin^{2}(θ_{1}))   
 b^{ }  =  a * r_{2 }* sin(θ_{2}) √(a^{2}  r_{2}^{2}cos^{2}(θ_{2}))   
 Note following trig identity: tan(θ) = sin(θ) / cos(θ) Therefore, tan(θ) cos(θ) = sin(θ) 
a^{2}^{ }  =  b^{2} * r_{1}^{2 }* cos^{2}(θ_{1}) b^{2}  r_{1}^{2}sin^{2}(θ_{1})   
 b^{2}^{ }  =  a^{2} * r_{2}^{2 }* sin^{2}(θ_{2}) a^{2}  r_{2}^{2}cos^{2}(θ_{2})   
 For convenience, use following substitutions:x_{1 }= r_{1}cos(θ_{1})  x_{1}^{2}= r_{1}^{2}cos^{2}(θ_{1})  y_{1 }= r_{1}sin(θ_{1})  y_{1} ^{2}= r_{1}^{2}sin^{2}(θ_{1})  x_{2 }= r_{2}cos(θ_{2})  x_{2}^{2}= r_{2}^{2}cos^{2}(θ_{2})  y_{2 }= r_{2}sin(θ_{2})  y_{2}^{2}= r_{2}^{2}sin^{2}(θ_{2})  
b^{2} * x_{1}^{2} b^{2}  y_{1}^{2}   = a^{2} =  b^{2} * x_{2}^{2} b^{2}  y_{2}^{2}   
   a^{2} * y_{1}^{2 } a^{2}  x_{1}^{2}   = b^{2} =  a^{2} * y_{2}^{2 } a^{2}  x_{2}^{2}   
 Since both radius vectors cross same ellipse, they can be used interchangeably as shown. Transitive identity property allow two outside expressions to equal b^{2} as well as each other. 
x_{1}^{2} b^{2}  y_{1}^{2}   =  x_{2}^{2} b^{2}  y_{2}^{2}   
   y_{1}^{2 } a^{2}  x_{1}^{2}   =  y_{2}^{2 } a^{2}  x_{2}^{2}   
 Deleting common term from both sides, above equations can be written as shown. 
x_{1}^{2}(b^{2}  y_{2}^{2})   =  x_{2}^{2}(b^{2}  y_{1}^{2}) 
x_{1}^{2}b^{2}  x_{1}^{2}y_{2}^{2}   =  x_{2}^{2}b^{2}  x_{2}^{2}y_{1}^{2} 
x_{1}^{2}b^{2}  x_{2}^{2}b^{2}   =  x_{1}^{2}y_{2}^{2}  x_{2}^{2}y_{1}^{2} 
b^{2}(x_{1}^{2}  x_{2}^{2})   =  x_{1}^{2}y_{2}^{2}  x_{2}^{2}y_{1}^{2} 
   y_{1}^{2 }(a^{2}  x_{2}^{2})  =  y_{2}^{2}(a^{2}  x_{1}^{2})   
  y_{1}^{2}a^{2}  y_{1}^{2}x_{2}^{2}  =  y_{2}^{2}a^{2}  y_{2}^{2}x_{1}^{2}   
  y_{1}^{2}a^{2}  y_{2}^{2}a^{2}  =  y_{1}^{2}x_{2}^{2}  y_{2}^{2}x_{1}^{2}   
  a^{2}(y_{1}^{2}  y_{2}^{2})  =  y_{1}^{2}x_{2}^{2}  y_{2}^{2}x_{1}^{2}   
 Rearrange as shown. 
b^{2}^{ }  =  x_{1}^{2}y_{2}^{2}  x_{2}^{2}y_{1}^{2} x_{1}^{2}  x_{2}^{2} 
 a^{2}^{ }  =  y_{1}^{2}x_{2}^{2}  y_{2}^{2}x_{1}^{2} y_{1}^{2}  y_{2}^{2} 
 FINALLY!!! We've isolated terms a^{2} and b^{2}. We can now substitute arbitrary values as shown in next row. 
b^{2}^{ }  =  0.54 AU^{2}AU^{2}  0.18AU^{2}AU^{2} 0.333 AU^{2} 
b^{ }  =  √1.08 AU^{2}  =  1.039 AU 
 a^{2}^{ }  =  (0.18AU^{2}  0.54 AU^{2})AU^{2} 0.107 AU^{2} 
a^{ }  =  √3.65 AU^{2}  =  1.834 AU 
 Arbitrary Example: Recall restriction r_{1} = 1.2 AU  θ_{1} = 30°  r_{2} = 1.0 AU  θ_{2}=45°  Recall substitutions  x_{1 }=1.04AU  y_{1}=.600AU  x_{2}= .707AU  y_{2}=.707AU_{}  x_{1}^{2}=1.08AU^{2}  y_{1}^{2}=.36AU^{2}  x_{2}^{2}=.5AU^{2}  y_{2}^{2}=.5AU^{2}  x_{1}^{2}y_{2}^{2 }= 0.54 AU^{2}AU^{2}  x_{2}^{2}y_{1}^{2}= 0.18 AU^{2}AU^{2}  x_{1}^{2 } x_{2}^{2 }= 0.333 AU^{2}  y_{1}^{2}  y_{2}^{2}= 0.107 AU^{2} 

For this entirely arbitrary example of two radius vectors from elliptic center to elliptic perimeter, we have calculated values for semimajor axis, a, and semiminor axis, b, to determine elliptic equation.
These vectors were limited only by convenience and a few common sense restrictions.
We can use now use values for a and b to calculate:
0 Comments:
Post a Comment
<< Home